aa r X i v : . [ m a t h . D S ] F e b JOINT ERGODICITY OF SEQUENCES
NIKOS FRANTZIKINAKIS
Abstract.
A collection of integer sequences is jointly ergodic if for every ergodicmeasure preserving system the multiple ergodic averages, with iterates given by thiscollection of sequences, converge in the mean to the product of the integrals. Wegive necessary and sufficient conditions for joint ergodicity that are flexible enough torecover most of the known examples of jointly ergodic sequences and also allow us toanswer some related open problems. An interesting feature of our arguments is thatthey avoid deep tools from ergodic theory that were previously used to establish similarresults. Our approach is primarily based on an ergodic variant of a technique pioneeredby Peluse and Prendiville in order to give quantitative variants for the finitary versionof the polynomial Szemerédi theorem. Introduction and main results
Introduction.
The study of multiple ergodic averages was initiated in the seminalwork of Furstenberg [23], where an ergodic theoretic proof of Szemerédi’s theorem onarithmetic progressions was given. Since then a variety of multiple ergodic averages hasbeen studied, resulting in new and far reaching combinatorial consequences. A rathergeneral family of problems is as follows: We are given a collection of integer sequences a , . . . , a ℓ : N → Z , and we are asked, for an arbitrary invertible measure preservingsystem ( X, µ, T ) and functions f , . . . , f ℓ ∈ L ∞ ( µ ) , to understand the limiting behaviorin L ( µ ) of the averages (with E n ∈ [ N ] we denote the average N P Nn =1 )(1) E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ as N → ∞ . The reader can find a large collection of related convergence results inthe survey [20], but we remark that even in the case where all the sequences are givenby integer polynomials the limit is not easy to compute. In particular, if the sequencessatisfy some non-trivial linear relations, then simple examples of ergodic compact Abeliangroup rotations show that the limit of the averages (1) is not equal to the product of theintegrals of the individual functions.The question we seek to study is under what conditions on the sequences a , . . . , a ℓ theiterates T a ( n ) , . . . , T a ℓ ( n ) , n ∈ N , behave independently enough, so that for every ergodicsystem (or totally ergodic system) the averages (1) converge in L ( µ ) to the product ofthe integrals of the individual functions. This gives rise to the following notions: Definition.
We say that the collection of sequences a , . . . , a ℓ : N → Z is(i) jointly ergodic for the system ( X, µ, T ) , if for all functions f , . . . , f ℓ ∈ L ∞ ( µ ) we have(2) lim N →∞ E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ = Z f dµ · . . . · Z f ℓ dµ where convergence takes place in L ( µ ) . (ii) jointly ergodic , if it is jointly ergodic for every ergodic system. Mathematics Subject Classification.
Primary: 37A45; Secondary: 28D05, 05D10, 11B25.
Key words and phrases.
Joint ergodicity, ergodic averages, recurrence, Hardy fields, fractional powers.The author was supported by the Hellenic Foundation for Research and Innovation, Project No: 1684. This should not be confused with the notion of joint ergodicity introduced in [2], which is a propertyof a collection of measure preserving transformations and not a property of a collection of sequences.
It is a direct consequence of Furstenberg’s correspondence principle [24] that if thecollection of sequences a , . . . , a ℓ is jointly ergodic, then every set of integers with positiveupper density contains patterns of the form m, m + a ( n ) , . . . , m + a ℓ ( n ) , for some m, n ∈ N . In fact, a stonger property holds (which fails when say a ( n ) = n, a ( n ) = 2 n , see[5]), namely, for every set of integers Λ we have(3) lim inf N →∞ E n ∈ [ N ] ¯ d (Λ ∩ (Λ + a ( n )) ∩ · · · ∩ (Λ + a ℓ ( n ))) ≥ ( ¯ d (Λ)) ℓ +1 . For ℓ = 1 , by appealing to the spectral theorem for unitary operators, it is an easyexercise to verify that a single sequence a : N → Z is (jointly) ergodic if and only if forevery t ∈ (0 , we have(4) lim N →∞ E n ∈ [ N ] e ( a ( n ) t ) = 0 where e ( t ) := e πit . In particular, the sequences [ n c ] , where c ∈ R + \ Z , and [ n α + nβ ] ,where α, β are non-zero real numbers such that α/β is irrational, are (jointly) ergodic.It also follows that a single sequence is jointly ergodic for every totally ergodic systemif and only if (4) holds for every irrational t ∈ [0 , , a condition that is known to besatisfied, for instance, by all non-constant integer polynomials.For ℓ ≥ , joint ergodicity is a much tougher property to establish. In [18] it was shownthat the collection of sequences [ n c ] , . . . , [ n c ℓ ] , where c , . . . , c ℓ ∈ R + \ Z are distinct,is jointly ergodic, and further examples involving polynomials with real coefficients andother sequences arising from smooth functions can be found in [10, 19, 29, 31]. Moreover,a collection of polynomial sequences p , . . . , p ℓ ∈ Z [ t ] is known to be jointly ergodic forall totally ergodic systems if and only if the polynomials are rationally independent [21];a typical case is the pair n, n , which was first handled in [25]. The proofs of mostof these results rely on deep tools from ergodic theory, such as the Host-Kra theoryof characteristic factors [27] and equidistibution results on nilmanifolds. But a generalcriterion for establishing joint ergodicity, like the one mentioned for ℓ = 1 , is still lacking,the reason being that we do not have a good substitute for the spectral theorem forunitary operators that would allow us to represent correlation sequences of the form R f · T n f · . . . · T n ℓ f ℓ dµ , for sparse values of n , . . . , n ℓ ∈ N , in a useful way.Using ergodic rotations on compact Abelian groups we see that a necessary conditionfor joint ergodicity of a collection of sequences a , . . . , a ℓ : N → Z is that for all t , . . . , t ℓ ∈ [0 , , not all of them zero, we have lim N →∞ E n ∈ [ N ] e ( a ( n ) t + · · · + a ℓ ( n ) t ℓ ) = 0 . The main objective of the current article is to verify in Theorem 1.1 that this condition isalso sufficient for joint ergodicity, assuming that the sequences a , . . . , a ℓ satisfy anothernecessary condition regarding seminorm estimates that is defined in the Section 1.3. Thenecessary conditions we give are flexible enough to allow us to recover painlessly allthe known results of jointly ergodic sequences mentioned above and enable us to provesome new ones. For example, in Theorem 1.6 we make progress towards a conjecturein [18, Problem 2] regarding joint ergodicity of Hardy field sequences, improving uponthe best known results in [10, 18], and in Theorem 1.7 we verify a conjecture from [10,Conjecture 6.1]. A special case of this last result asserts that a collection of sequencesof the form [ P ki =1 α i n b i ] , where α , . . . , α k ∈ Q , b , . . . , b k ∈ (0 , + ∞ ) are positive, isjointly ergodic for all totally ergodic systems if and if it is linearly independent; this waspreviously known when all the exponents are integers [21] or none of the exponents is aninteger [10].An interesting feature of our main result is that its proof avoids deep tools from ergodictheory and leads to vastly simpler proofs of several known results. In particular, our argu-ments do not rely on the Host-Kra theory of characteristic factors or on equidistributionresults on nilmanifolds, as previous results in the area do. Moreover, in Corollary 1.3 we OINT ERGODICITY OF SEQUENCES 3 show that our main result has some far-reaching consequences related to equidistributionproperties of general sequences on nilmanifolds. Our approach is to adapt and utilizein our ergodic theory setup, a technique developed by Peluse [35] and Peluse and Pren-diville [37] (see also [39] for an exposition of the technique) and used in order to establishquantitative results for finitary variants of special cases of the polynomial Szemerédi the-orem. Although the main ideas are elementary, they are a bit cumbersome to implementin full generality, so in order to facilitate reading, we first present our argument in thesimpler case of the multiple ergodic averages E n ∈ [ N ] T n f · T n g. It is a classical result of Furstenberg and Weiss that these averages converge in L ( µ ) , tothe product of the integrals for totally ergodic systems, and that the rational Kroneckerfactor is a characteristic factor for the above averages. We present in Section 2 a proof ofthis result using the general principles of the argument used in [37], but also take advan-tage of various simplifications that our infinitary setup allows (the complete argument isonly a few pages long). Subsequently, in Sections 3 and 4 we extend this method to givea proof of our main result, Theorem 1.1, which gives necessary and sufficient conditionsfor joint ergodicity. We then use it in Section 5 to give a result about characteristicfactors (Theorem 1.4) that also applies to arbitrary collections of rationally independentinteger polynomials. In Section 6, by applying the previous results we recover, usingvastly simpler arguments than previously used, known results for Hardy field sequencesand also address some related open problems.Finally, in Section 7 we deal with similar problems for flows. This section is indepen-dent from the previous ones, and the methodology used is completely different than theone used to cover discrete time averages. We prove pointwise convergence results thatapply to actions of not necessarily commuting transformations. We remark that evenfor commuting transformations, the corresponding results for discrete time actions arenot known, and for general non-commuting transformations they are known to be false.Our main result is given in Theorem 1.9, and representative special cases assert that forall ergodic measure preserving actions T t , S t , R t , t ∈ R , on a probability space ( X, X , µ ) and functions f, g, h ∈ L ∞ ( µ ) , the following limits exist lim y → + ∞ y Z y f ( T t x ) · g ( S t x ) dt, lim y → + ∞ y Z y f ( T t x ) · g ( S t x ) · h ( R t ) dt pointwise for µ -almost every x ∈ X , and are equal to the product of the integrals ofthe individual functions. We also establish analogous results without any ergodicityassumptions. When the actions T t and S t commute, the corresponding result for thefirst average was recently obtained in [14], using harmonic analysis techniques from [15].1.2. Definitions and notation.
With N we denote the set of positive integers andwith Z + the set of non-negative integers. With R + we denote the set of non-negativereal numbers. For t ∈ R we let e ( t ) := e πit . With [ t ] we denote the integer part of t andwith { t } the fractional part of t . With T we denote the one dimensional torus and weoften identify it with R / Z or with [0 , .For N ∈ N we let [ N ] := { , . . . , N } . Let a : N → C be a bounded sequence. If A is anon-empty finite subset of N we let E n ∈ A a ( n ) := 1 | A | X n ∈ A a ( n ) . If a, b : R + → R are functions we write a ( t ) ≺ b ( t ) if lim t → + ∞ a ( t ) /b ( t ) = 0 . We saythat the function a : R + → R has at most polynomial growth if there exists d ∈ N suchthat a ( t ) ≺ t d . OINT ERGODICITY OF SEQUENCES 4
Joint ergodicity of general sequences. A measure preserving system , or simply asystem , is a quadruple ( X, X , µ, T ) where ( X, X , µ ) is a probability space and T : X → X is an invertible, measurable, measure preserving transformation. We typically omit the σ -algebra X and write ( X, µ, T ) . Throughout, for n ∈ N we denote by T n the composition T ◦ · · · ◦ T ( n times) and let T − n := ( T n ) − and T := id X . Also, for f ∈ L ∞ ( µ ) and n ∈ Z we denote by T n f the function f ◦ T n .We say that the system ( X, µ, T ) is ergodic if the only functions f ∈ L ( µ ) that satisfy T f = f are the constant ones. It is totally ergodic , if ( X, µ, T d ) is ergodic for every d ∈ N .To facilitate discussion we introduce the following notions. Definition. If ( X, µ, T ) is a system we let Spec( T ) := { t ∈ [0 ,
1) :
T f = e ( t ) f for some non-zero f ∈ L ( µ ) } . For the definition of the ergodic seminorms ||| · ||| s we refer the reader to Section 3.1. Definition.
We say that the collection of sequences a , . . . , a ℓ : N → Z is:(i) good for seminorm estimates for the system ( X, µ, T ) , if there exists s ∈ N suchthat if f , . . . , f ℓ ∈ L ∞ ( µ ) and ||| f l ||| s = 0 for some l ∈ [ ℓ ] , then lim N →∞ E n ∈ [ N ] T a ( n ) f · . . . · T a l ( n ) f l = 0 where convergence takes place in L ( µ ) . It is good for seminorm estimates , if itis good for seminorm estimates for every ergodic system.(ii) good for equidistribution for the system ( X, µ, T ) , if for all t , . . . , t ℓ ∈ Spec( T ) ,not all of them , we have(5) lim N →∞ E n ∈ [ N ] e ( a ( n ) t + · · · + a ℓ ( n ) t ℓ ) = 0 . It is good for equidistribution if it is good for equidistribution for every system,or equivalently, if (5) holds for all t , . . . , t ℓ ∈ [0 , , not all of them .(iii) good for irrational equidistribution for the system ( X, µ, T ) , if for all t , . . . , t ℓ ∈ Spec( T ) , not all of them rational, equation (5) holds. It is good for irrationalequidistribution , if it is good for irrational equidistribution for every system, orequivalently, if (5) holds for all t , . . . , t ℓ ∈ [0 , , not all of them rational. Remarks. • Using a standard argument (see for example [19, Lemma 4.12]) we get thatif for the sequences a , . . . , a ℓ Property ( i ) holds for some s ∈ N and if ||| f l ||| s +1 = 0 for some l ∈ [ ℓ ] , then for every bounded sequence of complex numbers ( c n ) we have lim N →∞ E n ∈ [ N ] c n T a ( n ) f · . . . · T a l ( n ) f l = 0 in L ( µ ) . • If µ = R µ x dµ is the ergodic decomposition of the measure µ , it is known that if ||| f ||| s,µ = 0 , then ||| f ||| s,µ x = 0 for µ -almost every x ∈ X [28, Chapter 8, Proposition 18].As a consequence, if a collection of sequences is good for seminorm estimates for everyergodic system, then it is also good for seminorm estimates for arbitrary systems. • It is easy to verify that if the sequences a , . . . , a ℓ are good for irrational equidistri-bution, then the sequences , a , . . . , a ℓ are linearly independent. Although the converseis true for polynomial sequences, it is easy to see that it is not true in general. • If c , . . . , c k are distinct positive real numbers, then it is known that the sequences [ n c ] , . . . , [ n c k ] are good for seminorm estimates and irrational equidistribution. If inaddition the exponents c , . . . , c k are not integers, then the sequences are also good forequidistribution.It is easy to verify that if a collection of sequences is jointly ergodic for a system, thenit is good for seminorm estimates (with s = 1 ) and equidistribution for this system. Itis a rather surprising fact that the converse is also true; this is the context of our mainresult that we now state. OINT ERGODICITY OF SEQUENCES 5
Theorem 1.1.
Let ( X, µ, T ) be an ergodic system and a , . . . , a ℓ : N → Z be sequences.Then the following two conditions are equivalent: (i) The sequences a , . . . , a ℓ are jointly ergodic for ( X, µ, T ) . (ii) The sequences a , . . . , a ℓ are good for seminorm estimates and equidistributionfor ( X, µ, T ) . Remarks. • Even when ℓ = 1 and the sequence a is strictly increasing, we cannot omitthe seminorm condition from the equivalence. Indeed, if a system is weakly mixing butnot strongly mixing, then any sequence is good for equidistribution for this system, butclearly, there exists a strictly increasing sequence that is not ergodic for this system. Seethough Problem 1 below for a related open problem. • The equidistribution assumption is satisfied for all totally ergodic systems if andonly if (5) holds for all t , . . . , t ℓ ∈ Spec( T ) that are irrational or , but not all ofthem . In particular, this is satisfied by collections of linearly independent polynomialswith integer coefficients and zero constant terms (which are also known to be good forseminorm estimates), thus providing a simpler proof of the main result in [21]. See alsoTheorem 1.7 below for an application to a vastly more general collection of Hardy fieldsequences. • Even using the full force of the Host-Kra theory of characteristic factors [27] it isnot clear how to prove Theorem 1.1. So the more elementary approach we follow, whichis motivated by the recent work of Peluse and Prendiville [35, 37] on the polynomialSzemerédi theorem, offers substantial technical advantages. • It would be interesting to examine if the arguments used in [32] (which also cruciallyuse ideas from [35, 37]) can be modified to give similar necessary conditions so that (2)holds pointwise; of course, in this case one has to use stronger quantitative variants ofthe conditions used in ( ii ) .Next, we state a few consequences of our main result. The first is a strong multiplerecurrence property, that is not shared for example by collections of linear sequences, ormore general polynomial sequences. Corollary 1.2.
Let a , . . . , a ℓ : N → Z be sequences that are good for seminorm estimatesand equidistribution for the system ( X, µ, T ) . Then for every set A ∈ X we have lim N →∞ E n ∈ [ N ] µ ( A ∩ T − a ( n ) A ∩ · · · ∩ T − a ℓ ( n ) A ) ≥ ( µ ( A )) ℓ +1 . By invoking the correspondence principle of Furstenberg [24], one deduces that if thesequences a , . . . , a ℓ : N → Z are good for seminorm estimates and equidistribution, thenfor every set of integers Λ equation (3) holds.We also record a result where the assumption of seminorm estimates can be omitted. Itallows us to infer equidistribution properties for general nilsystems from the special caseof Abelian nilsystems; similar results were previously known for collections of polynomialsequences [33] (but with stronger equidistribution assumptions) and certain Hardy fieldsequences of at most polynomial growth [17, 40]. Corollary 1.3.
