Equidistribution of sparse sequences on nilmanifolds
aa r X i v : . [ m a t h . D S ] F e b EQUIDISTRIBUTION OF SPARSE SEQUENCES ON NILMANIFOLDS
NIKOS FRANTZIKINAKIS
Abstract.
We study equidistribution properties of nil-orbits ( b n x ) n ∈ N when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. Forexample, we show that if X = G/ Γ is a nilmanifold, b ∈ G is an ergodic nilrotation, and c ∈ R \ Z is positive, then the sequence ( b [ n c ] x ) n ∈ N is equidistributed in X for every x ∈ X .This is also the case when n c is replaced with a ( n ), where a ( t ) is a function that belongs tosome Hardy field, has polynomial growth, and stays logarithmically away from polynomials,and when it is replaced with a random sequence of integers with sub-exponential growth.Similar results have been established by Boshernitzan when X is the circle. Contents
1. Introduction and main results 12. Background on Hardy fields and nilmanifolds 83. A model equidistribution result 134. Single nil-orbits and Hardy sequences 165. Several nil-orbits and Hardy sequences 226. Random sequences of sub-exponential growth 26References 311.
Introduction and main results
Motivation.
A nilmanifold is a homogeneous space X = G/ Γ where G is a nilpotent Liegroup, and Γ is a discrete cocompact subgroup of G . For b ∈ G and x = g Γ ∈ X we define bx = ( bg )Γ. In recent years it has become clear that studying equidistribution properties of nil-orbits ( b n x ) n ∈ N , and their subsequences, is a central problem, with applications to various areasof mathematics that include combinatorics ([2], [42], [18], [14], [4], [19], [20]), ergodic theory([11], [28], [41], [29], [34], [16], [17], [30]), number theory ([1], [25], [3], [24]), and probabilitytheory ([13]).It is well known that for every b ∈ G and x ∈ X the sequence ( b n x ) n ∈ N is equidistributedin some nice algebraic set ([37], [35], [33]), and this is also the case when the parameter n is restricted to the range of some polynomial with integer coefficients ([39], [33]), or the setof prime numbers ([24]). Furthermore, very recently, quantitative equidistribution results forpolynomial nil-orbits have been established ([23]) and used as part of an ongoing project tofind asymptotics for the number of arithmetic progressions in the set of prime numbers ([22]).These quantitative estimates will also play a crucial role in the present article. Mathematics Subject Classification.
Primary: 22F30; Secondary: 37A17.
Key words and phrases.
Homogeneous space, nil man if ol d, equid is tribution, Hardy field.The author was partially supported by NSF grant DMS-0701027. he main objective of this article is to study equidistribution properties of nil-orbits ( b n x ) n ∈ N when the parameter n is restricted to some sparse sequence of integers that is not necessarilypolynomial. For example, we shall show that if b ∈ G has a dense orbit in X , meaning( b n Γ) n ∈ N = X , then for every x ∈ X the sequences(1)( b [ n √ ] x ) n ∈ N , ( b [ n log n ] x ) n ∈ N , ( b [ n √ n √ x ) n ∈ N , ( b [ n +(log n ) ] x ) n ∈ N , ( b [(log( n !)) k ] x ) n ∈ N , are all equidistributed in X . Furthermore, using a probabilistic construction we shall exhibitexamples of sequences with super-polynomial growth for which analogous equidistribution re-sults hold (explicit such examples are not known). Let us remark at this point, that since weshall work with sparse sequences of times taken along sequences whose range has typically neg-ligible intersection with the range of polynomial sequences, our results cannot be immediatelydeduced from known equidistribution results along polynomial sequences.We shall also study equidistribution properties involving several nil-orbits. For example,suppose that c , c , . . . , c k are distinct non-integer real numbers, all greater than 1, and b ∈ G has a dense orbit in X . We shall show that the sequence( b [ n c ] x , b [ n c ] x , . . . , b [ n ck ] x k ) n ∈ N is equidistributed in X k for every x , x , . . . , x k ∈ X .In a nutshell, our approach is to use the Taylor expansion of a function a ( t ) to partition therange of the sequence ([ a ( n )]) n ∈ N into approximate polynomial blocks of fixed degree, in sucha way that one can give useful quantitative estimates for the corresponding “Weyl type” sums.In order to carry out this plan, we found it very helpful to deal with the sequence ( a ( n )) n ∈ N first, thus leading us to study equidistribution properties of the sequence ( b a ( n ) x ) n ∈ N , where b s for s ∈ R can be defined appropriately.Before giving the exact results, let us also mention that an additional motivation for ourstudy is the various potential applications in ergodic theory and combinatorics. This direction ofresearch has already proven fruitful; very recently in [20] equidistribution results on nilmanifoldsplayed a key role in establishing a Hardy field refinement of Szemer´edi’s theorem on arithmeticprogressions and related multiple recurrence results in ergodic theory. However, in that article,equidistribution properties involving only conveniently chosen subsequences of the sequencesin question were studied. The problem of studying equidistribution properties of the full rangeof non-polynomial sequences, like those in (1), is more delicate, and is addressed for the firsttime in the present article. This turns out to be a crucial step towards an in depth study ofthe limiting behavior of multiple ergodic averages of the form1 N N X n =1 T [ a ( n )] f · . . . · T [ a ℓ ( n )] f ℓ , where ( a i ( n )) n ∈ N are real valued sequences that satisfy some regularity conditions. The re-maining steps of this project will be completed in a forthcoming paper ([15]).1.2. Equidistribution results.
Throughout the article we are going to work with the class ofreal valued functions H that belong to some Hardy field (see Section 2.1 for details). Workingwithin the class H eliminates several technicalities that would otherwise obscure the trans-parency of our results and the main ideas of their proofs. Furthermore, H is a rich enoughclass to enable one to deal, for example, with all the sequences considered in (1).In various places we evaluate an element b of a connected and simply connected nilpotentLie group G on some real power s . In Section 2.2 we explain why this operation is legitimate. hen writing a ( t ) ≺ b ( t ) we mean a ( t ) /b ( t ) → t → + ∞ . When writing a ( t ) ≪ b ( t ) wemean that | a ( t ) | ≤ C | b ( t ) | for some constant C for every large t . We also say that a function a ( t ) has polynomial growth if a ( t ) ≺ t k for some k ∈ N .1.2.1. A single nil-orbit.
If ( a ( n )) n ∈ N is a sequence of real numbers, and X = G/ Γ is a nil-manifold, with G connected and simply connected, we say that the sequence ( b a ( n ) x ) n ∈ N is equidistributed in a sub-nilmanifold X b of X , if for every F ∈ C ( X ) we havelim N →∞ N N X n =1 F ( b a ( n ) x ) = Z F dm X b where m X b denotes the normalized Haar measure on X b . Similarly, if the sequence ( a ( n )) n ∈ N has integer values, we can define a notion of equidistribution on every nilmanifold X , withoutimposing any connectedness assumption on G or X .A sequence ( a ( n )) n ∈ N of real numbers is pointwise good for nilsystems if for every nilman-ifold X = G/ Γ, where G is connected and simply connected, and every b ∈ G , x ∈ X ,the sequence ( b a ( n ) x ) n ∈ N has a limiting distribution , meaning, for every F ∈ C ( X ) the limitlim N →∞ N P Nn =1 F ( b a ( n ) x ) exists.We remark that for sequences ( a ( n )) n ∈ N with integer values, the connectedness assumptionsof the previous definition are superficial. Using the lifting argument of Section 2.2, one sees thatif a sequence of integers ( a ( n )) n ∈ N is pointwise good for nilsystems, then for every nilmanifold X = G/ Γ, b ∈ G , and x ∈ X , the sequence ( b a ( n ) x ) n ∈ N has a limiting distribution.Our first result gives necessary and sufficient conditions for Hardy sequences of polynomialgrowth to be pointwise good for nilsystems. Theorem 1.1.
Let a ∈ H have polynomial growth.Then the sequence ( a ( n )) n ∈ N (or the sequence ([ a ( n )]) n ∈ N ) is pointwise good for nilsystemsif and only if one of the following conditions holds: • | a ( t ) − cp ( t ) | ≻ log t for every c ∈ R and every p ∈ Z [ t ] ; or • a ( t ) − cp ( t ) → d for some c, d ∈ R and some p ∈ Z [ t ] ; or • | a ( t ) − t/m | ≪ log t for some m ∈ Z .Remarks. • The necessity of these conditions can be seen using rotations on the circle (see [9]).In the case were X = T their sufficiency was established in [9]. • Unlike the case of integer polynomial sequences, if p ∈ R [ t ], then the sequence ( b [ p ( n )] x ) n ∈ N may not be equidistributed in a finite union of sub-nilmanifolds of X . For example, when X = T (= R / Z ), the sequence ( − [ n √ / √ Z ) n ∈ N is equidistributed in the set (cid:8) t Z : { t } ∈ [0 , / √ (cid:9) .It seems sensible to assert that the first condition in Theorem 1.1 is satisfied by the “typical”function in H with polynomial growth. It turns out that in this “typical” case, restricting theparameter n of a nil-orbit ( b n Γ) n ∈ N to the range of the sequence ([ a ( n )]) n ∈ N , does not changeits limiting distribution: Theorem 1.2.
Let a ∈ H have polynomial growth and satisfy | a ( t ) − cp ( t ) | ≻ log t for every c ∈ R and p ∈ Z [ t ] . ( i ) If X = G/ Γ is a nilmanifold, with G connected and simply connected, then for every b ∈ G and x ∈ X the sequence ( b a ( n ) x ) n ∈ N is equidistributed in the nilmanifold ( b s x ) s ∈ R . ( ii ) If X = G/ Γ is a nilmanifold, then for every b ∈ G and x ∈ X the sequence ( b [ a ( n )] x ) n ∈ N is equidistributed in the nilmanifold ( b n x ) n ∈ N . emark. Suppose that we want the conclusion ( i ) (or ( ii )) to be true only for some fixed b ∈ G .Then our proof shows that the assumption can be relaxed to the following: | a ( t ) − cp ( t ) | ≻ log t for every p ∈ Z [ t ], and every c ∈ R of the form q/β where q ∈ Q and β is some non-zeroeigenvalue for the nilrotation by b (this means f ( bx ) = e ( β ) f ( x ) for some non-constant f ∈ L ( m X )). A special case of this stronger result (take G = R , Γ = Z , and b = 1) gives one of themain results in [8], stating that if a ∈ H has polynomial growth and satisfies | a ( t ) − p ( t ) | ≻ log t for every p ∈ Q [ t ], then the sequence ( a ( n ) Z ) n ∈ N is equidistributed in T .1.2.2. Several nil-orbits.