Let ( X, µ, T ) be an ergodic nilsystem. Then the following two conditionsare equivalent: (i) The sequences a , . . . , a ℓ are jointly ergodic for ( X, µ, T ) . (ii) The sequences a , . . . , a ℓ are good for equidistribution for ( X, µ, T ) . A k -step nilsystem is a system of the form ( X, m X , T a ) , where X = G/ Γ is a k -step nilmanifold (i.e. G is a k -step nilpotent Lie group and Γ is a discrete cocompact subgroup of G ), a ∈ G , T a : X → X isdefined by T a ( g Γ) := ( ag )Γ , g ∈ G , and m X is the normalized Haar measure on X . OINT ERGODICITY OF SEQUENCES 6
The previous result implies that if the sequences a , . . . , a ℓ are good for equidistributionfor the ergodic nilsystem ( X, µ, T ) , then for all f , . . . , f ℓ ∈ C ( X ) we have(6) lim N →∞ E n ∈ [ N ] f ( T a ( n ) x ) · . . . · f ℓ ( T a ℓ ( n ) x ) = Z f dµ · · · Z f ℓ dµ where convergence takes place in L ( µ ) . It is known (see [21, Example 5]) that theconvergence in (6) cannot be strengthened to pointwise convergence for every x ∈ X . Wedo not know if under the stated assumptions the convergence in (6) can be strengthenedto pointwise convergence for m X -almost every x ∈ X and all f , . . . , f ℓ ∈ C ( X ) .1.4. Characteristic factors for general sequences.
If some of the sequences a , . . . , a ℓ are integer polynomials, then Theorem 1.1 is not applicable to all ergodic systems becausethe equidistribution property is not satisfied. We will state a result about characteristicfactors that gives useful information in such cases and is strong enough to easily implyrelated convergence and multiple recurrence results. Definition.
Let ( X, µ, T ) be a system (not necessarily ergodic).(i) The rational Kronecker factor K rat ( T ) of a system ( X, µ, T ) is the L ( µ ) closureof the linear span of rational eigenfunctions of the system, meaning, functions f ∈ L ( µ ) such that T f = e ( t ) f for some t ∈ Q .(ii) We say that the rational Kronecker factor K rat ( T ) is a characteristic factorfor the sequences a , . . . , a ℓ : N → Z , if whenever at least one of the functions f , . . . , f ℓ ∈ L ∞ ( µ ) is orthogonal to K rat ( T ) , we have lim N →∞ E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ = 0 where convergence takes place in L ( µ ) .For the purposes of the next result we will also need a strengthening of a notionintroduced in the previous subsection. Definition.
We say that the collection of sequences a , . . . , a ℓ : N → Z is very good forseminorm estimates for the system ( X, µ, T ) , if there exists s ∈ N such that if f , . . . , f ℓ ∈ L ∞ ( µ ) and ||| f l ||| s = 0 for some l ∈ [ ℓ ] , then lim N →∞ E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ = 0 where convergence takes place in L ( µ ) . It is very good for seminorm estimates , if it isvery good for seminorm estimates for every ergodic system.It is easy to verify that if for a given collection of sequences the rational Kronecker fac-tor is characteristic for every measure preserving system, then this collection of sequencesis very good for seminorm estimates (with s = 2 ) and good for irrational equidistribution.It turns out that the converse is also true and is the context of our next result. Theorem 1.4.
Let a , . . . , a ℓ : N → Z be sequences. Then the following two conditionsare equivalent: (i) For every system ( X, µ, T ) the factor K rat ( T ) is a characteristic factor for thesequences a , . . . , a ℓ . (ii) The sequences a , . . . , a ℓ are very good for seminorm estimates and good forirrational equidistribution. Remarks. • Although it is not particularly hard to deduce this result from Theorem 1.1,we use in a crucial way the fact that characteristic factors of the averages we are interestedin are inverse limits of systems with finite rational spectrum, a property that we do notknow how to prove without appealing to the main result in [27]. • If in addition to the properties in Part ( ii ) we assume that for all rational t , . . . , t ℓ ∈ [0 , the averages E n ∈ [ N ] e ( a ( n ) t + · · · + a ℓ ( n ) t ℓ ) converge as N → ∞ , then it is easy OINT ERGODICITY OF SEQUENCES 7 to deduce from Theorem 1.4 that the averages (1) converge in the mean for all systemsand functions in L ∞ ( µ ) .Next, we give a consequence of the previous result to multiple recurrence. Definition.
We say that a collection of sequences a , . . . , a ℓ : N → Z has good divisibilityproperties , if for every r ∈ N we have(7) ¯ d ( { n ∈ N : a ( n ) ≡ , . . . , a ℓ ( n ) ≡ r ) } ) > . Corollary 1.5.
Let a , . . . , a ℓ : N → Z be sequences with good divisibility properties thatare very good for seminorm estimates and good for irrational equidistribution. Then forevery system ( X, µ, T ) , set A ∈ X , and every ε > , we have µ ( A ∩ T − a ( n ) A ∩ · · · ∩ T − a ℓ ( n ) A ) ≥ ( µ ( A )) ℓ +1 − ε for a set of n ∈ N with positive upper density (or lower density if in (7) we use d ). By invoking the correspondence principle of Furstenberg [24] one deduces that underthe previous assumptions on a , . . . , a ℓ , for every set of integers Λ and ε > we have ¯ d (Λ ∩ (Λ + a ( n )) ∩ · · · ∩ (Λ + a ℓ ( n ))) ≥ ( ¯ d (Λ)) ℓ +1 − ε for a set of n ∈ N with positive upper density. Moreover, arguing as in [22, Theorem 2.2] we get the following finitistic variant of this result: For every δ > and ε > there exists N ( δ, ε ) , such that for all N > N ( δ, ε ) , for every set of integers Λ ⊂ [ N ] with | Λ | ≥ δN ,there is some n ∈ N such that Λ contains at least (1 − ε ) δ ℓ +1 N patterns of the form m, m + a ( n ) , . . . , m + a ℓ ( n ) . Some articles in the literature, refer to such values of n ∈ N as “popular differences”.1.5. Joint ergodicity of Hardy field sequences.
In this subsection we record somerather straightforward applications of the previous results for collections of Hardy fieldsequences (defined in Section 6.1) that satisfy some growth assumptions that we definenext. Our method also gives alternative proofs of known results from [9, 18, 21, 29] thatavoid the theory of characteristic factors and equidistribution results on nilmanifolds.For the purposes of this sub-section we assume that all Hardy fields H considered havethe following property:(8) if a, b ∈ H , then a ◦ b − ∈ H and a ( · + h ) ∈ H for every h ∈ R + . It is known that there exists a Hardy field that has this property and contains alllogarithmic-exponential functions (the example of Pfaffian functions is mentioned in[11]); so this covers all the interesting examples considered in practice, like functionsof the form t c (log t ) c (log log t ) c where c , c , c ∈ R . Definition.
We say that a function a : R + → R (i) is tempered if t k log t ≺ a ( t ) ≺ t k +1 for some k ∈ Z + .(ii) stays logarithmically away from rational polynomials, if for every p ∈ Q [ t ] wehave lim t → + ∞ | a ( t ) − p ( t ) | log t = + ∞ . We denote by T the class of all tempered functions and by T + P the class of all linearcombinations of tempered and polynomial functions with real coefficients.It is known that if a sequence from a Hardy field has at most polynomial growthand stays logarithmically away from constant multiples of rational polynomials, then itis ergodic; this follows quite easily from the spectral theorem and [11], the details canbe found in [12, Theorem A]. A possible generalization of this result that covers finitecollections of Hardy field sequences is the following: It is crucial that Corollary 1.5 holds for all systems; it is not known how to deduce such finitisticconsequences from multiple recurrence results that are only known for all ergodic systems.
OINT ERGODICITY OF SEQUENCES 8
Conjecture 1 ([18], [20]) . Let a , . . . , a ℓ : [ c, + ∞ ) → R be functions from a Hardy fieldof at most polynomial growth such that every non-trivial linear combination of thesefunctions stays logarithmically away from rational polynomials. Then the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are jointly ergodic. A variant of the conjecture that uses appropriately chosen weighted averages in placeof Cèsaro averages was recently verified in [9, Theorem B]. In the form originally stated,the conjecture remains open and has been verified under either of the following, partiallycomplementary, additional assumptions: (a) The functions a , . . . , a ℓ are tempered and have different growth rates, meaning a ≺ a ≺ · · · ≺ a ℓ , see [19, Theorem 2.3] (see also [18, Theorem 2.6] for a simpler argumentthat works for a slightly more restricted class of functions).(b) The functions P ℓj =1 c j a ( k j ) j are in T + P for all c , . . . , c ℓ ∈ R and k , . . . , k ℓ ∈ Z + ,see [9, Corollary B1].The next result improves upon the previously mentioned results and establishes Con-jecture 1 when the functions a , . . . , a ℓ and their differences are in T + P . Its proof isprimarily based on Theorem 1.1. Theorem 1.6.
Let a , . . . , a ℓ : [ c, + ∞ ) → R be functions from a Hardy field. Suppose thatthe functions and their differences are in T + P and every non-trivial linear combinationof a , . . . , a ℓ stays logarithmically away from rational polynomials. Then the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are jointly ergodic. One easily verifies that the result applies for example if a , . . . , a ℓ are(a) functions of the form P ki =1 α i t b i where α , . . . , α k ∈ R , b , . . . , b k ∈ (0 , + ∞ ) , andevery non-trivial linear combination of a , . . . , a ℓ is not a rational polynomial (in [29]this was established when b , . . . , b k ∈ Z + ).(b) linearly independent functions of the form P ki =1 α i t b i (log t ) c i , where b i ∈ R + \ Z + , α i , c i ∈ R , i = 1 , . . . , k .Both cases were also previously covered by [9, Corollary B1]. Let us also remark thatour method of proof reduces Conjecture 1 to showing that under the assumptions ofthis conjecture the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are good for seminorm estimates; butverifying this property appears to be a challenging problem.We also give a result that applies to a larger class of sequences that includes polynomialsequences. The proof of Part ( i ) is based on Theorem 1.1 and the proof of Part ( ii ) isbased on Theorem 1.4. Theorem 1.7.
Let a , . . . , a ℓ : [ c, + ∞ ) → R be functions from a Hardy field. Supposethat the functions and their differences are in T + P , and that every non-trivial linearcombination of a , . . . , a ℓ , with at least one coefficient irrational, stays logarithmicallyaway from rational polynomials. Then (i) the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are jointly ergodic for totally ergodic systems. (ii) the rational Kronecker factor is characteristic for [ a ( n )] , . . . , [ a ℓ ( n )] . One easily verifies that the result applies for example if a , . . . , a ℓ are(a) linearly independent functions of the form P ki =1 α i t b i where α , . . . , α k ∈ Q and b , . . . , b k ∈ (0 , + ∞ ) (this generalizes the main result of [21], which covers the caseof integer polynomials), or, more generally, Note that conditions ( a ) and ( b ) are in general position; the collections { t c , t c − + log t } , { t c + t log t, t c − } , where c ∈ (1 , ∞ ) \ Z + , satisfy ( a ) but not ( b ) , and the collections { t a , t a + t b } , where a, b ∈ R + \ N and b < a , satisfy ( b ) but not ( a ) . An example of a collection of functions that satisfy the assumptions of Theorem 1.6 but is notcovered by previous results is { t a , t b , t a + t b + t k (log t ) c } , where a, b ∈ (1 , ∞ ) \ N are distinct, k ∈ N issmaller than max { a, b } , and c ∈ (0 , . OINT ERGODICITY OF SEQUENCES 9 (b) functions of the form P ki =1 α i t b i where α , . . . , α k ∈ R and b , . . . , b k ∈ (0 , + ∞ ) , suchthat every linear combination of a , . . . , a ℓ , with at least one coefficient irrational, isnot a rational polynomial. The previous special cases give an affirmative answer to [9, Conjecture 6.1].1.6.
Joint ergodicity for nilsystems.
Combining Corollary 1.3 and known equidis-tribution results on the circle from [3, 4, 13, 30], we can deduce some equidistributionresults on general nilmanifolds. These results do not seem to be accessible from theknown qualitative or quantitative equidistribution techniques of polynomial sequenceson nilmanifolds, since these techniques require the functions involved to have some de-rivative vanishing (in the sense that it converges to zero at infinity).
Theorem 1.8.
The following collections of sequences are jointly ergodic for every ergodicnilsystem: (i) [ n (log n ) c ] , . . . , [ n (log n ) cℓ ] , where c , . . . , c ℓ ∈ (0 , / are distinct. (ii) [ p ( n ) sin n ] , . . . , [ p ℓ ( n ) sin n ] , where , p , . . . , p ℓ ∈ Z [ t ] are linearly independentpolynomials. (iii) [ n k sin n ] , [ n k sin(2 n )] , . . . , [ n k sin( ℓn )] , where k, ℓ ∈ N . Remark.
We do not know if for these sequences equation (6) holds pointwise for every x ∈ X (or even for m X -almost every x ∈ X ) and all f , . . . , f ℓ ∈ C ( X ) .It follows from Theorem 1.1 that joint ergodicity (for arbitrary ergodic systems) of theprevious collections of sequences would follow if one establishes that they are good forseminorm estimates; see Problem 2 below.1.7. Joint ergodicity for flows. A -parameter measure preserving action (or flow)on a probability space ( X, X , µ ) is a family T t , t ∈ R , of invertible measure preservingtransformations, that satisfy T s + t = T s ◦ T t , s, t ∈ R ( T is the identity transformation),and such that the map R × X → X , defined by ( t, x ) T t x , is measurable ( X is equippedwith X and R with the Borel σ -algebra).We give a joint ergodicity result for not necessarily commuting measure preservingflows, when the iterates are given by functions that satisfy suitable growth conditions.It turns out that in the case of flows we can also prove pointwise convergence results;in contrast, for discrete time actions the corresponding results are false without anycommutativity assumptions (even for mean convergence) and for general commutingactions they are unknown. Theorem 1.9.
Let a , . . . , a ℓ : [ c, + ∞ ) → R + be functions from a Hardy field. Supposethat there exists δ > such that t δ ≺ a ( t ) and ( a j +1 ( t )) δ ≺ a j ( t ) ≺ ( a j +1 ( t )) − δ for j = 1 , . . . , ℓ − . Then for all measure preserving actions T t , . . . , T tℓ , t ∈ R , on aprobability space ( X, X , µ ) and f , . . . , f ℓ ∈ L ∞ ( µ ) , we have (9) lim y → + ∞ y Z y f ( T a ( t )1 x ) · . . . · f ℓ ( T a ℓ ( t ) ℓ x ) dt = ˜ f · · · ˜ f ℓ pointwise for µ -almost every x ∈ X , where for j = 1 , . . . , ℓ we denote by ˜ f j the orthogonalprojection of f j on the space of functions that are T tj -invariant for every t ∈ R . Remarks. • When ℓ = 2 and T , T commute, the result was established in [14] (for a ( t ) = t , a ( t ) = t , but for ℓ = 2 the argument given there probably extends to the moregeneral class of functions that we consider). The argument in [14] uses the transferenceprinciple of Calderón in order to get access to a harmonic analysis result from [15]; so itdoes not seem to extend to the case of non-commuting flows or to the case ℓ ≥ . Linear independence does not suffice in the case of real coefficients, to see this, take k = 1 and a ( t ) = αt where α is irrational, then the sequence [ a ( n )] is not ergodic for the (totally ergodic) system ( T , m T , R ) , where Rx = x + 1 /α , x ∈ T . OINT ERGODICITY OF SEQUENCES 10 • If the transformations T , . . . , T ℓ commute, and a , . . . , a ℓ are given by polynomials,then mean convergence for (56) was established in [1] without any growth assumptions(and previously in [38] when T = · · · = T ℓ ). Moreover, other related convergence resultsfor flows can be found in [8].The result applies for the collection of functions t c , . . . , t c ℓ where c , . . . , c ℓ ∈ (0 , + ∞ ) are distinct (or linear combinations of such functions, assuming they have differentgrowth), and also applies to some collections of fast growing functions like b t , . . . , b tℓ ,where b , . . . , b ℓ ∈ (1 , + ∞ ) are distinct. But our method does not work for collections offunctions with the same or “not substantially different growth”, like t , t + t and t, t log t ,or for functions with “very different growth”, like t, t .The method we use to prove Theorem 1.9 is different (and in fact much simpler) thanthe one used for the results on discrete time averages given on previous subsections.We take advantage of the fact that functions like t t are onto R + (a property thatcrucially fails in Z + ); this enables us to make a change of variables (see Lemma 7.2)and reduce matters to the case where all the iterates have sublinear growth. This is acrucial simplification and is what allows us to prove much stronger results than those fordiscrete-time actions.1.8. Open problems.
Theorem 1.1 shows that the sequences a , . . . , a ℓ : N → Z arejointly ergodic if and only if they are ( i ) good for seminorm estimates and ( ii ) good forequidistribution. It is a tantalizing problem to determine whether the equidistributionassumption alone suffices to deduce joint ergodicity (in which case condition ( i ) can beinferred from condition ( ii ) ). Although at first this may seem like an unlikely scenario,we have not been able to construct a counterexample, which leaves the following questionopen: Problem 1.
If the sequences a , . . . , a ℓ : N → Z are good for equidistribution, are theyjointly ergodic? Remark.
The corresponding question for pointwise convergence has a negative answereven when ℓ = 1 . For instance if a ( n ) := n + [(log n ) a ] , n ∈ N , where a ∈ (cid:0) , (cid:1) , thenit follows from [12, Theorems 3.2] that the sequence a ( n ) is good for the mean ergodictheorem (hence good for equidistribution), and from [12, Theorems 3.7] it follows thatit is bad for the pointwise ergodic theorem. So one has to impose stronger quantita-tive equidistribution assumptions on the sequences in order to get legitimate pointwiseconvergence criteria.In support of a positive answer to the previous question let us mention that the answeris positive for ℓ = 1 (this follows from the spectral theorem). Moreover, Corollary 1.3implies that if the sequences a , . . . , a ℓ are good for equidistribution, then they are jointlyergodic for nilsystems.We also record a problem regarding special sequences arising from smooth functionswithout a vanishing derivative. Problem 2.
Show that the collections of the sequences in Theorem 1.8 are good forseminorm estimates.
It is known that these collections of sequences are good for equidistribution (see theproof of Theorem 1.8 for relevant references), so solving this problem for any of theexamples given in Theorem 1.8 would imply (by Theorem 1.1) that the correspondingcollection of sequences is jointly ergodic. We remark that related multiple recurrenceresults and combinatorial consequences are largely unknown for these collections of se-quences.
OINT ERGODICITY OF SEQUENCES 11 The Furstenberg-Weiss theorem
The goal of this section is to explain the basic principles used in the proof of our mainresults in a simple yet interesting setup that allows us to avoid much of the technicalitiesthat appear in the proof of Theorem 1.1 and Theorem 1.4. We do this by giving a proofof the following result that was originally proved by Furstenberg and Weiss: Theorem 2.1 (Furstenberg-Weiss [25]) . Let ( X, µ, T ) be a system and f , f ∈ L ∞ ( µ ) .If either f or f is orthogonal to K rat ( T ) , then (10) lim N →∞ E n ∈ [ N ] T n f · T n f = 0 in L ( µ ) . It is easy to deduce from the previous result a related multiple recurrence statementand then, via the correspondence principle of Furstenberg, conclude that any set ofintegers with positive upper density contains patterns of the form m, m + n, m + n , forsome m, n ∈ N . This combinatorial result was first proved in [6] using ergodic theoryand a proof that avoids ergodic theory (and produces reasonable quantitative bounds)was given in [37].2.1. Proof strategy.
Let us briefly describe the main idea of the proof, which is anadaptation of a technique used in [37] to our ergodic setup. If (10) fails, then usingpretty standard arguments, we deduce that ||| ˜ f ||| > where (the next limit is a weaklimit) ˜ f := lim k →∞ E n ∈ [ N k ] T − n g k · T − n + n f for some g k ∈ L ∞ ( µ ) , bounded by , and N k → ∞ . The reader will find the details inSteps 1 and 2 below. This then easily implies that(11) lim inf N →∞ E n ∈ [ N ] ℜ (cid:16) Z ∆ n ˜ f · χ n dµ (cid:17) > , where ∆ n f = T n f · f , n ∈ N , and χ n , n ∈ N , are appropriate eigenfunctions of thesystem ( X, µ, T ) with unit modulus. Suppose for convenience that the operators ∆ n and E n ∈ [ N k ] commute; this is of course not true, but another convenient property holdsand can be used as a substitute. Using this simplifying assumption we have lim inf N →∞ E n ∈ [ N ] lim k →∞ E n ∈ [ N k ] ℜ (cid:16) Z T − n (∆ n g k ) · T − n + n (∆ n f ) · χ n dµ (cid:17) > . At this point we have a much simpler problem to work with. Indeed, after composingwith T n and using that T n χ n = e ( n α n ) · χ n , for some α n ∈ Spec( T ) , n ∈ N , andthen using the Cauchy-Schwarz inequality, we deduce that lim inf N →∞ E n ∈ [ N ] lim sup k →∞ (cid:13)(cid:13) E n ∈ [ N k ] e ( n α n ) · T n (∆ n f ) (cid:13)(cid:13) L ( µ ) > . Using the spectral theorem for unitary operators and then Fatou’s Lemma, we get thatfor some positive measures σ n on T , n ∈ N , with uniformly bounded total mass, wehave lim inf N →∞ E n ∈ [ N ] (cid:13)(cid:13)(cid:13)(cid:13) lim sup k →∞ (cid:12)(cid:12) E n ∈ [ N k ] e ( n α n + nt ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) L ( σ n ( t )) > . Using the good equidistribution properties of sequences of the form nα + n β (mod 1) for β irrational, we deduce that the numbers α n , n ∈ N , belong to a finite set of rationals.We conclude that for appropriate Λ N ⊂ [ N ] , N ∈ N , with lim inf N →∞ | Λ N | /N > , upon Another reason why we choose to prove Theorem 2.1 separately is in order to have an “elementaryproof” on record, since it is not covered by Theorem 1.1 (it is covered by Theorem 1.4, but this resultdepends on deep results from [27]).
OINT ERGODICITY OF SEQUENCES 12 replacing E n ∈ [ N ] with E n ∈ Λ N (which we can do by being more careful), equation (11)holds with χ n , n ∈ Λ N , that depend only on N ∈ N . It is then pretty straightforwardto deduce that ||| ˜ f ||| > . The details of the previous argument and the necessaryadjustments, are given in Steps 3 and 4. Given this, it is an easy matter to deduce inStep 5 that the functions f and f are not orthogonal to K rat ( T ) , contradicting theassumptions of Theorem 2.1.2.2. Step 1. (Characteristic factors) Our first step is to show that if (10) fails, then italso fails for some function f of special form. More precisely the following holds: Proposition 2.2.
Let ( X, µ, T ) be a system and f , f ∈ L ∞ ( µ ) be such that (12) E n ∈ [ N ] T n f · T n f in L ( µ ) as N → ∞ . Then there exist N k → ∞ and g k ∈ L ∞ ( µ ) , with k g k k L ∞ ( µ ) ≤ , k ∈ N , such that for (13) ˜ f := lim k →∞ E n ∈ [ N k ] T − n g k · T − n + n f , where the limit is a weak limit (note that then ˜ f ∈ L ∞ ( µ ) ), we have (14) E n ∈ [ N ] T n f · T n ˜ f in L ( µ ) as N → ∞ .Proof. We can assume that both f and f are bounded by . For fixed f ∈ L ∞ ( µ ) we let C = C ( f ) be the L ( µ ) closure of all linear combinations of all subsequential weak-limitsof sequences of the form E n ∈ [ N ] T − n g N · T − n + n f , where g N ∈ L ∞ ( µ ) , N ∈ N , are all bounded by .We first claim that if f is orthogonal to the subspace C , then E n ∈ [ N ] T n f · T n f → in L ( µ ) as N → ∞ . Indeed, if this is not the case, then there exist a > and N k → ∞ such that (cid:13)(cid:13)(cid:13) E n ∈ [ N k ] T n f · T n f (cid:13)(cid:13)(cid:13) L ( µ ) ≥ a, k ∈ N . If we define the functions g k := E n ∈ [ N k ] T n f · T n f , k ∈ N , which all have L ∞ ( µ ) -norm bounded by , we deduce that(15) E n ∈ [ N k ] Z g k · T n f · T n f dµ ≥ a , k ∈ N . By passing to a subsequence, we can assume that the averages E n ∈ [ N k ] T − n g k · T − n + n f ,which are functions bounded by , converge in the weak topology say to a function f ∈ C .Then composing with T − n in (15) we deduce that Z f · f dµ = 0 , contradicting our assumption that f is orthogonal to the subspace C . This proves ourclaim.From the previous claim we conclude that E n ∈ [ N ] T n f · T n ( f − E ( f |C )) → OINT ERGODICITY OF SEQUENCES 13 in L ( µ ) as N → ∞ , where E ( f |C ) denotes the orthogonal projection of f onto theclosed subspace C . Hence, if (12) holds, then E n ∈ [ N ] T n f · T n E ( f |C ) in L ( µ ) as N → ∞ . Using the definition of C and an approximation argument, we getthat there exist N k → ∞ and g k ∈ L ∞ ( µ ) , k ∈ N , all bounded by , such that for ˜ f asin (13) we have that (14) holds. Lastly, since f and g k , k ∈ N , all have L ∞ ( µ ) normbounded by , the same holds for ˜ f . This completes the proof. (cid:3) Step 2. (Seminorm estimates) We state some seminorm estimates that were provedin a slightly different form in [25]. The technique is standard and we sketch it forcompleteness.
Proposition 2.3.
Let ( X, µ, T ) be an ergodic system and f , f ∈ L ∞ ( µ ) be such that ||| f ||| = 0 . Then (16) lim N →∞ E n ∈ [ N ] T n f · T n f = 0 where convergence takes place in L ( µ ) .Proof. Using the van der Corput Lemma (see for example [25, Lemma 1.1]), composingwith T − n , and then using the Cauchy-Schwarz inequality, we get that it suffices to showthat for every m ∈ N we have lim N →∞ E n ∈ [ N ] T ( n + m ) − n f · T n − n f = 0 . Using the van der Corput lemma again, composing with T − n , and then using the Cauchy-Schwarz inequality, we further reduce matters to showing the following: If ||| f ||| = 0 ,then for all g, h ∈ L ∞ ( µ ) and all a, b, c ∈ N with a = b, a = c , we have(17) lim N →∞ E n ∈ [ N ] T an f · T bn g · T cn h = 0 . Since ||| f ||| = 0 , this follows from [34, Theorem 8]. (cid:3) Step 3. (Seminorms of averages of functions) Our next goal is to show that if theseminorm of an average of functions is positive, then some related positiveness propertyholds for the individual functions. This is a crucial property for our argument; we stateit here only in the form needed for the proof of Theorem 2.1, and prove the extensionneeded for Theorem 1.1 in Proposition 4.3 below.
Definition.
Let ( X, µ, T ) be a system.(i) We denote with E ( T ) the set of its eigenfunctions with modulus one, meaning,the functions f ∈ L ∞ ( µ ) such that | f | = 1 and T f = e ( t ) f for some t ∈ [0 , .(ii) If f ∈ L ∞ ( µ ) and n ∈ Z , we let ∆ n f := T n f · f . Proposition 2.4.
Let ( X, µ, T ) be an ergodic system, f n,k ∈ L ∞ ( µ ) , n, k ∈ N , be boundedby , and f ∈ L ∞ ( µ ) be defined by f := lim k →∞ E n ∈ [ N k ] f n,k , for some N k → ∞ , where the average is assumed to converge weakly. If ||| f ||| > , thenthere exist a > , a subset Λ of N with positive lower density, and χ n ∈ E ( T ) , n ∈ N ,such that ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > a, n ∈ Λ , The argument in [34] is non-trivial. One can avoid it by using that ||| f ||| = 0 implies the identity(17) (see for example [28, Chapter 21, Proposition 7]). The drawback is that the use of the -th seminormwould complicate our subsequent arguments; but still a proof that avoids deep machinery can be givenby using the argument in Section 4. OINT ERGODICITY OF SEQUENCES 14 and lim inf N →∞ E n ,n ′ ∈ Λ ∩ [ N ] lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z ∆ n − n ′ f n,k · T − n ′ ( χ n · χ n ′ ) dµ (cid:17) > . Remark.
We plan to apply this proposition in the next step for the function ˜ f givenby (13) in place of f . Proof.
By (28) we have ||| f ||| = lim N →∞ E n ∈ [ N ] ||| ∆ n f ||| > . We deduce from Proposi-tion 3.1 that there exist χ n ∈ E ( T ) , n ∈ N , such that lim inf N →∞ E n ∈ [ N ] ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > . Hence, there exist a > , and a subset Λ of N with positive lower density, such that ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > a, n ∈ Λ . Then we have lim inf N →∞ E n ∈ Λ ∩ [ N ] ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > . Since f = lim k →∞ E n ∈ [ N k ] f n,k (the limit is a weak limit), we deduce that lim inf N →∞ E n ∈ Λ ∩ [ N ] lim k →∞ E n ∈ [ N k ] ℜ (cid:16) Z T n f n,k · f · χ n dµ (cid:17) > . Since all the limits lim k →∞ E n ∈ [ N k ] exist, we can interchange the finite average E n ∈ Λ ∩ [ N ] with lim k →∞ E n ∈ [ N k ] , and after using the Cauchy-Schwarz inequality we deduce that lim inf N →∞ lim sup k →∞ E n ∈ [ N k ] Z | E n ∈ Λ ∩ [ N ] T n f n,k · χ n | dµ > . Expanding the square, composing with T − n ′ , and using that the limsup of a finite sumis at most the sum of the limsups, we get lim inf N →∞ E n ,n ′ ∈ Λ ∩ [ N ] lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z f n,k · T n − n ′ f n,k · T − n ′ ( χ n · χ n ′ ) dµ (cid:17) > . This completes the proof. (cid:3)
Step 4. (Going from ||| · ||| to ||| · ||| ) We will use the following elementary fact: Lemma 2.5.
Let N ∈ N and v , . . . , v N be elements of an inner product space of normat most . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N X n =1 v n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ N N X m =1 ℜ (cid:16) N N − m X n =1 h v n + m , v n i (cid:17) + 1 N , where ℜ ( z ) denotes the real part of the complex number z . Our next goal is to use Proposition 2.4 and the particular form of the iterates definingthe function ˜ f in order to establish the following result: Proposition 2.6.