We give an equidistribution result involving nil-orbits of several Hardysequences. We say that the functions a ( t ) , . . . , a ℓ ( t ) have different growth rates if the quotientof any two of these functions converges to ±∞ or to 0. Theorem 1.3.
Suppose that the functions a ( t ) , . . . , a ℓ ( t ) belong to the same Hardy field, havedifferent growth rates, and satisfy t k i log t ≺ a i ( t ) ≺ t k i +1 for some ak i ∈ N . ( i ) If X i = G i / Γ i are nilmanifolds, with G i connected and simply connected, then for every b i ∈ G i and x i ∈ X i , the sequence ( b a ( n )1 x , . . . , b a ℓ ( n ) ℓ x ℓ ) n ∈ N is equidistributed in the nilmanifold ( b s x ) s ∈ R × · · · × ( b sℓ x ℓ ) s ∈ R . ( ii ) If X i = G i / Γ i are nilmanifolds, then for every b i ∈ G i , and x i ∈ X i , the sequence ( b [ a ( n )]1 x , . . . , b [ a ℓ ( n )] ℓ x ℓ ) n ∈ N is equidistributed in the nilmanifold ( b n x ) n ∈ N × · · · × ( b nℓ x ℓ ) n ∈ N . Remark.
The preceding result contrasts the case of polynomial sequences, where differentgrowth does not imply simultaneous equidistribution for the corresponding nil-orbits. Forexample, there exists a connected nilmanifold X = G/ Γ and an ergodic element b ∈ G , suchthat the sequence ( b n Γ , b n Γ) n ∈ N is not even dense in X × X (see [16]). On the other hand, ourresult shows that if for instance a ( t ) = t √ , then for every nilmanifold X and ergodic element b ∈ G the sequence ( b [ a ( n )] Γ , b [( a ( n )) ] Γ) n ∈ N is equidistributed in X × X .It may very well be the case that the hypothesis of Theorem 1.3 can be relaxed to give amuch stronger result. The following is a closely related conjecture: Conjecture.
Let a ( t ) , . . . , a ℓ ( t ) be functions that belong to the same Hardy field and havepolynomial growth. Suppose further that every non-trivial linear combination a ( t ) of thesefunctions satisfies | a ( t ) − cp ( t ) | ≻ log t for every c ∈ R and p ∈ Z [ t ] .Then for every nilmanifold X = G/ Γ , b i ∈ G , and x i ∈ X , the sequence ( b [ a ( n )]1 x , . . . , b [ a ℓ ( n )] ℓ x ℓ ) n ∈ N is equidistributed in the nilmanifold ( b n x ) n ∈ N × · · · × ( b nℓ x ℓ ) n ∈ N . More general classes of functions.
We make some remarks about the extend of the func-tions our methods cover that do not necessarily belong to some Hardy field.The conclusions of Theorem 1.2 hold if for some k ∈ N the function a ( t ) is ( k + 1)-timesdifferentiable for large t ∈ R and satisfies:(2) a ( k +1) ( t ) → , and t | a ( k +1) ( t ) | → ∞ . (If a ∈ H , then (2) is equivalent to “ t k log t ≺ a ( t ) ≺ t k +1 ”.) To see this, one can repeat theproof of Theorem 1.3 in this particular setup. More generally, the conclusion of Theorem 1.2 olds for functions a ( t ) that satisfy the following less restrictive conditions: For some k ∈ N the function a ∈ C k +1 ( R + ) satisfies(3) | a ( k +1) ( t ) | decreases to zero , /t k ≺ a ( k ) ( t ) ≺ , and ( a ( k +1) ( t )) k ≺ ( a ( k ) ( t )) k +1 . (If a ∈ H , then (3) is equivalent to “ a ( t ) has polynomial growth and | a ( t ) − p ( t ) | ≻ log t forevery p ∈ R [ t ]”.) One can see this by repeating verbatim part of the proof of Theorem 1.2.The reader is advised to think of the second condition in (3) as the most important one andthe other two as technical necessities (for functions in H the second condition implies the othertwo).Theorem 1.3 can be proved for functions a i ( t ) that satisfy condition (2) for some k i ∈ N (callthis integer the type of a i ( t )), and also, for every k ∈ N every non-trivial linear combination ofthose functions a i ( t ) that have type k also satisfies (2).As for Theorem 1.1, unless one works within a “regular” class of functions like H , it seemshopeless to state a result with explicit necessary and sufficient conditions.1.2.4. Random sequences of sub-exponential growth.
So far, we have given examples of se-quences that are pointwise good for nilsystems and have polynomial growth. For Hardy se-quences of super-polynomial growth, it is indicated in [8] that no growth condition should sufficeto guarantee equidistribution on T . On the other hand, explicit sequences of super-polynomialgrowth like ( e √ n ) n ∈ N or ( e (log n ) ) n ∈ N are expected to be pointwise good for nilsystems, butproving this seems to be out of reach at the moment, even for rotations on T .Nevertheless, using a probabilistic argument, we shall show that there exist very sparselydistributed sequences that are pointwise good for nilsystems. In fact, loosely speaking, we shallsee that the only growth condition prohibiting the existence of such examples is exponentialgrowth.Our probabilistic setup is as follows. Let ( σ n ) n ∈ N be a decreasing sequence of reals in [0 , n in the set with probability σ n ∈ [0 , , Σ , P ) be a probability space, and ( X n ) n ∈ N be a sequence of0 − P ( { ω ∈ Ω : X n ( ω ) = 1 } ) = σ n . Given ω ∈ Ωwe construct the set of positive integers A ω by taking n ∈ A ω if and only if X n ( ω ) = 1. Bywriting the elements of A ω in increasing order we get a sequence ( a n ( ω )) n ∈ N .If σ n = 1 /n c where c ∈ [0 , n / (1 − c ) ). If σ n = 1 /n , then almost surely the resultingrandom sequence is bad for pointwise convergence results even for circle rotations (see [31]).Therefore, it makes sense to restrict our attention to the case where σ n ≻ /n . By choosing σ n appropriately, we can get examples of random sequences with any prescribed sub-exponentialgrowth.In [7] (and subsequently in [10]) it was shown that if lim n →∞ nσ n = ∞ , then almost surely,the random sequence ( a n ( ω )) n ∈ N is pointwise good for convergence of rotations on the circle.We extend this result to rotations on nilmanifolds by showing the following: Theorem 1.4.
Let ( σ n ) n ∈ N be a decreasing sequence of reals satisfying lim n →∞ nσ n = ∞ .Then almost surely, the random sequence ( a n ( ω )) n ∈ N is pointwise good for nilsystems.Remark. As it will become clear from the proof, the condition lim n →∞ nσ n = ∞ can be replacedwith the condition lim N →∞ P ≤ n ≤ N σ n log N = ∞ . Furthermore, our method of proof will show thatalmost surely, the limits lim N →∞ N P Nn =1 F ( b a n ( ω ) x ) and lim N →∞ N P Nn =1 F ( b n x ) are equalfor every nilmanifold X = G/ Γ, F ∈ C ( X ), b ∈ G , and x ∈ X . .3. Applications.
We give some rather straightforward applications of the preceding equidis-tribution results. We only sketch their proofs leaving some routine details to the reader. Foraesthetic reasons, we represent elements t Z of T = R / Z by t .The first is an equidistribution result on T , which we do not see how to handle using con-ventional exponential sum techniques for k ≥ Theorem 1.5.
Let a ∈ H have polynomial growth and satisfy | a ( t ) − cp ( t ) | ≻ log t for every c ∈ R and p ∈ Z [ t ] .Then for every k ∈ N and irrational β , the sequence ([ a ( n )] k β ) n ∈ N is equidistributed in T .Remark. A standard modification of our argument gives the following more general conclusion:for every q ∈ Z [ t ] non-constant and irrational β , the sequence (cid:0) q ([ a ( n )]) β (cid:1) n ∈ N is equidistributedin T . A similar extension holds for Theorems 1.7, 1.8, and Theorem 1.6 (with ℓ non-constantpolynomials). Proof (Sketch).
Suppose for convenience that k = 2. We define the transformation T : T → T by T ( x, y ) = (cid:0) x + β, y + 2 x + β (cid:1) . It is well known that the resulting system is isomorphic toa nilsystem (and the conjugacy map is continuous), and that this system is ergodic if β isirrational. Applying Theorem 1.2 we get that the sequence ( T [ a ( n )] (0 , n ∈ N is equidistributedin T . An easy computation shows that T n (0 ,
0) = ( nβ, n β ), therefore for every F ∈ C ( T )we have lim N →∞ N N X n =1 F ([ a ( n )] β, [ a ( n )] β ) = Z F dm T , where m T denotes the normalized Haar measure on T . Using this identity for F ( x, y ) = e ( ky ),where k is a non-zero integer, we getlim N →∞ N N X n =1 e ( k [ a ( n )] β ) = 0 . This shows that the sequence ([ a ( n )] β ) n ∈ N is equidistributed in T . (cid:3) Similarly, we can deduce from Theorem 1.3 the following result:
Theorem 1.6.
Let a ( t ) , . . . , a ℓ ( t ) be functions that belong to the same Hardy field, have dif-ferent growth rates, and satisfy t k log t ≺ a i ( t ) ≺ t k +1 for some k = k i ∈ N .Then for every l i ∈ N and irrationals β i , the sequence ([ a ( n )] l β , . . . , [ a ℓ ( n )] l ℓ β ℓ ) n ∈ N is equidistributed in T ℓ . Next we give an application to ergodic theory. We say that a sequence of integers ( a ( n )) n ∈ N is good for mean convergence , if for every invertible measure preserving system ( X, B , µ, T )and f ∈ L ( µ ) the averages N P Nn =1 f ( T a ( n ) x ) converge in L ( µ ) as N → ∞ . Using thespectral theorem for unitary operators, one can see that a sequence ( a ( n )) n ∈ N is good for meanconvergence if and only if for every t ∈ R the sequence ( a ( n ) t ) n ∈ N has a limiting distribution. Theorem 1.7.