Let ( X, µ, T ) be an ergodic system, ˜ f be as in (13) , and suppose that ||| ˜ f ||| > . Then ||| ˜ f ||| > .Proof. Recall that ˜ f := lim k →∞ E n ∈ [ N k ] T − n g k · T − n + n f , where the average converges weakly and the functions involved are uniformly bounded,say by . Applying Proposition 2.4 for f n,k := T − n g k · T − n + n f , n, k ∈ N , we get that OINT ERGODICITY OF SEQUENCES 15 there exist a > , a subset Λ of N with lower density at least a , and χ n ∈ E ( T ) , n ∈ N ,such that(18) ℜ (cid:16) Z ∆ n ˜ f · χ n dµ (cid:17) > a, n ∈ Λ , and(19) lim inf N →∞ E n ,n ′ ∈ Λ ∩ [ N ] lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z T − n (∆ n − n ′ g k ) · T − n + n (∆ n − n ′ f ) · T − n ′ ( χ n · χ n ′ ) dµ (cid:17) > . Our goal is to use (19) in order to show that the eigenfunctions χ n , n ∈ [ N ] , in (18)depend only on N ; this combined with Lemma 3.4 below, will enable us to deduce that ||| ˜ f ||| > .We analyse the second estimate first (the first will be used at the end of this proof).We compose with T n , use the identity T n χ n = e ( n α n ) · χ n , n, n ∈ N , which holds for some α n ∈ [0 , , n ∈ N , and then use the Cauchy-Schwarz inequality.We deduce that(20) lim inf N →∞ E n ,n ′ ∈ Λ ∩ [ N ] lim sup k →∞ (cid:13)(cid:13)(cid:13) E n ∈ [ N k ] e ( n ( α n − α n ′ )) T n (∆ n − n ′ f ) (cid:13)(cid:13)(cid:13) L ( µ ) > . Using the spectral theorem for unitary operators and Fatou’s Lemma, we deduce from(20) that there exist positive measures σ n ,n ′ on T , with total at most , such that(21) lim inf N →∞ E n ,n ′ ∈ Λ ∩ [ N ] (cid:13)(cid:13)(cid:13) g n ,n ′ (cid:13)(cid:13)(cid:13) L ( σ n ,n ′ ) > , where g n ,n ′ ( t ) := lim sup k →∞ | E n ∈ [ N k ] e ( nt + n ( α n − α n ′ )) | , n , n ′ ∈ N , t ∈ [0 , . We deduce from (21) that there exist a ′ > , n ′ ,N ∈ [ N ] , N ∈ N , such that lim inf N →∞ E n ∈ Λ ∩ [ N ] (cid:13)(cid:13)(cid:13) g n ,n ′ ,N (cid:13)(cid:13)(cid:13) L ( σ n ,n ′ ,N ) > a ′ . Hence, there exist subsets Λ N of Λ ∩ [ N ] , N ∈ N , such that lim inf N →∞ | Λ N | N > and (cid:13)(cid:13)(cid:13) g n ,n ′ ,N (cid:13)(cid:13)(cid:13) L ( σ n ,n ′ ,N ) > a ′ , n ∈ Λ N , N ∈ N . Since the measures have mass at most , this immediately implies that there exist t n ,N ∈ [0 , , n ∈ Λ N , N ∈ N , such that g n ,n ′ ,N ( t n ,N ) > a ′ , n ∈ Λ N , N ∈ N , or, equivalently,(22) lim sup k →∞ | E n ∈ [ N k ] e ( nt n ,N + n ( α n − α n ′ ,N )) | > a ′ , n ∈ Λ N , N ∈ N . Now using Weyl type estimates, as in Lemma 2.8 below, we deduce from (22) thatthere exists a finite set of rationals R = R ( a ′ ) in [0 , such that α n − α n ′ ,N ∈ R (mod 1) , n ∈ Λ ′ N , N ∈ N . OINT ERGODICITY OF SEQUENCES 16
We deduce from this that there exist subsets Λ ′ N of Λ N , N ∈ N , such that(23) lim inf N →∞ | Λ ′ N | N > , and r N ∈ R such that α n = β N := r N + α n ′ ,N (mod 1) for all N ∈ N and n ∈ Λ ′ N . For n ∈ Λ ′ N we then have that χ n is a T -eigenfunction with unit modulus and eigenvalue e ( β N ) , N ∈ N , namely,(24) T χ n = e ( β N ) · χ n , n ∈ Λ ′ N , N ∈ N . Using (18) and since Λ ′ N ⊂ Λ satisfies (23), we have that lim inf N →∞ E n ∈ Λ ′ N ℜ (cid:16) Z ∆ n ˜ f · χ n dµ (cid:17) > . We are going to deduce from this and (24) that ||| ˜ f ||| > . Composing with T n ′ ,averaging over n ′ ∈ [ N ] , using that T n ′ χ n = e ( n ′ β N ) · χ n , n ∈ Λ ′ N , N ∈ N , n ′ ∈ N , and then using the Cauchy-Schwarz inequality, we get lim inf N →∞ E n ∈ Λ ′ N Z | E n ′ ∈ [ N ] e ( n ′ β N ) · T n ′ + n ˜ f · T n ′ ˜ f | dµ > . Since (23) holds, we get lim inf N →∞ E n ∈ [ N ] Z | E n ′ ∈ [ N ] e ( n ′ β N ) · T n ′ + n ˜ f · T n ′ ˜ f | dµ > . Finally, using Lemma 2.5 and composing with T − n ′ , we deduce that lim inf N →∞ ℜ (cid:16) E n ,n ′ ,n ∈ [ N ] [ N ] ( n ′ + n ) e ( n β N ) Z ˜ f · T n ˜ f · T n ˜ f · T n + n ˜ f dµ (cid:17) > . Hence, for some n ′ ,N ∈ [ N ] , N ∈ N , we have lim inf N →∞ ℜ (cid:16) E n ,n ∈ [ N ] [ N ] ( n ′ ,N + n ) e ( n β N ) Z ˜ f · T n ˜ f · T n ˜ f · T n + n ˜ f dµ (cid:17) > . Using the case s = 2 of Lemma 3.4 below, for c N ( n , n ) := [ N ] ( n ′ ,N + n ) e ( n β N ) , n , n ∈ [ N ] , N ∈ N , which do not depend on the variable n , we deduce that ||| ˜ f ||| > ,completing the proof of the claim. (cid:3) Step 5. (Proof of Theorem 2.1) We are now ready to conclude the proof of Theo-rem 2.1. It is known that if a function is orthogonal to the rational Kronecker factor ofa system, then the same property holds with respect to almost every ergodic component(see for example [22, Theorem 3.2]). Hence, using the ergodic decomposition theorem,we can assume that the system is ergodic.We first show that if E ( f |K rat ( T )) = 0 , then (10) holds. Arguing by contradictionsuppose that (10) fails. Then if ˜ f is given by (13) we get by Propositions 2.2 and 2.3 that ||| ˜ f ||| > . We deduce from Proposition 2.6 that ||| ˜ f ||| > . It follows by Proposition 3.1below that there exists χ ∈ E ( T ) such that lim k →∞ E n ∈ [ N k ] Z T − n g k · T − n + n f · χ dµ = 0 . Composing with T n , using that T n χ = e ( n α ) · χ , n ∈ N , for some α ∈ [0 , , and usingthe Cauchy-Schwarz inequality, we get that E n ∈ [ N k ] e ( n α ) T n f in L ( µ ) as k → ∞ . Since lim N →∞ E n ∈ [ N ] e ( n α + nt ) = 0 if t is irrational and α ∈ R , using the spectral theorem, the bounded convergence theorem, and our assumption OINT ERGODICITY OF SEQUENCES 17 E ( f |K rat ( T )) = 0 (which implies that the spectral measure σ f has no rational pointmass), we deduce that the last limit is zero, a contradiction.Finally, we show that if E ( f |K rat ( T )) = 0 , then (10) holds. By the previous stepwe can assume that f belongs to the rational Kronecker factor of the system and byapproximation that it is an eigenfunction with eigenvalue e ( α ) for some α ∈ Q . Hence,it suffices to show that lim N →∞ E n ∈ [ N ] e ( nα ) T n f = 0 in L ( µ ) . Since lim N →∞ E n ∈ [ N ] e ( n t + nα ) = 0 if t is irrational and α ∈ R , as be-fore, using the spectral theorem, the bounded convergence theorem, and our assumption E ( f |K rat ( T )) = 0 , we deduce that the last limit is zero. This completes the proof ofTheorem 2.1.2.7. Weyl-type estimates.
We record some pretty standard Weyl-type estimates thatwere used in the previous argument. The following result was proved in Proposition 4.3in [26].
Lemma 2.7.
Let a > and d ∈ N . There exists Q = Q ( a, d ) > , C = C ( a, d ) > ,such that if | E n ∈ [ N ] e ( t n + · · · + t d n d ) | ≥ a for some N ∈ N and t , . . . , t d ∈ [0 , , then for every i ∈ [ d ] there exist non-negativeintegers p, q ≤ Q (depending on i ) such that (cid:12)(cid:12) t i − pq (cid:12)(cid:12) ≤ CN i . From this we deduce the following (we only use it for ℓ = 2 and p ( n ) = n , p ( n ) = n ): Lemma 2.8.
Let a > and d ∈ N . Then there exists a finite set of rationals R = R ( a, d ) with the following property: If p , . . . , p ℓ ∈ Z [ t ] are rationally independent polynomials ofdegree at most d such that for some l ∈ [ ℓ ] and t l ∈ [0 , we have lim sup N →∞ sup t i ∈ R ,i ∈ [ ℓ ] \{ l } | E n ∈ [ N ] e ( p ( n ) t + · · · + p ℓ ( n ) t ℓ ) | ≥ a, then t l ∈ R .Proof. Without loss of generality we can assume that l = ℓ = d and that all the polyno-mials have zero constant term. Let p j ( n ) = P ℓi =1 c i,j n i , j = 1 , . . . , ℓ , for some c i,j ∈ Z .The assumption gives that there exist N k → ∞ and t i,k , t ℓ ∈ [0 , , i = 1 , . . . , ℓ − , k ∈ N , such that | E n ∈ [ N k ] e ( p ( n ) t ,k + · · · + p ℓ − ( n ) t ℓ − ,k + p ℓ ( n ) t ℓ ) | ≥ a for all k ∈ N . Then for all k ∈ N we have | E n ∈ [ N k ] e ( ns ,k + · · · + n ℓ s ℓ,k ) | ≥ a where s i,k := ℓ − X j =1 c i,j t j,k + c i,ℓ t ℓ , i = 1 , . . . , ℓ, k ∈ N . The previous lemma implies that there exist positive integers
Q, C that depend onlyon a, ℓ such that for every k ∈ N and i ∈ { , . . . , ℓ } there exist non-negative integers p i,k , q k ≤ Q and m i,k ∈ Z , i = 1 , . . . , ℓ , k ∈ N , such that (cid:12)(cid:12)(cid:12) s i,k − p i,k q k − m i,k (cid:12)(cid:12)(cid:12) ≤ CN ik , i = 1 , . . . , ℓ. OINT ERGODICITY OF SEQUENCES 18
Since the polynomials p , . . . , p ℓ are rationally independent, the matrix ( c i,j ) i,j ∈ [ ℓ ] is in-vertible. We deduce that there exist l, l , . . . , l ℓ ∈ N , with size smaller than L = L ( Q ) ≥ Q , such that t ℓ = ℓ X i =1 l i l s i,k . Combining the previous two facts we get that there exist ˜ p k ∈ Z and non-negative integers ˜ q k ≤ L , k ∈ N , such that lim k →∞ ˜ p k ˜ q k = t ℓ . Since t ℓ ∈ [0 , and ˜ q k , k ∈ N , are positive integers bounded by L , it follows that t ℓ = pq for some non-negative integers p, q ≤ L . So we can take R to be the set of all rationalswith numerator and denominator at most L . (cid:3) More general results.
Before embarking into the proof of our main results let usmake some remarks about the extend to which the previous argument applies to moregeneral families of sequences.If in place of n, n we have sequences a , . . . , a ℓ that are good for seminorm estimatesand equidistribution, a similar, but technically more complicated argument can be usedto prove joint ergodicity and is given in the next two sections.If in place of n, n we have two rationally independent integer polynomials p , p ,then modulo a few additional technical complications, the previous argument can beadapted as in the next two sections in order to prove that the rational Kronecker factor ischaracteristic for p , p (for this extension we also need the Weyl estimates of Lemma 2.8).If in place of n, n we have rationally independent polynomials p , . . . , p ℓ , where ℓ ≥ ,or if we work in the more general setup of Theorem 1.4, then a new non-trivial obstaclearises. In order to overcome this problem we use a consequence of the main result in[27] that enables us to reduce matters to the case where the system is totally ergodic(the reduction is carried out in Section 5.1), a case that is covered by Theorem 1.1.Alternatively, we could have used an additional inductive argument, as in [36], and avoidthe use of deep results form [27], thus leading to a more “elementary proof” for the specialcase of Theorem 1.4 that covers all rational independent polynomials. We chose not to doso, firstly, because this would lead to a much more complicated argument and, secondly,because this approach probably requires to impose stronger equidistribution assumptions(of quantitative nature) than those used in Theorem 1.4.3.
Preparation for the main result
In this section we gather some basic notation and results about the ergodic semi-norms and also prove some elementary estimates that will be used later in the proof ofTheorem 1.1.3.1.
The ergodic seminorms.
Throughout, we use the following notation:
Definition.
Let ( X, µ, T ) be a system and f ∈ L ∞ ( µ ) . If n = ( n , . . . , n s ) ∈ Z s + , n ′ = ( n ′ , . . . , n ′ k ) ∈ Z k + , ǫ = ( ǫ , . . . , ǫ s ) ∈ { , } s , and z ∈ C , we let(i) ǫ · n := ǫ n + · · · + ǫ s n s ;(ii) | n | := n + · · · + n s ;(iii) C l z = z if l is even and C l z = z if l is odd; It is caused by the fact that we can no longer use the spectral theorem (as in the case ℓ = 2 ) inorder to carry out Step 4 of Section 2.5. For ℓ = 3 the problem occurs when f , f ∈ K rat ( T ) and wewant to deduce that an estimate of the form (38) below implies an estimate of the form (40). In the caseof sequences that are good for equidistribution, this problem does not arise because we can replace thefunctions f , f by constants. OINT ERGODICITY OF SEQUENCES 19 (iv) n ǫ := ( n ǫ , . . . , n ǫ s s ) , where n j := n j and n j := n ′ j for j = 1 , . . . , s ;(v) ∆ n f := ∆ n · · · ∆ n s f = Q ǫ ∈{ , } s C | ǫ | T ǫ · n f (here we allow n ∈ Z s ).Given a system ( X, µ, T ) we will use the seminorms ||| · ||| s , s ∈ N , that were introducedin [27] for ergodic systems and can be defined similarly for general systems (see forexample [28, Chapter 8, Proposition 16]). They are often refereed to as Gowers-Host-Kra seminorms and are inductively defined for f ∈ L ∞ ( µ ) as follows (for convenience wealso define ||| · ||| , which is not a seminorm): ||| f ||| := Z f dµ, and for s ∈ Z + we let(25) ||| f ||| s +1 s +1 := lim N →∞ E n ∈ [ N ] ||| ∆ n f ||| s s . The limit can be shown to exist by successive applications of the mean ergodic theoremand for f ∈ L ∞ ( µ ) and s ∈ Z + we have ||| f ||| s ≤ ||| f ||| s +1 (see [27] or [28, Chapter 8]). Itfollows immediately from the definition that(26) ||| f ||| s s = lim N →∞ · · · lim N s →∞ E n ∈ [ N ] · · · E n s ∈ [ N s ] Z ∆ n ,...,n s f dµ. It can be shown that we can take any s ′ ≤ s of the iterative limits to be simultaneouslimits (i.e. average over [ N ] s ′ and let N → ∞ ) without changing the value of the limit.This was originally proved in [27], for a much simpler proof see [7]. For s ′ = s this givesthe identity(27) ||| f ||| s s = lim N →∞ E n ∈ [ N ] s Z ∆ n f dµ, and for s ′ = s − it gives the identity(28) ||| f ||| s s = lim N →∞ E n ∈ [ N ] s − ||| ∆ n f ||| . Some key facts.
Recall that if ( X, µ, T ) is a system, with E ( T ) we denote the setof its eigenfunctions with modulus one. Proposition 3.1.
Let ( X, µ, T ) be an ergodic system and f ∈ L ∞ ( µ ) be a function with k f k L ∞ ( µ ) ≤ . Then ||| f ||| ≤ sup χ ∈E ( T ) ℜ (cid:16) Z f · χ dµ (cid:17) . Proof.
Let K ( T ) be the Kronecker factor of the system, meaning, the closed subspace of L ( µ ) spanned by all eigenfunctions of the system. It is well known (and not hard toprove, see for example [28, Chapter 8, Theorem 1]) that ||| f ||| = ||| ˜ f ||| where ˜ f := E ( f |K ( T )) . Since the system is ergodic, the subspace K ( T ) has an orthonor-mal basis of eigenfunctions of modulus one, say ( χ j ) j ∈ N . Then ˜ f = P ∞ j =1 c j χ j where c j := Z ˜ f · χ j dµ = Z f · χ j dµ, j ∈ N . Then we have ||| ˜ f ||| = ∞ X j =1 | c j | ≤ sup j ∈ N ( | c j | ) ∞ X j =1 | c j | = sup j ∈ N ( | c j | ) k f k L ( µ ) ≤ sup j ∈ N (cid:12)(cid:12)(cid:12) Z f · χ j dµ (cid:12)(cid:12)(cid:12) , where the first identity follows by orthonormality and direct computation, the secondidentity by the Parseval identity, and the last estimate since all functions involved are OINT ERGODICITY OF SEQUENCES 20 bounded by . The result now follows since the set E ( T ) is invariant under multiplicationby unit modulus constants. (cid:3) Proposition 3.2.
Let ( X, µ, T ) be an ergodic system and f ∈ L ∞ ( µ ) be such that ||| f ||| s +2 > for some s ∈ Z + . Then there exist χ n ∈ E ( T ) , n ∈ N , such that lim inf N →∞ E n ∈ [ N ] s ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > . Proof.
By (28) we have that lim N →∞ E n ∈ [ N ] s ||| ∆ n f ||| > . Using Proposition 3.1 we deduce that lim inf N →∞ E n ∈ [ N ] s sup χ ∈E ( T ) ℜ (cid:16) Z ∆ n f · χ dµ (cid:17) > . From this the asserted estimate readily follows. (cid:3)
We will use the following variant of the so called Gowers-Cauchy-Schwarz inequality:
Lemma 3.3.
Let ( X, µ, T ) be a system, and for s ∈ N let f ǫ ∈ L ∞ ( µ ) be bounded by for ǫ ∈ { , } s , and g n ∈ L ∞ ( µ ) for n ∈ N s . Let also , . . . , . Then for every N ∈ N we have (cid:12)(cid:12)(cid:12) E n ∈ [ N ] s Z Y ǫ ∈{ , } s T ǫ · n f ǫ · g n dµ (cid:12)(cid:12)(cid:12) s ≤ E n,n ′ ∈ [ N ] s Z ∆ n − n ′ f · T −| n | (cid:0) Y ǫ ∈{ , } s C | ǫ | g n ǫ (cid:1) dµ. Proof.
For notational simplicity we give the details only for s = 2 . The general casefollows in a similar manner by successively applying the Cauchy-Schwarz inequality withrespect to the variables n s , . . . , n , exactly as we do below for s = 2 . We have that (cid:12)(cid:12)(cid:12) E n ,n ∈ [ N ] Z f · T n f · T n f · T n + n f · g n ,n dµ (cid:12)(cid:12)(cid:12) is bounded by (we use that f , f are bounded by ) E n ∈ [ N ] Z (cid:12)(cid:12)(cid:12) E n ∈ [ N ] T n f · T n + n f · g n ,n (cid:12)(cid:12)(cid:12) dµ. After expanding the square we find that this expression is equal to E n ∈ [ N ] Z E n ,n ′ ∈ [ N ] T n f · T n ′ f · T n + n f · T n + n ′ f · g n ,n · g n ,n ′ dµ. After composing with T − n , exchanging E n ∈ [ N ] with E n ,n ′ ∈ [ N ] , using the Cauchy-Schwarz inequality, and that f is bounded by , we get that the square of the lastexpression is bounded by E n ,n ′ ∈ [ N ] Z (cid:12)(cid:12)(cid:12) E n ∈ [ N ] T n f · T n + n ′ − n f · T − n ( g n ,n · g n ,n ′ ) (cid:12)(cid:12)(cid:12) dµ. As before, we expand the square, and compose with T − n . We arrive at the expression E n ,n ,n ′ ,n ′ ∈ [ N ] Z f · T n ′ − n f · T n ′ − n f · T n ′ + n ′ − n − n f · T − n − n ( g n ,n · g n ,n ′ · g n ′ ,n · g n ′ ,n ′ ) dµ, which is equal to the right hand side of the asserted estimate when s = 2 (for n := ( n , n ) , n ′ := ( n ′ , n ′ ) ). Combining the previous two estimates gives the asserted bound for s = 2 . (cid:3) OINT ERGODICITY OF SEQUENCES 21
Lemma 3.4.
Let ( X, µ, T ) be an ergodic system and f ∈ L ∞ ( µ ) be such that ||| f ||| s = 0 for some s ∈ N . For j = 1 , . . . , s , N ∈ N , let b j,N ∈ ℓ ∞ ( N s ) be sequences that do notdepend on the variable n j and are bounded by , and let c N ( n ) := s Y j =1 b j,N ( n ) , n ∈ [ N ] s , N ∈ N . Then lim N →∞ (cid:13)(cid:13) E n ∈ [ N ] s c N ( n ) · ∆ n f (cid:13)(cid:13) L ( µ ) = 0 . Proof.
Using the Cauchy-Schwarz inequality, the fact that the sequence b s,N does notdepend on the variable n s and is bounded by , and the identity ∆ n ,...,n s f = T n s (∆ n ,...,n s − f ) · ∆ n ,...,n s − f , we get that it suffices to show that (for later convenience we rename the variable n s as n ′ s ) lim N →∞ E n ∈ [ N ] s − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E n ′ s ∈ [ N ] s − Y j =1 b j,N ( n, n ′ s ) · T n ′ s (∆ n f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) = 0 . Using Lemma 2.5 for the average over n ′ s and composing with T − n ′ s , we get that itsuffices to show that lim N →∞ E n ∈ [ N ] s − E n s ,n ′ s ∈ [ N ] [ N ] ( n s + n ′ s ) s − Y j =1 b j,N ( n, n ′ s + n s ) · b j,N ( n, n ′ s ) Z T n s (∆ n f ) · ∆ n f dµ = 0 , or, equivalently, that lim N →∞ E n ′ s ∈ [ N ] E ( n,n s ) ∈ [ N ] s [ N ] ( n s + n ′ s ) s − Y j =1 b j,N ( n, n ′ s + n s ) · b j,N ( n, n ′ s ) Z ∆ n,n s f dµ = 0 . This would follow if we show that lim N →∞ sup n ′ s ∈ [ N ] (cid:12)(cid:12)(cid:12) E ( n,n s ) ∈ [ N ] s [ N ] ( n s + n ′ s ) s − Y j =1 b j,N ( n, n ′ s + n s ) · b j,N ( n, n ′ s ) Z ∆ n,n s f dµ (cid:12)(cid:12)(cid:12) = 0 , or equivalently, that for any choice of n ′ s,N ∈ [ N ] , N ∈ N , we have lim N →∞ E ( n,n s ) ∈ [ N ] s [ N ] ( n s + n ′ s,N ) s − Y j =1 b j,N ( n, n ′ s,N + n s ) · b j,N ( n, n ′ s,N ) Z ∆ n,n s f dµ = 0 . Using the Cauchy-Schwarz inequality we get that it suffices to show that lim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ( n,n s ) ∈ [ N ] s s − Y j =1 b ′ j,N ( n, n s ) · ∆ n,n s f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) = 0 , where we let b ′ s − ,N ( n, n s ) := [ N ] ( n s + n ′ s,N ) · b s − ,N ( n, n ′ s,N + n s ) · b s − ,N ( n, n ′ s,N ) , ( n, n s ) ∈ [ N ] s , N ∈ N , and for j = 1 , . . . , s − , we let b ′ j,N ( n, n s ) := b j,N ( n, n ′ s,N + n s ) · b j,N ( n, n ′ s,N ) , ( n, n s ) ∈ [ N ] s , N ∈ N . Note that we now have a product of s − sequences, instead of s , and for j = 1 , . . . , s − the sequences b ′ j,N do not depend on the variable n j . Hence, we can continue like that OINT ERGODICITY OF SEQUENCES 22 (on the next step we eliminate the sequences b ′ s − ,N , etc), and after s steps we deducethat it suffices to show that lim N →∞ E n ∈ [ N ] s Z ∆ n f dµ = 0 . This follows immediately from our assumption ||| f ||| s = 0 and (27) and completes theproof. (cid:3) Joint ergodicity of general sequences
The primary goal of this section is to prove Theorem 1.1. At the end of the sectionwe also deduce Corollaries 1.2 and 1.3 from Theorem 1.1. For a better understanding ofthe argument, we advice the reader to read first the technically less complicated modelcase that was treated in Section 2.It is clear that Property ( i ) of Theorem 1.1 implies Property ( ii ) (use appropriate eigen-functions to prove the equidistribution property). So we only prove that Property ( ii ) implies Property ( i ) .4.1. Goal.
It will be more convenient to prove the following statement (the case l = ℓ implies Theorem 1.1): Proposition 4.1.
Let ( X, µ, T ) be an ergodic system and a , . . . , a ℓ : N → Z be sequencesthat are good for seminorm estimates and equidistribution for ( X, µ, T ) . Then for every l ∈ { , , . . . , ℓ } the following holds:( P l ) If f , . . . , f ℓ ∈ L ∞ ( µ ) are such that f j ∈ E ( T ) for j = l + 1 , . . . , ℓ , then (29) lim N →∞ E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ = Z f dµ · . . . · Z f ℓ dµ where convergence takes place in L ( µ ) . In the following steps we fix ℓ ∈ N , a system, and a collection of sequences, and weare going to prove that property ( P l ) of Proposition 4.1 holds by using (finite) inductionon l ∈ { , , . . . , ℓ } .4.2. Proof strategy.
Let us briefly describe the main idea of the proof, which is againmotivated by the technique used in [37]. If (29) fails, then in Step 1 below, using ourassumption that the sequences a , . . . , a ℓ are good for seminorm estimates for ( X, µ, T ) ,we deduce that ||| ˜ f l ||| s +2 > for some s ∈ Z + where ˜ f l is given by (34). This then impliesthat(30) lim inf N →∞ E n ∈ [ N ] s ℜ (cid:16) Z ∆ n ˜ f l · χ n dµ (cid:17) > , where χ n , n ∈ N s , are appropriate unit modulus eigenfunctions of the system ( X, µ, T ) .In Steps 2 and 3, using suitable manipulations of (30), the good equidistribution prop-erties of the sequences a , . . . , a ℓ , and the induction hypothesis, we deduce that theeigenfunctions χ n satisfy some algebraic relations, which allow us to conclude that theyare products of sequences in s − variables. In view of this, (30) combined withLemma 3.4 enables us to show that ||| ˜ f l ||| s +1 > . Iterating this step s + 1 times wededuce that R ˜ f l dµ > . Given this, it is an easy matter in Step 4 to contradict theassumptions of Proposition 4.1.Let us say a few words about the aforementioned “suitable manipulations” performedin Steps 2 and 3, since they are more complicated than those used for the case s = 1 in We crucially use in this step that the sequences a , . . . , a ℓ are good for equidistribution; if ℓ ≥ andwe only knew that they were good for irrational equidistribution (for example if they were polynomialsequences), then our argument would have been much more complicated (as is the case in [36]). OINT ERGODICITY OF SEQUENCES 23
Section 2. Suppose for simplicity that ℓ = l = s = 2 ; hence ||| ˜ f ||| > , and our goal is toshow that ||| ˜ f ||| > . In this case, our assumption implies that(31) lim inf N →∞ E n ∈ [ N ] ℜ (cid:16) Z ∆ n ˜ f · χ n dµ (cid:17) > , for some eigenfunctions χ n with modulus . Using the Cauchy-Schwarz inequality twice(as in Lemma 3.3) we deduce that (the symmetries of the ergodic seminorms are cruciallyused here) lim inf N →∞ E n,n ′ ∈ [ N ] lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z ∆ n − n ′ g k · T a ( n ) (∆ n − n ′ f ) · T a ( n ) χ n,n ′ dµ (cid:17) > , where for n = ( n , n ) ∈ [ N ] , n ′ = ( n ′ , n ′ ) ∈ [ N ] , the eigenfunctions χ n,n ′ are given by(32) χ n,n ′ = χ n ,n · χ n ,n ′ · χ n ′ ,n · χ n ′ ,n ′ . Note that the last estimate is substantially simpler to analyze than the ℓ = 2 case of(29), since in the inside average over the variable n , the iterate T a ( n ) is applied to aneigenfunction of the system and not to an arbitrary function in L ( µ ) . Using Property ( P ) of Proposition 4.1 (which is our induction hypothesis), we can assume that thefunction ∆ n − n ′ f is constant for all n, n ′ ∈ N . Using the good equidistribution propertiesof the sequence a , we deduce that for N ∈ N there exist constants n ′ N ∈ [ N ] and sets Λ N ⊂ [ N ] with lim inf N →∞ | Λ N | /N > , such that χ n,n ′ N is constant for n ∈ Λ N .Using (32), this gives that for those values of n ∈ [ N ] the -variable sequence χ n ,n is a product of two -variable sequences. If we use this information and an appropriatevariant of (31), where we average over Λ N instead of [ N ] (by being more careful we canguarantee that this holds), we deduce by Lemma 3.4 that ||| ˜ f ||| > .4.3. Step 1. (Characteristic factors) The first step is to show that if (29) fails, then italso fails for some function f l of special form. The next statement can be proved byadjusting the proof of Proposition 2.2 in a straightforward way, so we omit it. Proposition 4.2.
Let a , . . . , a ℓ : N → Z be sequences, ( X, µ, T ) be a system, and f , . . . , f ℓ ∈ L ∞ ( µ ) be such that (33) E n ∈ [ N ] T a ( n ) f · · · T a ℓ ( n ) f ℓ in L ( µ ) as N → ∞ . Let l ∈ [ ℓ ] . Then there exist N k → ∞ and g k ∈ L ∞ ( µ ) , with k g k k L ∞ ( µ ) ≤ , k ∈ N , such that for (34) ˜ f l := lim k →∞ E n ∈ [ N k ] T − a l ( n ) g k · Y j ∈ [ ℓ ] ,j = l T a j ( n ) − a l ( n ) f j , where the limit is a weak limit (note that then ˜ f ∈ L ∞ ( µ ) ), we have (35) E n ∈ [ N ] T a l ( n ) ˜ f l · Y j ∈ [ ℓ ] ,j = l T a j ( n ) f j in L ( µ ) as N → ∞ . Step 2. (Seminorms of averages of functions) Our next goal is to use Lemma 3.3in order to show that if the seminorm of an average of functions is positive, then somerelated positiveness property holds for the individual functions (the s = 1 case was provedin Proposition 2.4). Proposition 4.3.
Let ( X, µ, T ) be an ergodic system, f n,k ∈ L ∞ ( µ ) , k, n ∈ N , be boundedby , and f ∈ L ∞ ( µ ) be defined by f := lim k →∞ E n ∈ [ N k ] f n,k , OINT ERGODICITY OF SEQUENCES 24 for some N k → ∞ , where the average is assumed to converge weakly. If ||| f ||| s +2 > forsome s ∈ Z + , then there exist a > , a subset Λ of N s with positive lower density, and χ n ∈ E ( T ) , n ∈ Λ , such that ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > a, n ∈ Λ , and lim inf N →∞ E n,n ′ ∈ [ N ] s lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z ∆ n − n ′ f n,k · χ n,n ′ · Λ ′ ( n, n ′ ) dµ (cid:17) > , where χ n,n ′ := T −| n | (cid:0) Y ǫ ∈{ , } s C | ǫ | χ n ǫ (cid:1) , n, n ′ ∈ N s , and (recall that n ǫ is defined in Section 3.1) Λ ′ := { ( n, n ′ ) ∈ N s : n ǫ ∈ Λ for all ǫ ∈ { , } s } . Remarks. • The key point is that from a lower bound for expressions involving ∆ n f weinfer lower bounds for expressions involving ∆ n f n,k . • For s = 0 the conclusion is that there exists χ ∈ E ( T ) such that ℜ (cid:0) R f · χ dµ (cid:1) > . • We plan to apply this proposition in the next step for the function ˜ f l given by (34)in place of f . Proof.
By Proposition 3.2 we have that there exist a > , a subset Λ of N s with positivelower density, and χ n ∈ E ( T ) , n ∈ N s , such that ℜ (cid:16) Z ∆ n f · χ n dµ (cid:17) > a, n ∈ Λ . Hence, lim inf N →∞ E n ∈ [ N ] s ℜ (cid:16) Z ∆ n f · χ n · Λ ( n ) dµ (cid:17) > . Since ∆ n f = Q ǫ ∈{ , } s C | ǫ | T ǫ · n f , n ∈ N s , and f = lim k →∞ E n ∈ [ N k ] f n,k (the limit is aweak limit), we deduce that lim inf N →∞ lim k →∞ E n ∈ [ N k ] ℜ (cid:16) E n ∈ [ N ] s Z Y ǫ ∈{ , } k \{ } C | ǫ | T ǫ · n f · T n + ··· + n s f n,k · χ n · Λ ( n ) dµ (cid:17) > . For fixed k, n ∈ N we apply Lemma 3.3 for f := f n,k , f ǫ := C | ǫ | f for ǫ ∈ { , } s \ , and g n := χ n · Λ ( n ) , n ∈ N s , and deduce that lim inf N →∞ lim sup k →∞ E n ∈ [ N k ] E n,n ′ ∈ [ N ] s Z ∆ n − n ′ f n,k · T −| n | (cid:16) Y ǫ ∈{ , } s (cid:0) C | ǫ | χ n ǫ · Λ ( n ǫ ) (cid:1)(cid:17) dµ > . Since the limsup of a sum is at most the sum of the limsups, the second asserted estimatefollows immediately from this one. This completes the proof. (cid:3)
Step 3. (Going from |||·||| s +2 to |||·||| ) Our next goal is to use Proposition 4.3 and thefact that the sequences a , . . . , a ℓ are good for equidistribution for the system ( X, µ, T ) in order to prove the following result: Proposition 4.4.
Let ( X, µ, T ) be an ergodic system and a , . . . , a ℓ : N → Z be good forequidistribution for the system ( X, µ, T ) . Suppose that property ( P l − ) of Proposition 4.1holds for some l ∈ [ ℓ ] and let ˜ f l be as in (34) , where all related functions are bounded by and f l +1 , . . . , f ℓ ∈ E ( T ) . Suppose that ||| ˜ f l ||| s +2 > for some s ∈ Z + . Then R ˜ f l dµ = 0 . OINT ERGODICITY OF SEQUENCES 25
Proof.