Let a ∈ H have polynomial growth and k ∈ N .Then the sequence ([ a ( n )] k ) n ∈ N is good for mean convergence if and only if one of the threeconditions in Theorem 1.1 is satisfied.Remark. For k = 1 this result was established in [9]. roof (Sketch). The necessity of the conditions can be seen exactly as in the proof of Theo-rem 3.1. To prove the sufficiency, we apply Theorem 1.1 for some appropriate unipotent affinetransformations of some finite dimensional tori. We deduce that for every k ∈ N and t ∈ R thesequence ([ a ( n )] k t ) n ∈ N has a limiting distribution. As explained before, this implies that thesequence ([ a ( n )] k ) n ∈ N is good for mean convergence. (cid:3) Lastly, we give a recurrence result for measure preserving systems, and a correspondingcombinatorial consequence. We say that a sequence of integers ( a ( n )) n ∈ N is good for recurrence ,if for every invertible measure preserving system ( X, B , µ, T ) and set A ∈ B with µ ( A ) > µ ( A ∩ T − a ( n ) A ) > n ∈ N such that a ( n ) = 0. Using the correspondenceprinciple of Furstenberg ([21]) one can see that this notion is equivalent to the following one:A sequence of integers ( a ( n )) n ∈ N is intersective if every set of integers Λ with positive upperdensity contains two distinct elements x and y such that x − y = a ( n ) for some n ∈ N . Theorem 1.8.
Let a ∈ H have polynomial growth and satisfy | a ( t ) − cp ( t ) | ≻ log t for every c ∈ R and p ∈ Z [ t ] .Then for every k ∈ N the sequence ([ a ( n )] k ) n ∈ N is good for recurrence (or intersective).Remarks. • For k = 1 this result can be deduced from the equidistribution results in [8]. • A more tedious argument can be used to show that the following weaker assumptionsuffices: “ a ∈ H has polynomial growth and satisfies | a ( t ) − cp ( t ) | → ∞ for every c ∈ R and p ∈ Z [ t ]”. (By combining the spectral theorem and an argument similar to one used in theproof of Proposition 6.5 in [15], one can handle the case where and | a ( t ) − cp ( t ) | ≪ log t forsome c ∈ R and p ∈ Z [ t ].) Proof (Sketch).
We apply Theorem 1.2 for some appropriate unipotent affine transformationsof finite dimensional tori. We deduce that for every k ∈ N and t ∈ R the sequence ([ a ( n )] k t ) n ∈ N has the same limiting distribution as the sequence ( n k t ) n ∈ N . Using this and the spectraltheorem for unitary operators, we conclude that for every invertible measure preserving system( X, B , µ, T ) and set A ∈ B , we havelim N →∞ N N X n =1 µ ( A ∩ T − [ a ( n )] k A ) = lim N →∞ N N X n =1 µ ( A ∩ T − n k A ) . Since the last limit is known to be positive whenever µ ( A ) > a ( n )] k is good for recurrence. (cid:3) More delicate applications of the equidistribution results presented in Section 1.2 includestatements about multiple recurrence and convergence of multiple ergodic averages, and relatedcombinatorial consequences. Such results require much extra work and will be presented in aforthcoming paper ([15]).1.4.
Structure of the article.
In Section 2 we give the necessary background on Hardy fieldsand state some equidistribution results on nilmanifolds that will be used later.In Section 3 we work on a model equidistribution problem that helps us illustrate some of theideas needed to prove Theorems 1.1 and 1.2. We give a new proof of a result of Boshernitzanon equidistribution of the fractional parts of Hardy sequences of polynomial growth.In Section 4 we prove Theorems 1.1 and 1.2. The key ingredients are: (i) a reduction stepthat enables us to “remove” the integer parts and deal with equidistribution properties onnilmanifolds X = G/ Γ with G connected and simply connected, (ii) the proof technique of the odel problem described in Section 3, and (iii) some quantitative equidistribution results ofGreen and Tao.In Section 5 we prove Theorem 1.3. The proof strategy is similar with that of Theorems 1.2,with the exception of a key technical difference that is illustrated using a model equidistributionproblem.In Section 6 we prove Theorem 1.4. We adapt an argument of Bourgain that worked forcircle rotations to our more complicated non-Abelian setup.1.5. Notational conventions.
The following notation will be used throughout the article: N = { , , . . . } , T k = R k / Z k , T f = f ◦ T , e ( t ) = e πit , [ t ] denotes the integer part of t , { t } = t − [ t ], k t k = d ( t, Z ), E n ∈ A a ( n ) = | A | P n ∈ A a ( n ). By a ( t ) ≺ b ( t ) we mean lim t →∞ a ( t ) /b ( t ) = 0,by a ( t ) ∼ b ( t ) we mean lim t →∞ a ( t ) /b ( t ) is a non-zero real number, and by a ( t ) ≪ b ( t ) wemean | a ( t ) | ≤ C | b ( t ) | for some constant C for all large t . We use the symbol ≪ w ,...,w k whensome expression is majorized by some other expression and the implied constant depends onthe parameters w , . . . , w k . By o N →∞ ; w ,...,w k (1) we denote a quantity that goes to zero whenthe parameters w , . . . , w k are fixed and N → ∞ (when there is no danger of confusion we mayomit the parameters). We often write ∞ instead of + ∞ . For aesthetic reasons, we sometimesrepresent elements t Z of T = R / Z by t . Acknowledgement.
We thank the referee for providing constructive comments.2.
Background on Hardy fields and nilmanifolds
Hardy fields.
Let B be the collection of equivalence classes of real valued functionsdefined on some half line ( c, ∞ ), where we identify two functions if they agree eventually. A Hardy field is a subfield of the ring ( B, + , · ) that is closed under differentiation. With H wedenote the union of all Hardy fields . If a ∈ H is defined in [1 , ∞ ) (one can always choose sucha representative of a ( t )) we call the sequence ( a ( n )) n ∈ N a Hardy sequence .A particular example of a Hardy field is the set of all rational functions with real coefficients.Another example is the set LE that consists of all logarithmic-exponential functions ([26], [27]),meaning all functions defined on some half line ( c, ∞ ) by a finite combination of the symbols+ , − , × , : , log , exp, operating on the real variable t and on real constants. For example thefunctions t √ , t + t √ t log t , e t , e √ log log t / log( t + 1), are all elements of LE .We collect here some properties that illustrate the richness of H . More information aboutHardy fields can be found in the paper [8] and the references therein. • H contains the set LE and anti-derivatives of elements of LE . • H contains several other functions not in LE , like the functions Γ( t ), ζ ( t ), sin (1 /t ). • If a ∈ LE and b ∈ H , then there exists a Hardy field containing both a ( t ) and b ( t ). • If a ∈ LE , b ∈ H , and b ( t ) → ∞ , then a ◦ b ∈ H .If a ∈ LE , b ∈ H , and a ( t ) → ∞ , then b ◦ a ∈ H . • If a is a continuous function that is algebraic over some Hardy field, then a ∈ H .Using these properties it is easy to check that, for example, the sequences (log Γ( n )) n ∈ N ,( n √ ζ ( n )) n ∈ N , ((Li( n )) ) n ∈ N (Li( t ) = R t / ln s ds ), and the sequences that appear inside theinteger parts in (1), are Hardy sequences with polynomial growth. On the other hand, sequences The equivalence classes just defined are often called germs of functions . We choose to use the word functionwhen we refer to elements of B instead, with the understanding that all the operations defined and statementsmade for elements of B are considered only for sufficiently large values of t ∈ R . hat oscillate, like (sin n ) n ∈ N , ( n sin n ) n ∈ N , or the sequence ( e n + sin n ) n ∈ N are not Hardysequences.We mention some basic properties of elements of H relevant to our study. Every element of H has eventually constant sign (since it has a multiplicative inverse). Therefore, if a ∈ H , then a ( t ) is eventually monotone (since a ′ ( t ) has eventually constant sign), and the limit lim t →∞ a ( t )exists (possibly infinite). Since for every two functions a ∈ H , b ∈ LE ( b = 0), we have a/b ∈ H ,it follows that the asymptotic growth ratio lim t →∞ a ( t ) /b ( t ) exists (possibly infinite). This lastproperty is key, since it will often justify our use of l’Hopital’s rule. We are going to freely useall these properties without any further explanation in the sequel.
We caution the reader that although every function in H is asymptotically comparable withevery function in LE , some functions in H are not comparable. This defect of H will only playa role in one of our results (Theorem 1.3), and can be sidestepped by restricting our attentionto functions that belong to the same Hardy field.A key property of elements of H with polynomial growth is that we can relate their growthrates with the growth rates of their derivatives: Lemma 2.1.
Suppose that a ∈ H has polynomial growth. We have the following: ( i ) If t ε ≺ a ( t ) for some ε > , then a ′ ( t ) ∼ a ( t ) /t . ( ii ) If t − k ≺ a ( t ) for some k ∈ N , and a ( t ) does not converge to a non-zero constant, then a ( t ) / ( t (log t ) ) ≺ a ′ ( t ) ≪ a ( t ) /t .Remark. The assumption of polynomial growth is essential, to see this take a ( t ) = e t . To seethat the other assumptions in parts ( i ) and ( ii ) are essential take a ( t ) = log t for part ( i ), and a ( t ) = e − t , a ( t ) = 1 + 1 /t for part ( ii ). Proof.
First we deal with part ( i ). Applying l’Hopital’s rule we get(4) lim t →∞ ta ′ ( t ) a ( t ) = lim t →∞ (log | a ( t ) | ) ′ (log t ) ′ = lim t →∞ log | a ( t ) | log t . Since t ε ≺ a ( t ) for some ε > a ( t ) has polynomial growth, the last limit is a positive realnumber. Hence, a ′ ( t ) ∼ a ( t ) /t , proving part ( i ).Next we deal with part ( ii ). First notice that since a ( t ) does not converge to a non-zeroconstant we can assume that either | a ( t ) | → ∞ or | a ( t ) | → a ′ ( t ) ≪ a ( t ) /t . Since lim t →∞ log | a ( t ) | = ±∞ we can apply l’Hopital’s rule toget (4). Since a ( t ) has polynomial growth and t − k ≺ a ( t ) for some k ∈ N , we have that thelimit lim t →∞ log | a ( t ) | / log t is finite. Using (4) we conclude that the same holds for the limitlim t →∞ ( ta ′ ( t )) /a ( t ). It follows that a ′ ( t ) ≪ a ( t ) /t .Finally we show that a ( t ) / ( t (log t ) ) ≺ a ′ ( t ). Equivalently, it suffices to show that the limitlim t →∞ t (log t ) a ′ ( t ) a ( t )is infinite. Arguing by contradiction, suppose this is not the case. Then(log | a ( t ) | ) ′ ≪ t (log t ) , and integrating we get log | a ( t ) | ≪ t + c for some c ∈ R . It follows that log | a ( t ) | is bounded, which contradicts the fact that | a ( t ) | → ∞ or 0. This completes the proof. (cid:3) ollowing [8], for a non-negative integer k we say that:(i) The function a ∈ H has type k if a ( t ) ∼ t k .(ii) The function a ∈ H has type k + if t k ≺ a ( t ) ≺ t k +1 .It is easy to show the following: Lemma 2.2 (Boshernitzan [8]) . Suppose that a ∈ H has polynomial growth. Then ( i ) There exists a non-negative integer k such that a ( t ) has type either k or k + . ( ii ) If a ( t ) has type k , then a ( t ) = ct k + b ( t ) for some non-zero c ∈ R and b ∈ H with b ( t ) ≺ t k . Applying Lemma 2.1 repeatedly we get:
Corollary 2.3.
Suppose that a ∈ H has type k + for some non-negative integer k .Then for every l ∈ N with l ≤ k we have a ( l ) ( t ) ∼ a ( t ) /t l , and for every l ∈ N we have a ( t ) / ( t l (log t ) ) ≺ a ( l ) ( t ) ≪ a ( t ) /t l . Remark.
The conclusion fails for some functions of type k . Indeed, if a ( t ) = 1 + 1 /t , then a ′ ( t ) ≺ a ( t ) / ( t (log t ) ).2.2. Nilmanifolds.
Fundamental properties of rotations on nilmanifolds, related to our study,were studied in [1], [36], [35], and [33]. Below we summarize some facts that we shall use, allthe proofs can be found or deduced from [33] and [12].Given a topological group G , we denote its identity element by id G . By G we denotethe connected component of id G . If A, B ⊂ G , then [ A, B ] is defined to be the subgroupgenerated by elements of the form { [ a, b ] : a ∈ A, b ∈ B } where [ a, b ] = aba − b − . We definethe commutator subgroups recursively by G = G and G k +1 = [ G, G k ]. A group G is said tobe nilpotent if G k = { id G } for some k ∈ N . If G is a nilpotent Lie group and Γ is a discretecocompact subgroup, then the compact homogeneous space X = G/ Γ is called a nilmanifold .The group G acts on G/ Γ by left translation where the translation by a fixed element b ∈ G is given by T b ( g Γ) = ( bg )Γ. We denote by m X the normalized Haar measure on X , meaning,the unique probability measure that is invariant under the action of G by left translations andis defined on the Borel σ -algebra of X . We call the elements of G nilrotations . A nilrotation b ∈ G acts ergodically on X , if the sequence ( b n Γ) n ∈ N is dense in X . When the nilmanifold X is implicit we shall often simply say that a nilrotation b ∈ G is ergodic . It can be shownthat if b ∈ G is ergodic, then for every x ∈ X the sequence ( b n x ) n ∈ N is equidistributed in X .A nilrotation b ∈ G is totally ergodic , if for every r ∈ N the nilrotation b r is ergodic. If thenilmanifold X is connected it can be shown that every ergodic nilrotation is in fact totallyergodic. Example 2.4 (Heisenberg nilmanifold) . Let G be the nilpotent group that consists of all uppertriangular matrices of the form (cid:16) x z y (cid:17) with real entries. If we only allow integer entries weget a subgroup Γ of G that is discrete and cocompact. Then G/ Γ is a nilmanifold. It can beshown that a nilrotation b = (cid:18) α γ β (cid:19) is ergodic if and only if the numbers 1, α , and β arerationally independent.Let G be a connected and simply connected Lie group and exp : g → G be the exponentialmap, where g is the Lie algebra of G . Since G is a connected and simply connected nilpotentLie group, it is well known that the exponential map is a bijection. For b ∈ G and s ∈ R wedefine the element b s of G as follows: If X ∈ g is such that exp( X ) = b , then b s = exp( sX ). more intuitive way to make sense of the element b s is by thinking of G as a matrix group;then b s is the element one gets after replacing n by s in the formula giving the elements of thematrix b n . It is instructive to compare the two equivalent ways of defining b s in the followingexample. Example 2.5 (Heisenberg nilflow) . Let X be the Heisenberg nilmanifold. Then the expo-nential map is given by exp (cid:16) x z y (cid:17) = (cid:18) x z + xy y (cid:19) . As a consequence, if b = (cid:18) α γ β (cid:19) ,then exp (cid:18) α γ − αβ β (cid:19) = b , and a short computation shows that b s = (cid:18) sα sγ + s ( s − αβ sβ (cid:19) .Alternatively, one can find the same formula for b s after replacing n by s in the formula b n = (cid:18) nα nγ + n ( n − αβ nβ (cid:19) .Next we record some basic facts that we will frequently use:( Basic properties of b s ). If G is a connected and simply connected Lie group, then for b ∈ G the map s → b s is continuous, and for s, s , s ∈ R one has the identities b s + s = b s · b s ,( b s ) s = b s s , and ( gbg − ) s = gb s g − .( Ratner’s theorem, nilpotent case ) Let X = G/ Γ be a nilmanifold. Then for every b ∈ G theset X b = { b n Γ , n ∈ N } has the form H/ ∆, where H is a closed subgroup of G that contains b ,and ∆ = H ∩ Γ is a discrete cocompact subgroup of H . Furthermore, the sequence ( b n Γ) n ∈ N is equidistributed in X b .Likewise, if G is connected and simply connected, and b ∈ G , let Y b = { b s Γ , s ∈ R } . Then Y b has the form H/ ∆ where H is a closed connected and simply connected subgroup of G thatcontains all elements b s for s ∈ R , and ∆ is a discrete cocompact subgroup of H . Furthermore,the nilflow ( b s Γ) s ∈ R is equidistributed in Y b .( Change of base point formula ). Let X = G/ Γ be a nilmanifold. As mentioned before, forevery b ∈ G the nil-orbit ( b n Γ) n ∈ N is equidistributed in the set X b = { b n Γ , n ∈ N } . Usingthe identity b n g = g ( g − bg ) n we see that the nil-orbit ( b n g Γ) n ∈ N is equidistributed in the set g · X g − bg . A similar formula holds when G is connected and simply connected and we replacethe integer parameter n with the real parameter s and the nilmanifold X b with Y b .( Lifting argument ). In several instances it will be convenient for us to assume that a nil-manifold X has a representation G/ Γ with G connected and simply connected. To get thisextra assumption we argue as follows (see [33]): Since all our results deal with the action on X of finitely many elements of G we conclude that for the purposes of this paper, we can, andwill always assume that the discrete group G/G is finitely generated. In this case, one canshow that X = G/ Γ is isomorphic to a sub-nilmanifold of a nilmanifold ˜ X = ˜ G/ ˜Γ, where ˜ G is a connected and simply-connected nilpotent Lie group, with all translations from G “repre-sented” in ˜ G (for example if X = T then ˜ X = R / Z , and if X = ( Z × R ) / Z then ˜ X = R / Z ).Practically, this means that for every F ∈ C ( X ), b ∈ G , and x ∈ X , there exists ˜ F ∈ C ( ˜ X ),˜ b ∈ ˜ G , and ˜ x ∈ ˜ X , such that F ( b n x ) = ˜ F (˜ b n ˜ x ) for every n ∈ N .One should keep in mind though when using this lifting trick, that any assumption madeabout a nilrotation b acting on a nilmanifold X , is typically lost when passing to the liftednilmanifold ˜ X . Therefore, the above mentioned construction will be helpful only when ourworking assumptions impose no restrictions on a nilrotation. Example 2.6.
Let G be the non-connected nilpotent group that consists of all upper triangularmatrices of the form (cid:16) k z y (cid:17) where k ∈ Z and y, z ∈ R . If we also restrict the entries y and to be integers we get a subgroup Γ of G that is discrete and cocompact. In this case, theHeisenberg nilmanifold of Example 2.4 can serve as the lifting ˜ X of the nilmanifold X = G/ Γ.2.3.
Equidistribution on nilmanifolds.
We gather some equidistribution results of polyno-mial sequences on nilmanifolds that will be used later.2.3.1.
Qualitative equidistribution on nilmanifolds. If G is a nilpotent group, then a sequence g : N → G of the form g ( n ) = a p ( n )1 a p ( n )2 · · · a p k ( n ) k , where a i ∈ G , and p i are polynomials takinginteger values at the integers, is called a polynomial sequence in G . If the maximum degreeof the polynomials p i is at most d we say that the degree of g ( n ) is at most d . A polynomialsequence on the nilmanifold X = G/ Γ is a sequence of the form ( g ( n )Γ) n ∈ N where g : N → G is a polynomial sequence in G . Theorem 2.7 ( Leibman [33] ). Suppose that X = G/ Γ is a nilmanifold, with G connectedand simply connected, and ( g ( n )) n ∈ N is a polynomial sequence in G . Let Z = G/ ([ G, G ]Γ) and π : X → Z be the natural projection.Then the following statements are true: ( i ) The sequence ( g ( n ) x ) n ∈ N is equidistributed in a finite union of sub-nilmanifolds of X . ( ii ) For every x ∈ X the sequence ( g ( n ) x ) n ∈ N is equidistributed in X if and only if thesequence ( g ( n ) π ( x )) n ∈ N is equidistributed in Z . Quantitative equidistribution on nilmanifolds.
We shall frequently use a quantitativeversion of Theorem 2.7 that was obtained in [23]. In order to state it we need to review somenotions that were introduced in [23].Given a nilmanifold X = G/ Γ, the horizontal torus is defined to be the compact Abeliangroup Z = G/ ([ G, G ]Γ). If X is connected, then Z is isomorphic to some finite dimensionaltorus T l . By π : X → H we denote the natural projection map. A horizontal character χ : G → C is a continuous homomorphism that satisfies χ ( gγ ) = χ ( g ) for every γ ∈ Γ. Sinceevery character annihilates G , every horizontal character factors through Z , and thereforecan be thought of as a character of the horizontal torus. Since Z is identifiable with a finitedimensional torus T l (we assume that X is connected), χ can also be thought of as a characterof T l , in which case there exists a unique κ ∈ Z l such that χ ( t Z l ) = e ( κ · t ), where · denotes theinner product operation. We refer to κ as the frequency of χ and k χ k = | κ | as the frequencymagnitude of χ . Example 2.8.