We can assume that the functions f , . . . , f ℓ are bounded by . We will show thatif ||| ˜ f l ||| s +2 > for some s ∈ Z + , then ||| ˜ f l ||| s +1 > . Applying this successively s + 1 times,we deduce that ||| ˜ f l ||| > , or equivalently, that R ˜ f l dµ = 0 .Since ||| ˜ f l ||| s +2 > , we can use Proposition 4.3 for ˜ f l in place of f and f n,k := T − a l ( n ) g k · Y j ∈ [ ℓ ] ,j = l T a j ( n ) − a l ( n ) f j , k, n ∈ N . We deduce that there exist a > , a subset Λ of N s with positive lower density, χ n ∈ E ( T ) , n ∈ N s , such that(36) ℜ (cid:16) Z ∆ n ˜ f l · χ n dµ (cid:17) > a, n ∈ Λ , and(37) lim inf N →∞ E n,n ′ ∈ [ N ] s lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z T − a l ( n ) (∆ n − n ′ g k ) · Y j ∈ [ ℓ ] ,j = l T a j ( n ) − a l ( n ) (∆ n − n ′ f j ) · χ n,n ′ · Λ ′ ( n, n ′ ) dµ (cid:17) > , where χ n,n ′ := T −| n | (cid:0) Y ǫ ∈{ , } s C | ǫ | χ n ǫ (cid:1) , n, n ′ ∈ N d , and Λ ′ := { ( n, n ′ ) ∈ N d : n ǫ ∈ Λ for all ǫ ∈ { , } s } . Our goal is to use (37) in order to show that the eigenfunctions χ n in (36) satisfy certainalgebraic relations that enable us to deduce, using Lemma 3.4, that ||| ˜ f l ||| s +1 > .We thus start by analyzing (37). After composing with T a l ( n ) we get lim inf N →∞ E n,n ′ ∈ [ N ] s lim sup k →∞ E n ∈ [ N k ] ℜ (cid:16) Z ∆ n − n ′ g k · Y j ∈ [ ℓ ] ,j = l T a j ( n ) (∆ n − n ′ f j ) · T a l ( n ) χ n,n ′ · Λ ′ ( n, n ′ ) dµ (cid:17) > . We let g j,n,n ′ := ∆ n − n ′ f j , j ∈ [ ℓ ] , j = l, n, n ′ ∈ N s , and g l,n,n ′ := χ n,n ′ , n, n ′ ∈ N s . Using the Cauchy-Schwarz inequality we deduce that lim inf N →∞ E n,n ′ ∈ [ N ] s Λ ′ ( n, n ′ ) lim sup k →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E n ∈ [ N k ] ℓ Y j =1 T a j ( n ) g j,n,n ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) > . Since Λ ′ ( n, n ′ ) = Y ǫ ∈{ , } s Λ ( n ǫ ) ≤ Λ ( n ) and the set Λ has positive lower density, we get(38) lim inf N →∞ E n ′ ∈ [ N ] s E n ∈ Λ ∩ [ N ] s lim sup k →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E n ∈ [ N k ] ℓ Y j =1 T a j ( n ) g j,n,n ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) > . Note that since f j ∈ E ( T ) for j = l + 1 , . . . , ℓ , we have g j,n,n ′ ∈ E ( T ) for j = l + 1 , . . . , ℓ , n, n ′ ∈ N s . Moreover, since χ n ∈ E ( T ) for all n ∈ N s we get that g l,n,n ′ ∈ E ( T ) for all n, n ′ ∈ N s . Since by assumption property ( P l − ) of Proposition 4.1 holds, we get that OINT ERGODICITY OF SEQUENCES 26 the previous expression remains unchanged if for j = 1 , . . . , l − we replace the functions g j,n,n ′ , n, n ′ ∈ N s , by constants (namely, their integrals). Next, note that(39) T a l ( n ) χ n = e ( a l ( n ) α n ) χ n , n ∈ N , n ∈ N s , for some α n ∈ Spec( T ) , n ∈ N s . Hence, we have that T a l ( n ) g l,n,n ′ = e ( a l ( n ) β n,n ′ ) g l,n,n ′ , where β n,n ′ := X ǫ ∈{ , } s ( − | ǫ | α n ǫ . (For instance, if s = 2 , we have β n,n ′ = β n ,n ,n ′ ,n ′ := α n ,n − α n ′ ,n − α n ,n ′ + α n ′ ,n ′ .)For j = l + 1 , . . . , ℓ , n, n ′ ∈ N s , the eigenvalues of the eigenfunctions g j,n,n ′ can beexpressed in the form e ( α j,n,n ′ ) , where α j,n,n ′ ∈ Spec( T ) . Then (recall the for j < l thefunctions g j,n,n ′ are assumed to be constant)(40) lim inf N →∞ E n ′ ∈ [ N ] s E n ∈ Λ ∩ [ N ] s lim sup k →∞ | E n ∈ [ N k ] e ( a l ( n ) β n,n ′ + ℓ X j = l +1 a j ( n ) α j,n,n ′ ) | > . From (40) we deduce that there exist integers n ′ N ∈ [ N ] s , and subsets Λ N of Λ ∩ [ N ] s with(41) lim inf N →∞ | Λ N | N s > , N ∈ N , and such that lim sup k →∞ | E n ∈ [ N k ] e ( a l ( n ) β n,n ′ N + ℓ X j = l +1 a j ( n ) α j,n,n ′ N ) | > , n ∈ Λ N . Since the sequences a , . . . , a ℓ are good for equidistribution for ( X, µ, T ) and { β n,n ′ N } , α j,n,n ′ N ∈ Spec( T ) , j = l + 1 , . . . , ℓ , n ∈ Λ N , N ∈ N , we deduce that β n,n ′ N = 0 (mod 1) , n ∈ Λ N , N ∈ N . Then if n ǫN := ( n ǫ , . . . , n ǫ s s ) , where n j := n j and n j := n ′ j,N for j = 1 , . . . , s , we get(42) α n = − X ǫ ∈{ , } s \ ( − | ǫ | α n ǫN (mod 1) , n ∈ Λ N , N ∈ N . (For s = 2 we get α n ,n = α n ′ ,N ,n + α n ,n ′ ,N − α n ′ ,N ,n ′ ,N (mod 1) .) The importantpoint is that we expressed α n as a sum of sequences that depend on s − variables,chosen from the variables n , . . . , n s .Since (36) holds for all n ∈ Λ N ⊂ Λ , N ∈ N , and (41) holds, we deduce that lim inf N →∞ E n ∈ Λ N ℜ (cid:16) Z ∆ n ˜ f l · χ n dµ (cid:17) > . Finally we are going to combine this with (42) in order to deduce that ||| ˜ f l ||| s +1 > .After composing with the transformation T n ′ s , averaging over n ′ s ∈ [ N ] , and using that T n ′ s χ n = e ( n ′ s α n ) · χ n , n ∈ N s , n ′ s ∈ N , When we deal with polynomial sequences a , . . . , a ℓ , we cannot replace the functions g j,n,n ′ byconstants but by their projection to the rational Kronecker factor, and this leads to serious complications.We bypass them in the proof of Theorem 1.4 by appealing to Theorem 5.1 below, which allows us toreduce matters to the case where the rational Kronecker factor is trivial. OINT ERGODICITY OF SEQUENCES 27 and then using the Cauchy-Schwarz inequality, we deduce that lim inf N →∞ E n ∈ Λ N (cid:13)(cid:13)(cid:13) E n ′ s ∈ [ N ] T n ′ s l (∆ n ˜ f l ) · e ( n ′ s α n ) (cid:13)(cid:13)(cid:13) L ( µ ) > . Using Lemma 2.5 for the average over n ′ s and composing with T − n ′ s on the integrals thatarise, we deduce that lim inf N →∞ ℜ (cid:16) E ( n,n s +1 ,n ′ s ) ∈ [ N ] s +2 c N ( n, n s +1 , n ′ s ) · Λ N ( n ) Z ∆ ( n,n s +1 ) ˜ f l dµ (cid:17) > , where(43) c N ( n, n s +1 , n ′ s ) := [ N ] ( n ′ s + n s +1 ) · e ( n s +1 α n ) , n ∈ [ N ] s , n s +1 , n ′ s ∈ [ N ] , N ∈ N . Hence, for some n ′ s,N ∈ [ N ] , N ∈ N , we have lim inf N →∞ ℜ (cid:16) E ( n,n s +1 ) ∈ [ N ] s +2 c N ( n, n s +1 , n ′ s,N ) · Λ N ( n ) Z ∆ ( n,n s +1 ) ˜ f l dµ (cid:17) > and using the Cauchy-Schwarz inequality we deduce that lim inf N →∞ (cid:13)(cid:13)(cid:13) E ( n,n s +1 ) ∈ [ N ] s +1 c N ( n, n s +1 , n ′ s,N ) · Λ N ( n ) · ∆ ( n,n s +1 ) ˜ f l (cid:13)(cid:13)(cid:13) L ( µ ) > . Using (42) and (43) we get that for all N ∈ N and n ∈ Λ N we have c N ( n, n s +1 ) = s Y j =1 b j,N ( n, n s +1 ) , n ∈ Λ N , n s +1 ∈ [ N ] , N ∈ N , where b j,N do not depend on the variable n j , for j = 1 , . . . , s , N ∈ N , and are bounded by . Since also Λ N ( n ) does not depend on the variable n s +1 , we deduce from Lemma 3.4that ||| ˜ f l ||| s +1 > . This completes the proof of Proposition 4.4. (cid:3)
Step 4. (Proof of Proposition 4.1) Let the ergodic system ( X, µ, T ) , the positiveinteger ℓ , and the sequences a , . . . , a ℓ be fixed. We will prove that property ( P l ) ofProposition 4.1 holds by induction on l ∈ { , . . . , ℓ } .For ℓ = 1 , as we mentioned in the introduction, it is a consequence of the spectraltheorem that the asserted properties ( P and P ) hold. So we can assume that ℓ ≥ .We first consider the case l = 0 . In this case our assumption is that T f j = e ( t j ) f j for some t j ∈ Spec( T ) , j = 1 , . . . , ℓ . Note first that if t j = 0 for j = 1 , . . . , ℓ , then byergodicity we have that T f j = R f j dµ , j = 1 , . . . , ℓ , in which case (29) is obvious. Hence,in order to show that (29) holds, it suffices to show that lim N →∞ E n ∈ [ N ] e ( a ( n ) t + · · · + a ℓ ( n ) t ℓ ) = 0 for t , . . . , t ℓ ∈ Spec( T ) , not all of them zero. This holds since the sequences a , . . . , a ℓ are assumed to be good for equidistribution for the system ( X, µ, T ) .For l ∈ [ ℓ ] we assume that property ( P l − ) of Proposition 4.1 holds and we are goingto show that property ( P l ) holds. Note that in order to prove that (29) holds it sufficesto assume that at least one of the functions f j , for j ∈ [ ℓ ] \ { l } , has zero integral. Indeed,if we write f j = ˜ f j + R f j dµ where ˜ f j = f j − R f j dµ , j ∈ [ ℓ ] \ { l } , and expand the producton the average into a sum of ℓ − terms, we get a sum of ℓ − − averages with functionsthat have the required property, plus the term Q j ∈ [ ℓ ] \{ l } R f j dµ · E n ∈ [ N ] T a l ( n ) f l , whichby the ℓ = 1 case we know that it converges in L ( µ ) to Q j ∈ [ ℓ ] R f j dµ .So under this additional assumption, our goal is to show that(44) lim N →∞ E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ = 0 OINT ERGODICITY OF SEQUENCES 28 where convergence takes place in L ( µ ) . Arguing by contradiction, suppose that E n ∈ [ N ] T a ( n ) f · . . . · T a l ( n ) f ℓ in L ( µ ) as N → ∞ . Using Propositions 4.2 we get that the same thing holds with thefunction ˜ f l , defined by (34), in place of f l . Recall that(45) ˜ f l = lim k →∞ E n ∈ [ N k ] T − a l ( n ) g k · Y j ∈ [ ℓ ] ,j = l T a j ( n ) − a l ( n ) f j , for some N k → ∞ and some g k ∈ L ∞ ( µ ) , k ∈ N , where all functions are bounded by ,and the limit is a weak limit.Since f l +1 , . . . , f ℓ ∈ E ( T ) , we deduce that there exists a sequence of complex numbers ( c n ) , of modulus one, such that E n ∈ [ N ] c n T a ( n ) f · . . . · T a l ( n ) ˜ f l in L ( µ ) as N → ∞ . Using the fact that the sequences a , . . . , a ℓ are good for semi-norm estimates for the system ( X, µ, T ) and the remark after the relevant definition inSection 1.3, we deduce that ||| ˜ f l ||| s > , for some s ∈ N .Since the assumptions of Proposition 4.4 are satisfied, we deduce that R ˜ f l dµ = 0 .Using (45), we deduce that lim k →∞ E n ∈ [ N k ] Z T − a l ( n ) g k · Y j ∈ [ ℓ ] ,j = l T a j ( n ) − a l ( n ) f j dµ = 0 . Composing with T a l ( n ) and using the Cauchy-Schwarz inequality we get that E n ∈ [ N k ] Y j ∈ [ ℓ ] ,j = l T a j ( n ) f j in L ( µ ) as k → ∞ . Since at least one of the functions f j , for j ∈ [ ℓ ] \{ l } , has zero integral,and f l +1 , . . . , f ℓ ∈ E ( T ) , using property ( P l − ) of Proposition 4.1 (with f l := 1 ∈ E ( T ) )we get that the last limit is zero, a contradiction. We conclude that property ( P l ) ofProposition 4.1 holds. This completes the induction and the proof of Proposition 4.1.4.7. Proof of Corollary 1.2.
Let I ( T ) be the closed subspace of T -invariant functions.For f ∈ L ( µ ) , we denote by E ( f |I ( T )) the orthogonal projection, in L ( µ ) , of f onto I ( T ) . Using a standard argument we deduce from Theorem 1.1 that the limit lim N →∞ E n ∈ [ N ] µ ( A ∩ T − a ( n ) A ∩ · · · ∩ T − a ℓ ( n ) A ) is equal to Z ( E ( A |I ( T ))) ℓ +1 dµ ≥ (cid:16) Z E ( A |I ( T )) dµ (cid:17) ℓ +1 = ( µ ( A )) ℓ +1 . Proof of Corollary 1.3.
It is clear that ( i ) implies ( ii ) .We show that ( ii ) implies ( i ) . Let ( X, µ, T ) be an ergodic k -step nilsystem for some k ∈ N . By [28, Chapter 12, Theorem 17] we have that if f ∈ L ∞ ( µ ) is such that ||| f ||| k +1 =0 , then f = 0 . Hence, any collection of sequences is good for seminorm estimates for ( X, µ, T ) . Since, by assumption the sequences a , . . . , a ℓ are good for equidistribution for ( X, µ, T ) , it follows from Theorem 1.1 that they are jointly ergodic for ( X, µ, T ) .5. Characteristic factors of general sequences
In this section we prove Theorem 1.4 and use it to deduce Corollary 1.5.
OINT ERGODICITY OF SEQUENCES 29
Proof of Theorem 1.4.
In this subsection we show how we can deduce Theo-rem 1.4 from Theorem 1.1. For this, we will need some deeper tools from ergodic theorythan those need in the proof of Theorem 1.1. In [27] it is shown that for every ergodicsystem ( X, X , µ, T ) and s ∈ N there exists a T -invariant sub- σ -algebra Z s of X withthe property that for f ∈ L ∞ ( µ ) we have E ( f |Z s ) = 0 if and only if ||| f ||| s +1 = 0 . Weneed some structural information about the system ( X, Z s , µ, T ) that is a corollary ofthe main result in [27]. We remark that this is the only part of the article that we makeuse of this structural theory. Definition.
We say that a system ( X, µ, T ) has finite rational spectrum if Spec( T ) ∩ Q is finite, or, equivalently, if there exists k ∈ N such that the ergodic components of thesystem ( X, µ, T k ) have trivial rational spectrum. Theorem 5.1 (Host-Kra [27]) . Let ( X, µ, T ) be an ergodic system such that X = Z s forsome s ∈ N . Then ( X, µ, T ) is an inverse limit of systems with finite rational spectrum. Remark.
The main result in [27] states much more, namely, ( X, µ, T ) is an inverse limitof nilsystems, but we will not need this.We are also going to use the following fact: Lemma 5.2.
Suppose that the sequences a , . . . , a ℓ : N → Z are good for irrationalequidistribution and seminorm estimates. For k ∈ N and i = 1 , . . . , ℓ let b i ( n ) := k − X r =0 k Z + r ( a i ( n )) a i ( n ) − rk , n ∈ N . Then the sequences b , . . . , b ℓ are good for irrational equidistribution and seminorm esti-mates.Proof. We first establish the equidistribution statement. Let t , . . . , t ℓ ∈ [0 , , not all ofthem rational. We want to show that lim N →∞ E n ∈ [ N ] e (cid:0) ℓ X i =1 b i ( n ) t i (cid:1) = 0 . By direct computation we see that it suffices to show that lim N →∞ E n ∈ [ N ] F ( a ( n ) , . . . , a ℓ ( n )) e (cid:0) ℓ X i =1 a i ( n ) t i k (cid:1) = 0 , where F : Z ℓk → C is defined by F ( t , . . . , t ℓ ) := e (cid:0) − ℓ X i =1 k X r =1 k Z + r ( x i ) rk t i (cid:1) , t , . . . , t ℓ ∈ Z k . Using the Fourier expansion of F on Z ℓk we see that it suffices to show that lim N →∞ E n ∈ [ N ] e (cid:0) ℓ X i =1 a i ( n ) t i + r i k (cid:1) = 0 for all r , . . . , r ℓ ∈ { , . . . , k − } . Since, by assumption, the sequences a , . . . , a ℓ are goodfor irrational equidistribution, and t i + r i k is irrational for some i ∈ { , . . . , ℓ } , the neededidentity follows.Next, we establish the statement about seminorm estimates. Let ( X, µ, T ) be a systemand k ∈ N . Our plan is to show that the sequences b , . . . , b ℓ are good for seminormestimates for the system ( X, µ, T ) by using that (by our assumption) the sequences OINT ERGODICITY OF SEQUENCES 30 a , . . . , a ℓ are good for seminorm estimates for the “ k -th root” of the system ( X, µ, T ) ,which is defined as follows: We consider the system ( X k , µ k , T k ) where X k := X × { , . . . , k − } , µ k = µ × ν k , ν k := δ + · · · + δ k − k , and for x ∈ X we let T k ( x, i ) = ( x, i + 1) , i = 0 , . . . , k − , T k ( x, k −
1) = (
T x, . The key property is that T kk ( x, i ) = ( T x, i ) , x ∈ X, i ∈ { , . . . , k − } , hence for f ∈ L ∞ ( µ ) we have(46) ( T kk ( f ⊗ x, i ) = ( T f )( x ) , x ∈ X, i ∈ { , . . . , k − } . Applying our assumption for the system ( X k , µ k , T k ) , and taking into account theremarks following the definition in Section 1.3 for the good seminorm property, we getthat there exists an s ∈ N (we can assume that s ≥ ) such that if g , . . . , g ℓ ∈ L ∞ ( µ k ) and ||| g l ||| s,T k = 0 for some l ∈ [ ℓ ] , then for every bounded sequence ( c n ) we have(47) lim N →∞ E n ∈ [ N ] c n T a ( n ) k g · . . . · T a l ( n ) k g l = 0 in L ( µ k ) .We claim that this s ∈ N produces good seminorm estimates for the sequences b , . . . , b ℓ for the system ( X, µ, T ) . To see this let f , . . . , f ℓ ∈ L ∞ ( µ ) be such that ||| f l ||| s,T = 0 forsome l ∈ [ ℓ ] . It suffices to show that lim N →∞ E n ∈ [ N ] T b ( n ) f · . . . · T b l ( n ) f l = 0 in L ( µ ) . Let g i ∈ L ∞ ( µ k ) be defined by g i := f i ⊗ , i = 1 , . . . , ℓ , and recall that µ k = µ × ν k . By (46) it suffices to show that lim N →∞ E n ∈ [ N ] T kb ( n ) k g · . . . · T ka l ( n ) k g l = 0 in L ( µ k ) , or equivalently, that lim N →∞ E n ∈ [ N ] T a ( n ) k ( T − r ( n ) k g ) · . . . · T a l ( n ) k ( T − r l ( n ) k g l ) = 0 in L ( µ k ) , for some sequences r , . . . , r l : N → { , . . . , k − } . Note that the last averagecan be written as a sum of k l weighted averages of the form(48) E n ∈ [ N ] c n T a ( n ) k ( T − r k g ) · . . . · T a l ( n ) k ( T − r l k g l ) for some r , . . . , r l ∈ { , . . . , k − } and c n ∈ { , } , n ∈ N . Hence, it suffices to showthat averages of the form (48) converge to in L ( µ k ) and N → ∞ .Since g l = f l ⊗ and µ k = µ × ν k , our assumption ||| f l ||| s,T = 0 and (46) give that ||| g l ||| s,T kk = ||| f l ||| s,T = 0 . Since s ≥ , by [34, Theorem 2] we have that ||| g l ||| s,T kk = 0 implies that ||| g l ||| s,T k = 0 , hence we also have ||| T − r l k g l ||| s,T k = ||| g l ||| s,T k = 0 . It then followsfrom (47) that the averages (48) converge to in L ( µ k ) as N → ∞ . This completes theproof. (cid:3) Proof of Theorem 1.4.