Let X be the Heisenberg nilmanifold (see Example 2.4). The map χ : G → C defined by χ (cid:16) x z y (cid:17) = e ( kx + ly ), where k, l ∈ Z , is a horizontal character of G . The map φ (cid:16) x z y (cid:17) = ( x Z , y Z ) induces an identification of the horizontal torus with T . Under thisidentification, χ is mapped to the character ˜ χ ( x Z , y Z ) = e ( kx + ly ) of T .Suppose that p : Z → R is a polynomial sequence of degree k , then p can be uniquelyexpressed in the form p ( n ) = P ki =0 (cid:0) ni (cid:1) α i where α i ∈ R . We define(5) k e ( p ( n )) k C ∞ [ N ] = max ≤ i ≤ k ( N i k α i k )where k x k = d ( x, Z ).Given N ∈ N , a finite sequence ( g ( n )Γ) ≤ n ≤ N is said to be δ - equidistributed , if (cid:12)(cid:12)(cid:12) N N X n =1 F ( g ( n )Γ) − Z X F dm X (cid:12)(cid:12)(cid:12) ≤ δ k F k Lip( X )12 or every Lipschitz function F : X → C , where k F k Lip( X ) = k F k ∞ + sup x,y ∈ X,x = y | F ( x ) − F ( y ) | d X ( x, y )for some appropriate metric d X on X . We can now state the equidistribution result that weshall use. It is a direct consequence of Theorem 2.9 in [23] (we have suppressed some distractingquantitative details that will be of no use for us): Theorem 2.9 ( Green & Tao [23] ). Let X = G/ Γ be a nilmanifold with G connected andsimply connected, and d ∈ N .Then for every small enough δ > there exist M = M X,d,δ ∈ R with the following property:For every N ∈ N , if g : Z → G is a polynomial sequence of degree at most d such that the finitesequence ( g ( n )Γ) ≤ n ≤ N is not δ -equidistributed, then for some non-trivial horizontal character χ with k χ k ≤ M we have (6) k χ ( g ( n )) k C ∞ [ N ] ≤ M, where χ is thought of as a character of the horizontal torus Z = T l and g ( n ) as a polynomialsequence in T l . Example 2.10.
It is instructive to interpret the previous result in some special case. Let X = T (with the standard metric), and suppose that the polynomial sequence on T is givenby p ( n ) = ( n d α + q ( n )) Z where d ∈ N , α ∈ R , and q ∈ Z [ x ] with deg( q ) ≤ d −
1. In thiscase Theorem 2.9 reads as follows: There exists
M > N ∈ N and δ small enough, if the finite sequence (cid:0) ( n d α + q ( n )) Z (cid:1) ≤ n ≤ N is not δ -equidistributed in T , then k kα k ≤ M/N d for some non-zero k ∈ Z with | k | ≤ M .3. A model equidistribution result
Before delving into the proof of the various equidistribution results on nilmanifolds we findit instructive to deal with a much simpler equidistribution problem on the circle. This modelproblem will motivate some of the ideas used later. We shall give a new proof for the followingresult:
Theorem 3.1 ( Boshernitzan [8]) . Let a ∈ H have polynomial growth.Then the sequence ( a ( n ) Z ) n ∈ N is equidistributed in T if and only if for every p ∈ Q [ t ] wehave | a ( t ) − p ( t ) | ≻ log t . Our strategy will be to use the Taylor expansion of the function a ( t ) to partition the rangeof the sequence ( a ( n )) n ∈ N into blocks that are approximately polynomial and then use classicalresults to estimate the corresponding exponential sums over these blocks. This argument can beadapted to the non-Abelian setup we are interested in, the main reason being that “Weyl type”sums involving polynomial block sequences of fixed degree on nilmanifolds can be effectivelyestimated using a rather sophisticated application of the van der Corput difference trick (thisis done in [23]), and with a bit of care one can piece together these estimates to get usableresults. The following simple example best illustrates our method: Example 3.2.
Suppose that a ( t ) = t log t . We shall show that the sequence ( n log n Z ) n ∈ N is equidistributed in T . Using Lemma 3.3 below, it suffices to show that for every non-zerointeger k we have lim N →∞ E N Let ( a ( n )) n ∈ N be a bounded sequence of complex numbers. Suppose that lim N →∞ (cid:0) E N ≤ n ≤ N + l ( N ) a ( n ) (cid:1) = 0 for some positive function l ( t ) with l ( t ) ≺ t . Then lim N →∞ E ≤ n ≤ N a ( n ) = 0 . Proof. We can cover the interval [1 , N ] by a union I N of non-overlapping intervals of the form[ k, k + l ( k )]. Since l ( t ) ≺ t and the sequence ( a ( n )) n ∈ N is bounded, we have thatlim N →∞ E ≤ n ≤ N a ( n ) = lim N →∞ E n ∈ I N a ( n ) . Using our assumption, one easily gets that the limit lim N →∞ E n ∈ I N a ( n ) is zero, finishing theproof. (cid:3) A modification of the argument used in Example 3.2 gives the following more general result: Lemma 3.4. Suppose that for some m ∈ N the function a ∈ C m +1 ( R + ) satisfies | a ( m +1) ( t ) | is decreasing , /t m ≺ a ( m ) ( t ) ≺ , ( a ( m +1) ( t )) m ≺ ( a ( m ) ( t )) m +1 . Then the sequence ( a ( n ) Z ) n ∈ N is equidistributed in T .Proof. It suffices to show that for every non-zero integer k we havelim N →∞ E ≤ n ≤ N e ( ka ( n )) = 0 . Since our assumptions are also satisfied for ka ( t ) in place of a ( t ) whenever k = 0, we canassume that k = 1.By Lemma 3.3 it is enough to show that the averages(9) E N ≤ n ≤ N + l ( N ) e ( a ( n ))converge to zero as N → ∞ for some positive function l ( t ) that satisfies l ( t ) ≺ t . Using the Taylor expansion of a ( t ) around the point t = N we get(10) a ( N + n ) = a ( N ) + na ′ ( N ) + · · · + n m m ! a ( m ) ( N ) + n m +1 ( m + 1)! a ( m +1) ( ξ n ) The choice of l ( t ) will depend on the function a ( t ). For example, if a ( t ) = t log t we need to assume that t / ≺ l ( t ) ≺ t / , and if a ( t ) = (log t ) we need to assume that t/ log t ≺ l ( t ) ≺ t/ √ log t . or some ξ n ∈ [ N, N + n ]. Since | a ( m +1) ( t ) | is decreasing we have | a ( m +1) ( ξ n ) | ≤ | a ( m +1) ( N ) | .It follows that if l ( t ) also satisfies ( l ( t )) m +1 a ( m +1) ( t ) ≺ , then the averages in (9) are equal to E ≤ n ≤ l ( N ) e (cid:16) a ( N ) + na ′ ( N ) + · · · + n m m ! a ( m ) ( N ) (cid:17) + o N →∞ (1) . Next, using Example 2.10 (or Weyl’s estimates; see e.g. [40]) we get that the last averagesconverge to zero as N → ∞ if1 ≺ ( l ( t )) m (cid:13)(cid:13)(cid:13) a ( m ) ( t ) (cid:13)(cid:13)(cid:13) = ( l ( t )) m | a ( m ) ( t ) | , the last equality being valid for every large t since | a ( m ) ( t ) | → N → ∞ as longas we can establish the existence of a function l ( t ) satisfying the following conditions(11) l ( t ) ≺ t and ( l ( t )) m +1 | a ( m +1) ( t ) | ≺ ≺ ( l ( t )) m | a ( m ) ( t ) | . Since by assumption ( a ( m +1) ( t )) mm +1 ≺ a ( m ) ( t ) and 1 /t m ≺ a ( m ) ( t ), we can indeed find afunction l ( t ) that satisfies max (cid:0) ( a ( m +1) ( t ) (cid:1) mm +1 , /t m ) ≺ / ( l ( t )) m ≺ | a ( m ) ( t ) | , and so (11)holds. This completes the proof. (cid:3) The previous lemma applies to a wide variety of functions. For example the functions(log t ) , t log t , t / , t √ t / satisfy the stated assumptions. In fact our next lemma showsthat Lemma 3.4 comes rather close to establishing Theorem 3.1. Lemma 3.5. Let a ∈ H have polynomial growth and satisfy | a ( t ) − p ( t ) | ≻ log t for every p ∈ R [ t ] .Then the function a ( t ) satisfies the assumptions of Lemma 3.4 for some m ∈ N . As aconsequence, the sequence ( a ( n ) Z ) n ∈ N is equidistributed in T .Proof. By Lemma 2.2 the function a ( t ) has type k or k + for some non-negative integer k . Weshall show that the assumptions of Lemma 3.4 are satisfied for m = k + 1. We can assume thatthe function a ( k ) ( t ) is eventually positive, if this not the case we work with the function − a ( t ).Since a ( t ) ≺ t k +1 , it follows from Corollary 2.3 that the functions a ( k +1) ( t ) and a ( k +2) ( t )converge to zero. Furthermore, since both functions are elements of H the convergence ismonotone.We show that a ( k +1) ( t ) ≻ /t k +1 . Suppose first that a ( t ) has type k + for some positiveinteger k . By Corollary 2.3 we have a ( k +1) ( t ) ≻ a ( t ) / ( t k +1 (log t ) ) ≫ / (cid:0) t (log t ) (cid:1) ≻ /t k +1 . Suppose now that a ( t ) has type 0 + , in which case we shall show that a ′ ( t ) ≻ /t . Arguingby contradiction, suppose that this is not the case. Since a ′ ( t ) is eventually positive, weconclude that for large values of t we have 0 ≤ a ′ ( t ) ≤ c /t for some non-negative constant c .Integrating we get that for large values of t we have 0 ≤ a ( t ) ≤ c log t + c for some constants c , c , contradicting our assumption | a ( t ) | ≻ log t . Lastly, suppose that a ( t ) has type k for somenon-negative integer k . Since a ( t ) stays away from polynomials, we conclude from Lemma 2.2that a ( t ) = p ( t ) + b ( t ), for some p ∈ R [ t ] of degree k , and some b ∈ H of type l + for somenon-negative integer l with l < k . Arguing as before, we conclude that b ( k +1) ( t ) ≻ /t k +1 .Since a ( k +1) ( t ) = b ( k +1) ( t ), we get a ( k +1) ( t ) ≻ /t k +1 . t remains to show that ( a ( k +2) ( t )) k +1 ≺ ( a ( k +1) ( t )) k +2 . By Lemma 2.1 we know that a ( k +2) ( t ) ≪ a ( k +1) ( t ) /t . Using this, and the previously established estimate a ( k +1) ( t ) ≻ /t k +1 , we get ( a ( k +2) ( t )) k +1 ≪ ( a ( k +1) ( t )) k +1 /t k +1 ≺ ( a ( k +1) ( t )) k +2 . This completes the proof. (cid:3) We now complete the proof of Theorem 3.1 Proof of Theorem 3.1. We first prove the sufficiency of the conditions. Combining Lemma 3.4and Lemma 3.5 we cover the case where | a ( t ) − p ( t ) | ≻ log t for every p ∈ R [ t ]. It remainsto deal with the case where a ( t ) = p ( t ) + e ( t ) for some p ∈ R [ t ] that has at least one non-constant coefficient irrational and e ( t ) ≪ log t . Since e ( n + 1) − e ( n ) → e ′ ( t ) → N as a union of non-overlapping intervals ( I m ) m ∈ N such that | I m | → ∞ and max n ,n ∈ I m | e ( n ) − e ( n ) | ≤ /m .Combining this with the fact that the sequence ( p ( n ) Z ) n ∈ N is well distributed in T (meaninglim N − M →∞ E M ≤ n ≤ N e ( kp ( n )) = 0 for every non-zero k ∈ Z ), we deduce that the sequence( a ( n )) n ∈ N is equidistributed in T .To prove the necessity of the conditions suppose that 1 ≺ a ( t ) ≪ log t ; the general case can beeasily reduced to this one. We shall show that the sequence ( a ( n )) n ∈ N cannot be equidistributedin T . The key property we shall use is that a ( n + 1) − a ( n ) ≪ /n . (This estimate is aconsequence of the mean value theorem and the estimate a ′ ( t ) ≤ c/t for large enough t whichcan be proved as in Lemma 2.1.) For convenience we assume that a ( n + 1) − a ( n ) < /n is satisfied for every n ∈ N , and the sequence ( a ( n ) Z ) n ∈ N is increasing. The general case issimilar. Arguing by contradiction, suppose that the sequence ( a ( n ) Z ) n ∈ N is equidistributed.Let n m be the first integer that satisfies a ( n m ) > m . Since a ( n m ) < a ( n ) < a ( n m ) + n/n m and a ( n m ) is very close to an integer for large m , approximately all the integers in [ n m , n m / { a ( n ) } ≤ / 2. Furthermore, because of the equidistribution property, for large m ∈ N ,approximately half of the integers in [1 , n m ] satisfy { a ( n ) } ≤ / 2. Therefore, for large m ∈ N ,approximately two thirds of the integers in [1 , n m / 2] satisfy { a ( n ) } ≤ / 2, contradicting ourequidistribution assumption. (cid:3) Single nil-orbits and Hardy sequences In this section we are going to prove Theorems 1.1 and 1.2.4.1. A reduction. We start with some initial maneuvers that will allow us to reduce Theo-rem 1.2 to a more convenient statement.First we give a result that enables us to translate distributional properties of sequences ofthe form ( b a ( n ) x ) n ∈ N to sequences of the form ( b [ a ( n )] x ) n ∈ N . Lemma 4.1. Let ( a ( n )) n ∈ N be a sequence of real numbers such that for every nilmanifold X = G/ Γ , with G connected and simply connected, and every b ∈ G , the sequence ( b a ( n ) Γ) n ∈ N is equidistributed in the nilmanifold ( b s Γ) s ∈ R .Then for every nilmanifold X = G/ Γ , every b ∈ G and x ∈ X , the sequence ( b [ a ( n )] x ) n ∈ N isequidistributed in the nilmanifold ( b n x ) n ∈ N .Proof. Let X = G/ Γ be a nilmanifold b ∈ G and x ∈ X . We start with some reductions.By using the lifting argument of Section 2.2, we can assume that G is connected and simplyconnected. Furthermore, by changing the base point and using the formula in Section 2.2, wecan assume that x = Γ. et X b be the nilmanifold ( b n Γ) n ∈ N and m X b be the corresponding normalized Haar measure.It suffices to show that for every F ∈ C ( X ) we have(12) lim N →∞ E ≤ n ≤ N F ( b [ a ( n )] Γ) = Z X b F dm X b . So let F ∈ C ( X ). To begin with, we use our assumption in the following case˜ X = ˜ G/ ˜Γ where ˜ G = R × G, ˜Γ = Z × Γ , and ˜ b = (1 , b ) . (Notice that ˜ G is connected and simply connected.) We conclude that for every ˜ H ∈ C ( ˜ X )(13) lim N →∞ E ≤ n ≤ N ˜ H (˜ b a ( n ) ˜Γ) = Z ˜ X ˜ b ˜ H dm ˜ X ˜ b , where ˜ X ˜ b is the nilmanifold ( s Z , b s Γ) s ∈ R , and m ˜ X ˜ b is the corresponding normalized Haar mea-sure.Next we claim that (13) can be applied for the function ˜ F : ˜ X → C defined by(14) ˜ F ( t Z , g Γ) = F ( b −{ t } g Γ) . We caution the reader that the function ˜ F may be discontinuous. The set of discontinuities of˜ F is a subset of the sub-nilmanifold { Z } × X . Near a point ( Z , g Γ) of { Z } × X the function˜ F comes close to the value F ( g Γ) or the value F ( b − g Γ). For δ > / F δ ∈ C ( ˜ X ) that agree with ˜ F on ˜ X δ = I δ × X , where I δ = { t Z : k t k ≥ δ } ,and are uniformly bounded by 2 k F k ∞ . Our assumption gives that the sequence ( a ( n ) Z ) n ∈ N isequidistributed in T . Since ˜ b a ( n ) = ( a ( n ) , b a ( n ) ), we deduce that ˜ b a ( n ) ˜Γ ∈ ˜ X δ for a set of n ∈ N with density 1 − δ . As a consequence,(15) lim sup N →∞ E ≤ n ≤ N | ˜ F (˜ b a ( n ) ˜Γ) − ˜ F δ (˜ b a ( n ) ˜Γ) | ≤ k F k ∞ δ. By assumption, (13) holds when one uses the functions ˜ F δ in place of the function ˜ H . Usingthese identities for every δ > 0, and letting δ → 0, we get using (15) that (13) also holds forthe discontinuous function ˜ F defined in (14) (to get that R ˜ F δ dm ˜ X ˜ b → R ˜ F dm ˜ X ˜ b we use that m ˜ X ˜ b ( { } × X ) = 0, which holds since { } × X is a proper sub-nilmanifold of ˜ X ˜ b ). This verifiesour claim.Applying (13) for the function ˜ F defined in (14), and noticing that˜ F (˜ b a ( n ) ˜Γ) = F ( b −{ a ( n ) } b a ( n ) Γ) = F ( b [ a ( n )] Γ) , we get lim n →∞ E ≤ n ≤ N F ( b [ a ( n )] Γ) = Z ˜ X ˜ b ˜ F dm ˜ X ˜ b = Z ˜ X ˜ b F ( b −{ s } g Γ) dm ˜ X ˜ b ( s Z , g Γ) . Since b −{ s } b s Γ = b [ s ] Γ, the map ( s Z , g Γ) → b −{ s } g Γ sends the nilmanifold ˜ X ˜ b onto the nilman-ifold X b = ( b n Γ) n ∈ N . On X b we define the measure m by letting Z X b F dm = Z ˜ X b F ( b −{ s } g Γ) dm ˜ X ˜ b ( s Z , g Γ)for every F ∈ C ( X b ). We claim that m = m X b . Indeed, a quick computation shows thatthe measure m is invariant under left translation by b . As it is well known, any rotation b isuniquely ergodic on its orbit closure X b , hence m = m X b . This establishes (12) and completesthe proof. (cid:3) he previous lemma shows that part ( ii ) of Theorem 1.2 follows from part ( i ). It turns outthat dealing with part ( i ) presents significant technical advantages (in fact we do not see howto establish part ( ii ) directly).Next we show that in order to prove part ( i ) of Theorem 1.2 it suffices to establish thefollowing result: Proposition 4.2. Let a ∈ H have polynomial growth and satisfy | a ( t ) − cp ( t ) | ≻ log t for every c ∈ R and p ∈ Z [ t ] . Let X = G/ Γ be a nilmanifold, with G connected and simply connected,and suppose that b ∈ G acts ergodically on X .Then the sequence ( b a ( n ) Γ) n ∈ N is equidistributed in X . To carry out this reduction we shall use the following lemma: Lemma 4.3. Let X = G/ Γ be a nilmanifold with G connected and simply connected.Then for every b ∈ G there exists s ∈ R such that the element b s acts ergodically on thenilmanifold ( b s Γ) s ∈ R .Proof. By Ratner’s theorem (see Section 2.2), we have ( b s Γ) s ∈ R = H/ ∆, where H is a connectedand simply connected closed subgroup of G that contains all the elements b s , s ∈ R , and∆ = H ∩ Γ. By Theorem 2.7 it suffices to check that b s acts ergodically on the horizontaltorus G/ ([ G, G ] | Γ), which we can assume to be T k for some k ∈ N . Equivalently, this amountsto showing that if β Z k ∈ T k , where β ∈ R k , then there exists s ∈ R such that ( ns β Z k ) n ∈ N =( sβ Z k ) s ∈ R . One can check (we omit the routine details) that it suffices to choose s suchthat the number 1 /s is rationally independent of any non-zero integer combination of thecoordinates of β . This completes the proof. (cid:3) Putting together Lemma 4.1 and Lemma 4.3 we get the advertised reduction: Proposition 4.4. In order to prove Theorem 1.2 it suffices to prove Proposition 4.2.Proof. Using Lemma 4.1, we see that part ( ii ) of Theorem 1.2 follows from part ( i ).To establish part ( i ) we argue as follows. Let b ∈ G . By Lemma 4.3 there exists non-zero s ∈ R such that the element b s acts ergodically on the nilmanifold ( b s Γ) s ∈ R . Using Proposi-tion 4.2 for the element b s and the function a ( s ) /s , we get that the sequence ( b a ( n ) Γ) n ∈ N isequidistributed in the nilmanifold ( b s Γ) s ∈ R . (cid:3) We now turn our attention to the proof of Proposition 4.2.4.2. Proof of Proposition 4.2. The