It is straightforward to verify that Property ( i ) implies Prop-erty ( ii ) (the seminorm property holds with s = 2 and one can use appropriate rota-tions on T ℓ to verify the equidistribution property). So we only prove that Property ( ii ) implies Property ( i ) .Let µ = R µ x dµ be the ergodic decomposition of the measure µ . Since E ( f |K rat ( µ )) =0 implies that E ( f |K rat ( µ x )) = 0 for µ x almost every x ∈ X (see for example [22,Theorem 3.2]), we can assume that the system ( X, µ, T ) is ergodic. OINT ERGODICITY OF SEQUENCES 31
We claim that it suffices to show that the Kronecker factor K ( T ) is characteristic formean convergence of the averages(49) E n ∈ [ N ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ , meaning that the previous averages converge to in L ( µ ) as N → ∞ if at least one ofthe functions is orthogonal to K ( T ) (recall that K ( T ) is L ( µ ) -closure of the linear spanof the eigenfunctions of the system). Indeed, if this is the case, then by approximation,we can assume that all functions are eigenfunctions; hence, for i = 1 , . . . , ℓ we have T a i ( n ) f i = e ( a i ( n ) t i ) f i , n ∈ N , for some t i ∈ [0 , , not all of them rational (since at least one of the functions isorthogonal to K rat ( T ) ). In this case, the needed convergence to zero follows from ourassumption that the sequences a , . . . , a ℓ are good for irrational equidistribution.So it remains to show that if E ( f i |K ( T )) = 0 , for some i ∈ { , . . . , ℓ } , then the averages(49) converge to in L ( µ ) . Without loss of generality we can assume that i = 1 , hence E ( f |K ( T )) = 0 . Since the sequences a , . . . , a ℓ are very good for seminorm estimatesand E ( f |Z s ( T )) = 0 implies ||| f ||| s +1 ,T = 0 , we can assume that all functions f , . . . , f ℓ are Z s ( T ) -measurable for some s ∈ N . In this case we can assume that X = Z s ( T ) , hence using Theorem 5.1 and an approximation argument we can assume that the system ( X, µ, T ) has finite rational spectrum, in which case there exists k ∈ N such the system ( X, µ, T k ) has trivial rational spectrum.For i = 1 , . . . , ℓ we let b i ( n ) := k − X r =0 k Z + r ( a i ( n )) a i ( n ) − rk , n ∈ Z . By Lemma 5.2, the sequences b , . . . , b ℓ are good for irrational equidistribution and semi-norm estimates. Then for i = 1 , . . . , ℓ and S := T k we have that T a i ( n ) f i = k − X r =0 k Z + r ( a i ( n )) ( T k ) ai ( n ) − rk ( T r f i ) = k − X r =0 k Z + r ( a i ( n )) S b i ( n ) ( T r f i ) , n ∈ N . We insert this identity in (49) and expand the product. We deduce that it suffices toshow the following: If the (not necessarily ergodic) system ( X, µ, S ) has trivial rationalspectrum, the sequences b , . . . , b ℓ : N → Z are good for irrational equidistibution andseminorm estimates, and E ( f |K ( T )) = 0 , then for every bounded sequence ( c n ) we have lim N →∞ E n ∈ [ N ] c n · S b ( n ) f · . . . · S b ℓ ( n ) f ℓ = 0 in L ( µ ) .To prove this, we first remark that since the system ( X, µ, S ) has trivial rationalspectrum, the same holds for the system ( X × X, µ × µ, S × S ) . Furthermore, it is knownthat E ( f |K ( S )) = 0 implies that E ( f ⊗ f |I ( S × S )) = 0 . Combining these facts (whoseproof follows for example from [24, Lemma 4.18]) we deduce that if µ × µ = Z ( µ × µ ) ( x,y ) d ( µ × µ ) is the ergodic decomposition of the measure µ × µ with respect to the transformation S × S , then for ( µ × µ ) -almost every ( x, y ) ∈ X × X the system ( X × X, ( µ × µ ) ( x,y ) , S × S ) is ergodic, has trivial rational spectrum, and R f ⊗ f d ( µ × µ ) ( x,y ) = 0 . By Theorem 1.1we deduce that for ( µ × µ ) -almost every ( x, y ) ∈ X × X we have E n ∈ [ N ] ( S × S ) b ( n ) ( f ⊗ f ) · . . . · ( S × S ) b ℓ ( n ) ( f ℓ ⊗ f ℓ ) → L (( µ × µ ) ( x,y ) ) The assumption that the sequences a , . . . , a ℓ are very good for seminorm estimates (versus simply“good”) is needed in order to get this reduction. OINT ERGODICITY OF SEQUENCES 32 as N → ∞ . This implies that E n ∈ [ N ] ( S × S ) b ( n ) ( f ⊗ f ) · . . . · ( S × S ) b ℓ ( n ) ( f ℓ ⊗ f ℓ ) → L ( µ × µ ) as N → ∞ , and using the Cauchy-Schwarz inequality we deduce that lim N →∞ E n ∈ [ N ] Z ( f ,N ⊗ f ,N ) · ( S × S ) b ( n ) ( f ⊗ f ) · . . . · ( S × S ) b ℓ ( n ) ( f ℓ ⊗ f ℓ ) d ( µ × µ ) = 0 for all functions f ,N ∈ L ∞ ( µ ) , N ∈ N , that are uniformly bounded. Hence, lim N →∞ E n ∈ [ N ] (cid:12)(cid:12)(cid:12) Z f ,N · S b ( n ) f · . . . · S b ℓ ( n ) f ℓ dµ (cid:12)(cid:12)(cid:12) = 0 , which implies using the Cauchy-Schwarz inequality that lim N →∞ E n ∈ [ N ] c n Z f ,N · S b ( n ) f · . . . · S b ℓ ( n ) f ℓ dµ = 0 for every bounded sequence ( c n ) . If we let f ,N := E n ∈ [ N ] c n · S b ( n ) f · . . . · S b ℓ ( n ) f ℓ , N ∈ N , we deduce that lim N →∞ E n ∈ [ N ] c n · S b ( n ) f · . . . · S b ℓ ( n ) f ℓ = 0 in L ( µ ) . This completes the proof. (cid:3) Proof of Corollary 1.5.
Let A ∈ X and ε > . Let K r denote the closed subspaceof L ( µ ) consisting of all T r -invariant functions. Then there exists r ∈ N such that(50) k E ( A |K rat ) − E ( A |K r ) k L ( µ ) ≤ εℓ . Let S r = { n ∈ N : a ( n ) ≡ , . . . , a ℓ ( n ) ≡ r ) } . By assumption, we have ¯ d ( S r ) > , hence there exist N k → ∞ such that(51) lim k →∞ | S r ∩ [ N k ] | N k > . First, we claim that K rat ( T ) is a characteristic factor for the averages(52) E n ∈ S r ∩ [ N k ] T a ( n ) f · . . . · T a ℓ ( n ) f ℓ , meaning, if E ( f j |K rat ( T )) = 0 for some j ∈ { , . . . , ℓ } , then the averages converge tozero in L ( µ ) as k → ∞ . To see this, note that for every k ∈ N the averages (52) areequal to(53) N k | S r ∩ [ N k ] | · E n ∈ [ N k ] ( T × R ) a ( n ) ( f ⊗ g ) · . . . · T a ℓ ( n ) ( f ℓ ⊗ g ) , where R is the shift transformation on a cyclic group of order r and g is the indicatorfunction of the identity element in this cyclic group. Then E ( f j ⊗ g |K rat ( T × R )) = 0 ,and since by assumption the sequences a , . . . , a ℓ are very good for seminorm estimatesand good for irrational equidistribution, we get by Theorem 1.4 that the averages in (52)converge to in L ( µ ) as k → ∞ . We deduce from this and (51) that the averages (53)converge to in L ( µ ) as k → ∞ , completing the proof of our claim.It follows from what we just proved that the limit lim inf k →∞ E n ∈ S r ∩ [ N k ] µ ( A ∩ T − a ( n ) A ∩ · · · ∩ T − a ℓ ( n ) A ) is equal to the limit lim inf k →∞ E n ∈ S r ∩ [ N k ] Z E ( A |K rat ( T )) · T a ( n ) E ( A |K rat ( T )) · . . . · T a ℓ ( n ) E ( A |K rat ( T )) dµ. OINT ERGODICITY OF SEQUENCES 33
Using (50) and telescoping, we get that the last limit is greater or equal than lim inf k →∞ E n ∈ S r ∩ [ N k ] Z E ( A |K r ( T )) · T a ( n ) E ( A |K r ( T )) · . . . · T a ℓ ( n ) E ( A |K r ( T )) dµ − ε. Note that for n ∈ S r we have T a j ( n ) E ( A |K r ( T )) = E ( A |K r ( T )) , j = 1 , . . . , ℓ , hencethe last limit is equal to Z ( E ( A |K r ( T ))) ℓ +1 dµ ≥ (cid:16) Z E ( A |K r ( T )) dµ (cid:17) ℓ +1 = ( µ ( A )) ℓ +1 . Combining the above estimates we get that lim inf k →∞ E n ∈ S r ∩ [ N k ] µ ( A ∩ T − a ( n ) A ∩ · · · ∩ T − a ℓ ( n ) A ) ≥ ( µ ( A )) ℓ +1 − ε, completing the proof of Corollary 1.5.6. Joint ergodicity of special sequences
In this section we prove the results of Sections 1.5 and 1.6.6.1.
Definition of Hardy fields.
Let B be the collection of equivalence classes of realvalued functions defined on some half line [ c, + ∞ ) , where we identify two functions ifthey agree eventually. A Hardy field H is a subfield of the ring ( B, + , · ) that is closedunder differentiation. For the purposes of this section we assume that all Hardy fields H considered are contained in some other Hardy field H ′ that satisfies property (8). Aparticular example of such a Hardy field H is the collection of logarithmic-exponentialfunctions , meaning all functions defined on some half line [ c, + ∞ ) by a finite combinationof the symbols + , − , × , : , log , exp , operating on the real variable t and on real constants;linear combinations of functions of the form t a (log t ) b c t , a, b ∈ R , c > , are examplesof such functions. The reader can find more information about them in [11] and thereferences therein.6.2. Good equidistribution properties for Hardy field sequences.
We will usethe following equidistribution result:
Theorem 6.1 (Boshernitzan [11]) . Let a : [ c, + ∞ ) → R be a Hardy field function withat most polynomial growth. Then the sequence ( a ( n )) is equidistributed on T if and onlyif it stays logarithmically away from rational polynomials (see definition in Section 1.5). We will also use the following reduction, variants of which have been frequently usedin the literature. We give its proof for completeness.
Lemma 6.2.
Let a , . . . , a ℓ : [ c, + ∞ ) → R be such that for every t , . . . , t ℓ ∈ R , not allof them zero, we have (54) lim N →∞ E n ∈ [ N ] e ( a ( n ) t + · · · + a ℓ ( n ) t ℓ ) = 0 . Then the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are good for equidistribution. Remark.
If we assume that (54) holds for all t , . . . , t ℓ ∈ R , not all of them rational, thena similar argument gives that the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are good for irrationalequidistribution. The equivalence classes just defined are often called “germs of functions”. We choose to use theword function when we refer to elements of B instead, with the understanding that all the operationsdefined and statements made for elements of B are considered only for sufficiently large values of t ∈ R . OINT ERGODICITY OF SEQUENCES 34
Proof.
We first remark that it suffices to show the following: If F : T → C and G : T ℓ → R are Riemann-integrable, then for all t , . . . , t ℓ ∈ [0 , , not all of them , we have(55) lim N →∞ E n ∈ [ N ] F ( a ( n ) t + · · · + a ℓ ( n ) t ℓ ) G ( a ( n ) , . . . , a ℓ ( n )) = Z F dm T · Z G dm T ℓ . Indeed, if this is the case, then using (55) for the continuous function F ( x ) := e ( x ) , x ∈ T , and the Riemann integrable function G ( x , . . . , x ℓ ) := e ( −{ x } t − · · · − { x ℓ } t ℓ ) , x , . . . , x ℓ ∈ T , we get that lim N →∞ E n ∈ [ N ] e ([ a ( n )] t + · · · + [ a ℓ ( n )] t ℓ ) = 0 for all t , . . . , t ℓ ∈ [0 , , not all of them .We move now to the proof of (55). After approximating from above and below bycontinuous functions, we can assume that F ∈ C ( T ) and G ∈ C ( T ℓ ) . After a furtherapproximation by trigonometric polynomials we can assume that F ( x ) = e ( kx ) , x ∈ T ,and G ( x , . . . , x ℓ ) = e ( k x + · · · + k ℓ x ℓ ) , x , . . . , x ℓ ∈ T , for some k, k , . . . , k ℓ ∈ Z . If k = k = · · · = k ℓ = 0 , then the identity is obvious. Hence, it suffices to show that forall k, k , . . . , k ℓ ∈ Z , not all of them zero, and t , . . . , t ℓ ∈ [0 , , not all of them zero, wehave lim N →∞ E n ∈ [ N ] e (( k + kt ) a ( n ) + · · · + ( k ℓ + kt ℓ ) a ℓ ( n )) = 0 . Since k + kt , . . . , k ℓ + kt ℓ are not all of them zero, this follows from our assumptions,completing the proof. (cid:3) Combining the previous two results we get the following:
Proposition 6.3.
Let a , . . . , a ℓ : [ c, + ∞ ) → R be functions of at most polynomial growthfrom a Hardy field such that every non-trivial linear combination of these functions stayslogarithmically away from rational polynomials. Then the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are good for equidistribution. In a similar fashion, using the variant recorded on the remark following Lemma 6.2,we get the following:
Proposition 6.4.
Let a , . . . , a ℓ : [ c, + ∞ ) → R be functions of at most polynomial growthfrom a Hardy field such that every non-trivial linear combination of these functions, withat least one irrational coefficient, stays logarithmically away from rational polynomials.Then the sequences [ a ( n )] , . . . , [ a ℓ ( n )] are good for irrational equidistribution. Good seminorm estimates for Hardy field sequences.
We will use the fol-lowing known result. Proposition 6.5.
Let a , . . . , a ℓ : [ c, + ∞ ) → R be functions from a Hardy field such thatthe functions and their pairwise differences are non-constant functions in T + P . Thenthe sequences [ a ( n )] , . . . , [ a ℓ ( n )] are very good for seminorm estimates. If T is replaced with the class of functions a : R + → R that satisfy the slightly morerestrictive growth condition t k + ε ≺ a ( t ) ≺ t k +1 for some k ∈ Z + , then the argument usedto prove [18, Theorem 2.9] can be applied without any change to prove Proposition 6.5.For the more extended class of functions used above one can employ the argument usedto prove [10, Theorem 4.2] without essential changes. We omit the details.6.4. Proof of Theorem 1.6.
By Proposition 6.5 the sequences [ a ( n )] , . . . , [ a ℓ ( n )] aregood for seminorm estimates, and by Proposition 6.3 they are also good for equidistri-bution. Hence by Theorem 1.1 they are jointly ergodic. As far as we know this result is proved in detail only under the assumption (8), which is the reasonwhy we impose this assumption on all Hardy fields considered in this section.