following lemma is the key ingredient in the proof ofProposition 4.2: Lemma 4.5. Suppose that for some k ∈ N the function a ∈ C k +1 ( R + ) satisfies | a ( k +1) ( t ) | is decreasing , /t k ≺ a ( k ) ( t ) ≺ , ( a ( k +1) ( t )) k ≺ ( a ( k ) ( t )) k +1 . Let X = G/ Γ be a nilmanifold, with G connected and simply connected, and suppose that b ∈ G acts ergodically on X .Then the sequence ( b a ( n ) Γ) n ∈ N is equidistributed in X .Proof. Let F ∈ C ( X ) with zero integral. We want to show thatlim N →∞ E ≤ n ≤ N F ( b a ( n ) Γ) = 0 . y Lemma 3.3 it suffices to show that the averages(16) E N ≤ n ≤ N + l ( N ) F ( b a ( n ) Γ)converge to zero as N → ∞ for some positive function l ( t ) that satisfies l ( t ) ≺ t .Using the Taylor expansion of a ( t ) around the point x = N we have(17) a ( N + n ) = a ( N ) + na ′ ( N ) + · · · + n k k ! a ( k ) ( N ) + n k +1 ( k + 1)! a ( k +1) ( ξ n )for some ξ n ∈ [ N, N + n ]. Since | a ( k +1) ( t ) | is decreasing we have | a ( k +1) ( ξ n ) | ≤ | a ( k +1) ( N ) | . Itfollows that if the function l ( t ) satisfies( l ( t )) k +1 a ( k +1) ( t ) ≺ , then the averages (16) are equal to E ≤ n ≤ l ( N ) F (cid:16) b p N ( n ) Γ (cid:17) + o N →∞ (1)where p N ( n ) = a ( N ) + na ′ ( N ) + · · · + n k k ! a ( k ) ( N ) . Our objective now is to show that for every δ > 0, for large values of N , the finite sequence( b p N ( n ) Γ) ≤ n ≤ l ( N ) is δ -equidistributed in X . This would immediately imply that the averagesin (16) converge to zero as N → ∞ .So let δ > 0. Notice first that since b p N ( n ) = b ,N b n ,N · · · b n k k,N , where b i,N = b a ( i ) ( N ) /i ! for i = 0 , , . . . , k , for every fixed N ∈ N the sequence ( b p N ( n ) ) n ∈ N is a polynomial sequence in G . Since X = G/ Γ and G is connected and simply connected,we can apply Theorem 2.9 (for δ small enough). We conclude that if the finite sequence( b p N ( n ) Γ) ≤ n ≤ l ( N ) is not δ -equidistributed in X , then there exists a constant M (dependingonly on δ , X , and k ), and a horizontal character χ with k χ k ≤ M such that(18) (cid:13)(cid:13)(cid:13) χ ( b p N ( n ) ) (cid:13)(cid:13)(cid:13) C ∞ [ l ( N )] ≤ M. Let π ( b ) = ( β Z , . . . , β s Z ), where β i ∈ R , be the projection of b on the horizontal torus T s (notice that s is bounded by the dimension of X ). Since b acts ergodically on X the real numbers1 , β , . . . , β s must be rationally independent. For t ∈ R we have π ( b t ) = ( t ˜ β Z , . . . , t ˜ β s Z ) forsome ˜ β i ∈ R with ˜ β i Z = β i Z . As a consequence, we have χ ( b p N ( n ) ) = e (cid:16) p N ( n ) s X i =1 l i ˜ β i (cid:17) for some l i ∈ Z with | l i | ≤ M . From this, the definition of p N ( t ), and the definition of k·k C ∞ [ N ] (see (5)), we get that (cid:13)(cid:13)(cid:13) χ ( b p N ( n ) ) (cid:13)(cid:13)(cid:13) C ∞ [ l ( N )] ≥ ( l ( N )) k (cid:13)(cid:13)(cid:13) a ( k ) ( N ) β (cid:13)(cid:13)(cid:13) , here β is a non-zero (we use the rational independence of the ˜ β i ’s here) real number thatbelongs to the finite set B = n k ! s X i =1 l i ˜ β i : | l i | ≤ M o . Combining this estimate with (18), and using that (cid:13)(cid:13) a ( k ) ( N ) β (cid:13)(cid:13) = | a ( k ) ( N ) β | for large N (sinceby assumption a ( k ) ( t ) → l ( N )) k | a ( k ) ( N ) β | ≤ M for some β ∈ B . It follows that if the function l ( t ) satisfies1 ≺ ( l ( t )) k a ( k ) ( t ) , then (19) fails for large N , and as a result the finite sequence ( b p N ( n ) Γ) ≤ n ≤ l ( N ) is δ -equidistributedin X for every large N .Summarizing, we have shown that the averages (16) converge to 0 when N → ∞ , as long aswe can find a positive function l ( t ) that satisfies the following growth conditions l ( t ) ≺ t and ( l ( t )) k +1 a ( k +1) ( t ) ≺ ≺ ( l ( t )) k a ( k ) ( t ) . As in the proof of Lemma 3.4, one checks that the existence of such a function is guaranteedby our assumption, concluding the proof. (cid:3) Notice that by Lemma 3.5 the previous result applies to every function a ∈ H that haspolynomial growth and satisfies | a ( t ) − p ( t ) | ≻ log t for every p ∈ R [ t ]. In order to deal withthe remaining cases of Proposition 4.2 we need one more lemma. Its proof follows the samestrategy as in the proof of Lemma 4.5, so in order to avoid unnecessary repetition our argumentwill be rather sketchy. Lemma 4.6. Let a ∈ H satisfy a ( t ) = p ( t ) + e ( t ) , where e ( t ) ≪ log t , and p ∈ R [ t ] is not ofthe form cq ( t ) + d with c, d ∈ R and q ∈ Z [ t ] . Let X = G/ Γ be a nilmanifold, with G connectedand simply connected, and suppose that b ∈ G acts ergodically on X .Then the sequence ( b a ( n ) Γ) n ∈ N is equidistributed in X .Remark. Our argument can easily be adapted to cover every function e ∈ H that satisfies e ( t ) ≺ t , but the case treated suffices for our purposes. Proof. Arguing as in Lemma 4.5, it suffices to show that for every F ∈ C ( X ) with zero integral,the averages(20) E N ≤ n ≤ N + √ N F ( b a ( n ) Γ)converge to zero as N → ∞ .Using Lemma 2.1 we conclude that the function | e ′ ( t ) | is decreasing and e ′ ( t ) ≺ /t − ε forevery ε > 0. Using the mean value theorem we conclude that for n ∈ [1 , √ N ] we have e ( N + n ) = e ( N ) + o N →∞ (1)and as a result the averages in (20) are equal to E ≤ n ≤√ N F (cid:16) b p ( N + n )+ e ( N ) Γ (cid:17) + o N →∞ (1) . Hence, our proof will be complete if we show that for every δ > 0, and every large N , thefinite sequence ( b p ( N + n )+ e ( N ) Γ) ≤ n ≤√ N is δ -equidistributed in X . Suppose that this is not thecase. We are going to use Theorem 2.9 to derive a contradiction. The key property to be used s that for every non-zero real number β the polynomial βp ( t ) has at least one non-constantcoefficient irrational. Arguing as in Lemma 4.5, we deduce that there exists a constant M (depending only on δ , X and the degree of p ), and a finite set B of irrational numbers, suchthat for infinitely many positive integers N we have √ N k β k ≤ M for some β ∈ B . This is a contradiction and the proof is complete. (cid:3) Combining the last two lemmas it is now easy to prove Proposition 4.2. Proof of Proposition 4.2 (Conclusion of proof of Theorem 1.2). Using Lemmas 3.5 and 4.5 wecover the case where | a ( t ) − p ( t ) | ≻ log t for every p ∈ R [ t ]. The remaining cases are coveredby Lemma 4.6. (cid:3) Next we prove Theorem 1.1. It is a direct consequence of Theorem 1.2 and the followinglemma: Lemma 4.7. Let a ∈ H satisfy a ( t ) − p ( t ) → for some p ∈ R [ t ] .Then the sequences ( a ( n )) n ∈ N and ([ a ( n )]) n ∈ N are pointwise good for nilsystems.Proof. Let X = G/ Γ be a nilmanifold, with G connected and simply connected, and b ∈ G .The sequence ( a ( n )) n ∈ N is pointwise good for nilsystems if and only if the same holds for thesequence ( p ( n )) n ∈ N . Let p ( t ) = c + c t + · · · + c k t k for some non-negative integer k and c i ∈ R .Since b p ( n ) = b · b n · . . . · b n k k , where b i = b c i , we have that ( b p ( n ) ) n ∈ N is a polynomial sequencein G . It follows by Theorem 2.7 that the sequence ( p ( n )) n ∈ N is pointwise good for nilsystems.Next we deal with the sequence ([ a ( n )]) n ∈ N . Suppose first that p ( t ) − p (0) ∈ Q [ t ]. Then p ( t ) = r ˜ p ( t ) + c for some r ∈ N , c ∈ R , and ˜ p ∈ Z [ t ] with p (0) = 0. For i = 0 , . . . , r − a ( rn + i )] = q i ( n ) for some q i ∈ Z [ t ]. Using this, the result follows from Theorem 2.7.It remains to deal with the case where the polynomial p has an irrational non-constantcoefficient. We are going to use a strategy similar to the one used in the proof of Lemma 4.1.Let ˜ X = ˜ G/ ˜Γ where ˜ G = R × G, ˜Γ = Z × Γ , and ˜ b = (1 , b ) . Given F ∈ C ( X ) we define ˜ F : ˜ X → C by˜ F ( t Z , g Γ) = F ( b −{ t } g Γ) . (We caution the reader that ˜ F may not be continuous). Notice that F ( b [ a ( n )] Γ) = F ( b −{ a ( n ) } b a ( n ) Γ) = ˜ F (˜ b a ( n ) ˜Γ) , and as a result it suffices to show that the averages(21) E ≤ n ≤ N ˜ F (˜ b a ( n ) ˜Γ)converge as N → ∞ . We verify this as follows. For δ > / 2) there existfunctions ˜ F δ ∈ C ( ˜ X ) that agree with ˜ F on ˜ X δ = I δ × X , where I δ = (cid:8) t Z : k t k ≥ δ (cid:9) , and areuniformly bounded by 2 k F k ∞ . Since the polynomial p has a non-constant irrational coefficient,the sequence ( p ( n ) Z ) n ∈ N is equidistributed in T , and as a result ˜ b p ( n ) ˜Γ ∈ ˜ X δ for a set of n ∈ N with density 1 − δ . It follows that(22) lim sup N →∞ E ≤ n ≤ N | ˜ F (˜ b a ( n ) ˜Γ) − ˜ F δ (˜ b a ( n ) ˜Γ) | ≤ k F k ∞ δ. As shown in the first part of our proof, the sequence ( a ( n )) n ∈ N is pointwise good for nilsystems,hence