OINT ERGODICITY OF SEQUENCES 35
Proof of Theorem 1.7.
By Proposition 6.5 the sequences [ a ( n )] , . . . , [ a ℓ ( n )] arevery good for seminorm estimates and by Proposition 6.4 they are good for irrationalequidistribution. Hence, by Theorem 1.1 they are jointly ergodic for totally ergodicsystems and by Theorem 1.4 the rational Kronecker factor is characteristic for thesesequences.6.6. Proof of Theorem 1.8.
By Corollary 1.3 it suffices to show that these collectionsof sequences are good for equidistribution. By Lemma 6.2 it suffices to show that, afterremoving the integer parts, every non-trivial linear combination of the given collectionsof sequences is equidistributed on the circle.For the sequences in Part ( i ) this follows by combining [13, Lemma 9] (which is themain result in [30]) with [13, Lemma 16].For the sequences in Parts ( ii ) and ( iii ) this follows from [3, Theorem 2.2] and [4,Theorem 4.1]. 7. Joint ergodicity for flows
In this section we prove Theorem 1.9, which we repeat for convenience.
Theorem 7.1.
Let a , . . . , a ℓ : [ c, + ∞ ) → R + be functions from a Hardy field. Supposethat there exists δ > such that t δ ≺ a ( t ) and ( a j +1 ( t )) δ ≺ a j ( t ) ≺ ( a j +1 ( t )) − δ for j = 1 , . . . , ℓ − . Then for all measure preserving actions T t , . . . , T tℓ , t ∈ R , on aprobability space ( X, X , µ ) and f , . . . , f ℓ ∈ L ∞ ( µ ) , we have (56) lim y → + ∞ y Z y f ( T a ( t )1 x ) · . . . · f ℓ ( T a ℓ ( t ) ℓ x ) dt = ˜ f · · · ˜ f ℓ pointwise for µ -almost every x ∈ X , where for j = 1 , . . . , ℓ we denote by ˜ f j the orthogonalprojection of f j on the space of functions that are T tj -invariant for every t ∈ R . We start with the following crucial change of variables property (a variant of thisproperty also appears in [1]):
Lemma 7.2.
Let a : R + → R + be a function from a Hardy field such that t δ ≺ a ( t ) forsome δ > . Let f ∈ L ∞ ( m R ) and suppose that the following limit exists lim y → + ∞ y Z y f ( t ) dt. Then also the following limit exists lim y → + ∞ y Z y f ( a ( t )) dt and the two limits are equal. Remark.
More generally, our argument works if there exist δ, M > such that thefunctions a, a − are three times differentiable, their derivatives are non-zero on [ M, + ∞ ) ,and also a ( t ) ≥ t δ and a ( t ) ta ′ ( t ) is bounded for t ≥ M . Proof.
We can assume that the first limit is . Let F ( t ) = 1 t Z t f ( s ) ds, t > . Let ε > . Using our assumption we have that there exists M > such that | F ( t ) | ≤ ε for t > M and also a, a − ∈ C ([ M, + ∞ )) . Since f is bounded, it suffices to show that lim y → + ∞ y R ya − ( M ) f ( a ( t )) dt = 0 . We assume that y is large enough so that a ( y ) ≥ M .Using the change of variables s = a ( t ) we get y Z ya − ( M ) f ( a ( t )) dt = 1 y Z a ( y ) M f ( t ) · ( a − ) ′ ( t ) dt. OINT ERGODICITY OF SEQUENCES 36
Since f ( t ) = ( tF ( t )) ′ for Lebesgue almost every t ∈ R , we have y Z a ( y ) M f ( t ) · ( a − ) ′ ( t ) dt = 1 y Z a ( y ) M ( tF ( t )) ′ · ( a − ) ′ ( t ) dt. Integration by parts ( tF ( t ) and ( a − ) ′ ( t ) are absolutely continuous on [ M, a ( y )] ) givesthat the last integral is equal to(57) a ( y ) F ( a ( y ))( a − ) ′ ( a ( y )) y − Cy − y Z a ( y ) M tF ( t ) · ( a − ) ′′ ( t ) dt for some C ∈ R + .The first term in (57) is equal to a ( y ) F ( a ( y ))( a − ) ′ ( a ( y )) y = R a ( y )0 f ( t ) dtya ′ ( y ) = R a ( y )0 f ( t ) dta ( y ) · a ( y ) ya ′ ( y ) . Notice that since a ( t ) is a Hardy field function with a ( t ) ≻ t δ , we have that lim y → + ∞ a ( y ) ya ′ ( y ) = lim y → + ∞ log y log a ( y ) < δ . Moreover, by assumption we have lim y → + ∞ y R y f ( t ) dt = 0 and lim y → + ∞ a ( y ) = + ∞ .We deduce that the first term in (57) converges to as y → + ∞ .It remains to show that the limsup as y → + ∞ of last term in (57) is bounded by aconstant multiple of ε . Since | F ( t ) | ≤ ε for t > M , it suffices to show that the limsup as y → + ∞ of the expression y Z a ( y ) M t · | ( a − ) ′′ ( t ) | dt is bounded by a quantity that is independent of ε . Since ( a − ) ′′ has eventually constantsign, say in [ M , + ∞ ) , we can replace M with M and remove the absolute value. Afterdoing so, integration by parts leads to the expression y (cid:16) a ( y ) · ( a − ) ′ ( a ( y )) − C − Z a ( y ) M ( a − ) ′ ( t ) dt (cid:17) for some C ∈ R + . The last expression is equal to a ( y ) ya ′ ( y ) − C y − y − a − ( M ) y . As we showed before, the limit of the first term is bounded and the limit of the othertwo terms as y → + ∞ is . This completes the proof. (cid:3) We review some basic facts from the spectral theory of unitary R -actions that we willuse. Proofs of the stated facts can be found for example in [16] (we use Theorem 9.58 andthe variant of Theorems 9.17 that applies to flows). If ( X, µ, T t ) , t ∈ R + , is a measurepreserving flow, then for every f ∈ L ( µ ) there exists a positive and bounded measure σ f on R , that is called the spectral measure of f , such that(58) Z T t f · f dµ = Z e ( ts ) dσ f ( s ) , t ∈ R . More generally, for every f, g ∈ L ( µ ) there exists a complex measure σ f,g , with boundedvariation, such that Z T t f · g dµ = Z e ( ts ) dσ f,g ( s ) , t ∈ R . OINT ERGODICITY OF SEQUENCES 37
Furthermore, if h ∈ L ∞ ( R ) , then there exists a bounded operator h ( T ) : L ( µ ) → L ( µ ) that commutes with T t , t ∈ R , and satisfies Z h ( T ) f · g dµ = Z h dσ f,g for all f, g ∈ L ( µ ) . We then have(59) ( h + h )( T ) = h ( T ) + h ( T ) , h , h ∈ L ∞ ( µ ) , and(60) dσ h ( T ) f = h dσ f , k h ( T ) f k L ( µ ) = k h k L ( σ f ) . We will use the following fact:
Lemma 7.3.
Let ( X, µ, T t ) , t ∈ R + , be a measure preserving flow. Then the set G := { f ∈ L ( µ ) : σ f has compact support } is dense in L ( µ ) .Proof. Let f ∈ L ( µ ) . Using the previous notation we have by (60) that for n ∈ N thespectral measure of the function f n = [ − n,n ] ( T ) f is supported on the interval [ − n, n ] ,hence f n ∈ G . Moreover, since R ( T ) f = f , using (59) and (60) we get k f − f n k L ( µ ) = (cid:13)(cid:13) R \ [ − n,n ] ( T ) f (cid:13)(cid:13) L ( µ ) = (cid:13)(cid:13) R \ [ − n,n ] (cid:13)(cid:13) L ( σ f ) → as n → ∞ , since σ f is a bounded measure. (cid:3) Lemma 7.4.
Let b : R + → R + be a function from a Hardy field that satisfies t δ ≺ b ( t ) ≺ t − δ for some δ > , and ( X, µ, T t ) , t ∈ R + , be a measure preserving flow. Then forevery f ∈ L ∞ ( µ ) and c ∈ R we have (61) lim y → + ∞ y Z y | f ( T b ( t + c ) x ) − f ( T b ( t ) x ) | dt = 0 for µ -almost every x ∈ X . Remark.
More generally, our argument works if the function b satisfies the propertiesmentioned in the remark following Lemma 7.2 and also b ( t ) ≺ t − δ and b ′ monotonicallydecreases to as t → + ∞ . Proof.
We can assume that c ≥ . Our assumptions imply that there exists t > such that for t ≥ t we have ≤ b ′ ( t ) ≤ , b ′ ( t ) is decreasing, and b ( t ) ≤ t − δ . Since f ∈ L ∞ ( µ ) it suffices to show that(62) lim y → + ∞ y Z yt | f ( T b ( t + c ) x ) − f ( T b ( t ) x ) | dt = 0 for µ -almost every x ∈ X .Let ε > . If G is the dense subset of L ( µ ) given by Lemma 7.3, we have a decompo-sition f = f + f where f ∈ G and k f k L ( µ ) ≤ ε .For g ∈ L ( µ ) let A y ( g ) := 1 y Z yt | g ( T b ( t + c ) x ) − g ( T b ( t ) x ) | dt, y ≥ t . We first deal with the contribution of f . We clearly have lim sup y → + ∞ A y ( f ) ≤ y → + ∞ y Z yt | f ( T b ( t ) x ) | dt. OINT ERGODICITY OF SEQUENCES 38
Hence, using Lemma 7.2 and the pointwise ergodic theorem for flows we deduce that(63) Z lim sup y → + ∞ A y ( f ) dµ ≤ Z (cid:16) lim y → + ∞ y Z yt | f ( T t x ) | dt (cid:17) dµ = 4 Z | f | dµ ≤ ε . Next, we deal with the contribution of f . Since f ∈ G , there exists M > such thatthe spectral measure σ f of f is supported on the set [ − M, M ] . Using the Fubini-Tonellitheorem we get k A y ( f ) k L ( µ ) = 1 y Z yt Z X | f ( T b ( t + c ) x ) − f ( T b ( t ) x ) | dµ dt. Using (58) and the Fubini-Tonelli theorem again, we get that the last expression is equalto Z M − M y Z yt | e ( sb ( t + c )) − e ( sb ( t )) | dt dσ f ( s ) . After bounding the integrant pointwise, using the mean value theorem, the fact that b ′ is non-negative, and decreasing for t ≥ t , and that c ≥ , we get that the last expressionis bounded by a constant multiple of Z M − M y Z yt (cid:0) scb ′ ( t )) dt dσ f ( s ) ≤ M c k f k L ( µ ) y Z yt ( b ′ ( t )) dt. Since ≤ b ′ ( t ) ≤ for t ≥ t we have for y ≥ t that Z yt (cid:0) b ′ ( t )) dt ≤ Z yt b ′ ( t ) dt = b ( y ) − b ( t ) . Since b ( y ) ≤ y − δ for y > t , combining the above we get for C := M c ||| f ||| L ( µ ) that k A y ( f )( x ) k L ( µ ) ≤ Cy δ for all y > t . Using the Borel-Cantelli lemma, we get that for a > /δ we have lim N →∞ A N a ( f )( x ) = 0 for µ -almost every x ∈ X . From this we deduce that(64) lim y → + ∞ A y ( f )( x ) = 0 for µ -almost every x ∈ X . Indeed, if N y ∈ Z + is such that N ay ≤ y < ( N y + 1) a , then (cid:13)(cid:13)(cid:13) A y ( f ) − A N ay ( f ) (cid:13)(cid:13)(cid:13) L ∞ ( µ ) ≤ k f k L ∞ ( µ ) | − N ay /y | → as y → + ∞ .Combining (63) and (64) we get that Z lim sup y → + ∞ A y ( f ) dµ ≤ ε . Since ε is arbitrary, we get that (62) holds for µ -almost every x ∈ X , completing theproof. (cid:3) Proof of Theorem 1.9.
We prove the statement by induction on ℓ . For ℓ = 1 the resultfollows from Lemma 7.2 and the pointwise ergodic theorem for flows. Suppose that thestatement holds for ℓ − , we shall show that it holds for ℓ .Without loss of generality we can assume that k f j k L ∞ ( µ ) ≤ for j = 1 , . . . , ℓ . We let A y ( f ℓ )( x ) := 1 y Z y f ( T a ( t )1 x ) · . . . · f ℓ ( T a ℓ ( t ) ℓ x ) dt, y ∈ R + . Let ε > . Using a standard Hilbert space argument we get a decomposition f ℓ = f ℓ, + f ℓ, + f ℓ, , where f ℓ, = ˜ f ℓ , k f ℓ, k L ( µ ) ≤ ε , and f ℓ, belongs to the linear subspace spanned by thefunctions T cℓ h − h , c ∈ R , h ∈ L ( µ ) . After approximating h in L ( µ ) by functions in OINT ERGODICITY OF SEQUENCES 39 L ∞ ( µ ) and incorporating the error in f ℓ, , we can assume that h ∈ L ∞ ( µ ) . Furthermore,we have k f ℓ, k L ∞ ( µ ) ≤ k f ℓ k L ∞ ( µ ) ≤ and k f ℓ, k L ( µ ) ≤ ε . An application of theCauchy-Schwarz inequality and the Fubini-Tonelli theorem shows that for µ -almost every x ∈ X the quantities A y ( f ℓ,j ) , j = 1 , , , are well defined finite numbers.Our goal is to show (72) below. We first deal with the contribution of the term f ℓ, .Since T tℓ f ℓ, = f ℓ, = ˜ f ℓ for every t ∈ R + , the induction hypothesis gives(65) lim y → + ∞ A y ( f ℓ, ) = ˜ f · · · ˜ f ℓ . for µ -almost every x ∈ X .Next we deal with the contribution of the term f ℓ, . For every y > , using theCauchy-Schwarz inequality we have | A y ( f ℓ, )( x ) | ≤ y Z y | f ℓ, ( T a ℓ ( t ) ℓ x ) | dt. Using Lemma 7.2 and the pointwise ergodic theorem for flows we get(66) Z lim sup y → + ∞ | A y ( f ℓ, ) | dµ = k f ℓ, k L ( µ ) ≤ ε. It remains to deal with the contribution of the term f ℓ, . We claim that(67) lim y → + ∞ | A y ( f ℓ, )( x ) | = 0 for µ -almost every x ∈ X . By Lemma 7.2, it suffices to show that for µ -almost every x ∈ X we have(68) lim y → + ∞ y Z y ℓ − Y j =1 f j ( T b j ( t ) j x ) · f ℓ, ( T tℓ x ) dt = 0 where b j := a j ◦ a − ℓ for j = 1 , . . . , ℓ − .In order to establish (68) it suffices to verify that if f ℓ, := T cℓ h − h, for some c ∈ R and h ∈ L ∞ ( µ ) , then (68) holds for µ -almost every x ∈ X . So let c > and h ∈ L ∞ ( µ ) . After inserting f ℓ, = T cℓ h − h in (68) and using the change of variables t t − c in the first of the two integrals, we get that it suffices to show that(69) lim y → + ∞ y Z y (cid:16) ℓ − Y j =1 f j ( T b j ( t − c ) j x ) − ℓ − Y j =1 f j ( T b j ( t ) j x ) (cid:17) · h ( T tℓ x ) dt = 0 for µ -almost every x ∈ X . It suffices to show that for d = 1 , . . . , ℓ − we have(70) lim y → + ∞ y Z y (cid:16) d Y j =1 f j ( T b j ( t − c ) j x ) ℓ − Y j = d +1 f j ( T b j ( t ) j x ) − d − Y j =1 f j ( T b j ( t − c ) j x ) ℓ − Y j = d f j ( T b j ( t ) j x ) (cid:17) · h ( T tℓ x ) dt = 0 We remark that since not all Hardy fields are closed under composition and compositional inversion,we cannot assume that the functions b j belong to some Hardy field. Nevertheless, one can easily verifythat these functions satisfy the necessary assumptions mentioned on the remark following Lemma 7.2,so we are entitled to apply this lemma. OINT ERGODICITY OF SEQUENCES 40 for µ -almost every x ∈ X . Finally, notice that our growth assumptions give that t δ ′ ≺ b d ( t ) ≺ t − δ ′ for some δ ′ > . Hence, by Lemma 7.4, for d = 1 , . . . , ℓ − , we have(71) lim y → + ∞ y Z y | f d ( T b d ( t − c ) d x ) − f d ( T b d ( t ) d x ) | dt = 0 for µ -almost every x ∈ X . Hence, using the Cauchy-Schwarz inequality, equation (71),and the fact that all the functions f j and h are bounded, we get that (70) holds for µ -almost every x ∈ X . Combning the above we deduce that (67) holds.From (65), (66), (67), we deduce that(72) Z lim sup y → + ∞ | A y ( f ℓ ) − ˜ f · · · ˜ f ℓ | dµ ≤ ε. Since ε was arbitrary, we deduce that lim y → + ∞ A y ( f ℓ )( x ) = ˜ f · · · ˜ f ℓ for µ -almost every x ∈ X , completing the proof. (cid:3) References [1] T. Austin. Norm convergence of continuous-time polynomial multiple ergodic averages.
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