the averages (21) converge when one uses the functions ˜ F δ in place of the function ˜ F . sing this and (22), we deduce that the averages in (21) form a Cauchy sequence, and hencethey converge as N → ∞ . This proves that the sequence ([ a ( n )]) n ∈ N is pointwise good fornilsystems and completes the proof. (cid:3) Proof of Theorem 1.1. The sufficiency of the conditions follows immediately from Theorem 1.2and Lemma 4.7, with the exception of the case where | a ( t ) − t/m | ≪ log t for some non-zerointeger m . As noticed in [9] (proof of Theorem 3.3), this last case is easily reduced to the case a ( t ) = t . In this particular instance the result is well known (e.g. [33]).The necessity of the conditions can be seen by working with rational rotations on the circle,for the details see [9]. (cid:3) Several nil-orbits and Hardy sequences In this section we shall prove Theorem 1.3. A crucial part of our argument will be differentthan the one used to prove of Theorem 1.2, so we find it instructive to start with a modelequidistribution problem that illustrates the key technical difference.5.1. A model equidistribution problem. We shall give yet another proof of the followingspecial case of Theorem 3.1:“If a ∈ H satisfies ( t log t ) ≺ a ( t ) ≺ t , then the sequence ( a ( n )) n ∈ N is equidistributed in T .”We shall take the following fact for granted:“If a ∈ H satisfies log t ≺ a ( t ) ≺ t , then the sequence ( a ( n )) n ∈ N is equidistributed in T .”So suppose that a ∈ H satisfies ( t log t ) ≺ a ( t ) ≺ t . It suffices to show that for everynon-zero k ∈ Z we have(23) lim N →∞ E ≤ n ≤ N e ( ka ( n )) = 0 . For convenience we assume that k = 1. For every fixed R ∈ N we have(24) E ≤ n ≤ RN e ( a ( n )) = E ≤ n ≤ N (cid:0) E ≤ r ≤ R e ( a ( Rn + r )) (cid:1) + o N →∞ (1) . For n = 1 , , . . . , we use the Taylor expansion of a ( t ) around the point t = Rn . Since a ′′ ( t ) → r ∈ [1 , R ] that a ( Rn + r ) = a ( Rn ) + ra ′ ( Rn ) + o n →∞ ; R (1) . It follows that the averages in (24) are equal to(25) E ≤ n ≤ N A R,n + o N →∞ ; R (1) , where A R,n = E ≤ r ≤ R e ( a ( Rn ) + ra ′ ( Rn )) . For fixed ε > E ≤ n ≤ N | A R,n | as follows E ≤ n ≤ N ( n : k a ′ ( Rn ) k≤ ε · | A R,n | ) + E ≤ n ≤ N ( n : k a ′ ( Rn ) k >ε · | A R,n | ) = Σ ,R,N,ε + Σ ,R,N,ε . We estimate Σ ,R,N,ε . By Lemma 2.1 we have that log t ≺ a ′ ( Rt ) ≺ t . It follows that thesequence ( a ′ ( Rn ) Z ) n ∈ N is equidistributed in T , and as a consequence | ≤ n ≤ N : k a ′ ( Rn ) k ≤ ε | N = 2 ε + o N →∞ ; R (1) . Therefore, Σ ,R,N,ε ≤ ε + o N →∞ ; R (1).We estimate Σ ,R,N,ε . We have | A R,n | = | E ≤ r ≤ R e ( ra ′ ( Rn )) | . e estimate the geometric series in the standard fashion; computing the sum and using theestimate | sin πt | ≥ k t k , we find that (whenever a ′ ( Rn ) is not an integer) | A R,n | ≤ R k a ′ ( Rn ) k . It follows that Σ ,R,N,ε ≤ / (2 Rε ).Combining the estimates for Σ ,R,N,ε and Σ ,R,N,ε we get E ≤ n ≤ N | A R,n | ≤ ε + 12 Rε + o N →∞ ; R (1) . Letting first N → ∞ , then R → ∞ , and then ε → 0, we getlim R →∞ lim N →∞ E ≤ n ≤ N | A R,n | = 0 . As explained before, this implies (23) and completes the proof.5.2. A reduction. As was the case with the proof of Theorem 1.2, we start with some initialmaneuvers that enable us to reduce Theorem 1.3 to a more convenient statement. Since thisstep can be completed with straightforward modifications of the arguments used in Section 4.1,we omit the proofs.First notice that in order to prove Theorem 1.3 we can assume that X = · · · = X ℓ = X .Indeed, consider the nilmanifold ˜ X = X × · · · × X ℓ . Then ˜ X = ˜ G/ ˜Γ, where ˜ G = G × · · · × G ℓ is connected and simply connected, ˜Γ = Γ × · · · × Γ ℓ is a discrete cocompact subgroup of ˜ G ,each b i can be thought of as an element of ˜ G , and each x i as an element of ˜ X . Lemma 5.1. Let ( a ( n )) n ∈ N , . . . , ( a ℓ ( n )) n ∈ N be sequences of real numbers. Suppose that forevery nilmanifold X = G/ Γ , with G connected and simply connected, and every b , . . . , b ℓ ∈ G ,the sequence ( b a ( n )1 Γ , . . . , b a ℓ ( n ) ℓ Γ) n ∈ N is equidistributed in the nilmanifold ( b s Γ) s ∈ R × · · · × ( b sℓ Γ) s ∈ R .Then for every nilmanifold X = G/ Γ , every b , . . . , b ℓ ∈ G , and x , . . . , x ℓ ∈ X , the sequence ( b [ a ( n )]1 x , . . . , b [ a ℓ ( n )] ℓ x ℓ ) n ∈ N is equidistributed in the nilmanifold ( b n x ) n ∈ N × · · · × ( b nℓ x ℓ ) n ∈ N . The previous lemma shows that part ( ii ) of Theorem 1.3 follows from part ( i ). Lemma 5.2. Let X = G/ Γ be a nilmanifold with G connected and simply connected.Then for every b , . . . , b ℓ ∈ G there exists s ∈ R such that for i = 1 , . . . , ℓ the element b s i acts ergodically on the nilmanifold ( b si Γ) s ∈ R . Using Lemmas 5.1 and 5.2, we see as in section Section 4.1, that Theorem 1.3 reduces toproving the following result: Proposition 5.3. Suppose that the functions a ( t ) , . . . , a ℓ ( t ) belong to the same Hardy field,have different growth rates, and satisfy t k log t ≺ a i ( t ) ≺ t k +1 for some k = k i ∈ N .Then given nilmanifolds X i = G i / Γ i , i = 1 , . . . , ℓ , with G i connected and simply connected,and elements b , . . . , b ℓ ∈ G i acting ergodically on X i , the sequence ( b a ( n )1 Γ , . . . , b a ℓ ( n ) ℓ Γ ℓ ) n ∈ N is equidistributed in the nilmanifold X × · · · × X ℓ . .3. Proof of Proposition 5.3. Since there is a key technical difference in the proofs ofProposition 5.3 and Proposition 4.2, we are going to give all the details. We are going toadapt the proof technique of the model equidistribution result of Section 5.1 to our particularnon-Abelian setup. Proof of Proposition 5.3. For convenience of exposition we assume that X = · · · = X ℓ = X ,the proof in the general case is similar. Let F ∈ C ( X ℓ ) with zero integral. We want to showthat(26) lim N →∞ E ≤ n ≤ N F ( b a ( n )1 Γ , . . . , b a ℓ ( n ) ℓ Γ) = 0 . For every fixed R ∈ N we have(27) E ≤ n ≤ RN F ( b a ( n )1 Γ , . . . , b a ℓ ( n ) ℓ Γ) = E ≤ n ≤ N (cid:0) E ≤ r ≤ R F ( b a ( nR + r )1 Γ , . . . , b a ℓ ( nR + r ) ℓ Γ) (cid:1) + o N →∞ ; R (1) . For n = 1 , , . . . , we use the Taylor expansion of the functions a i ( t ) around the point t = Rn .Since t k i log t ≺ a i ( t ) ≺ t k i +1 for some k i ∈ N , Lemma 2.1 gives that a ( k i +1) i ( t ) → 0. Hence, for r ∈ [1 , R ] we have that a i ( Rn + r ) = p i,R,n ( r ) + o n →∞ ; R (1) , where(28) p i,R,n ( r ) = a i ( Rn ) + ra ′ i ( Rn ) + · · · + r k i k i ! a ( k i ) i ( Rn ) . It follows that the averages in (27) are equal to(29) E ≤ n ≤ N A R,n + o N →∞ ; R (1) , where A R,n = E ≤ r ≤ R F ( b p ,R,n ( r )1 Γ , . . . , b p ℓ,R,n ( r ) ℓ Γ) . Our objective now is to show that for every δ > 0, for all large values of R , the finite sequence( b p ,R,n ( r )1 , . . . , b p ℓ,R,n ( r ) ℓ ) ≤ r ≤ R is δ -equidistributed in X ℓ for most values of n . This will enableus to show that the averages in (29) converge to zero as N → ∞ .So let δ > 0. As in the proof of Proposition 4.2 we verify that for fixed R, n ∈ N thesequence ( b p ,R,n ( r )1 , . . . , b p ℓ,R,n ( r ) ℓ ) r ∈ N is a polynomial sequence in G k . Since X ℓ = G ℓ / Γ ℓ , and G ℓ is connected and simply connected, we can apply Theorem 2.9 (for small δ ). We get that ifthe finite sequence ( b p ,R,n ( r )1 Γ , . . . , b p ℓ,R,n ( r ) ℓ Γ) ≤ r ≤ R is not δ -equidistributed in X ℓ , then thereexists a constant M (depending only on δ , X , and the k i ’s), and a non-trivial horizontalcharacter χ of X ℓ , with k χ k ≤ M , and such that(30) (cid:13)(cid:13)(cid:13) χ ( b p ,R,n ( r )1 , . . . , b p ℓ,R,n ( r ) ℓ ) (cid:13)(cid:13)(cid:13) C ∞ [ R ] ≤ M. For i = 1 , . . . , ℓ , let π ( b i ) = ( β i, Z , . . . , β i,s Z ), where β i,j ∈ R , be the projection of b i on thehorizontal torus T s of X (notice that s is bounded by the dimension of X ). Since each b i acts ergodically on X , the set of real numbers { , β i, , . . . , β i,s } is rationally independent for i = 1 , . . . , ℓ . For t ∈ R we have π ( b ti ) = ( t ˜ β i, Z , . . . , t ˜ β i,s Z ) for some ˜ β i,j ∈ R with ˜ β i,j Z = β i,j Z .As a consequence(31) χ ( b p ,R,n ( r )1 , . . . , b p ℓ,R,n ( r ) ℓ ) = e (cid:16) s X i =1 (cid:0) p i,R,n ( n ) s X j =1 l i,j ˜ β i,j (cid:1)(cid:17) or some l i,j ∈ Z with | l i,j | ≤ M . Let k min = min { k , . . . , k ℓ } and k max = max { k , . . . , k ℓ } . Itfollows from (28), (31), and the definition of k·k C ∞ [ R ] (see (5)), that there exists k ∈ N with k min ≤ k ≤ k max such that (cid:13)(cid:13)(cid:13) χ ( b p ,R,n ( r )1 , . . . , b p ℓ,R,n ( r ) ℓ ) (cid:13)(cid:13)(cid:13) C ∞ [ R ] ≥ R k (cid:13)(cid:13)(cid:13) X i ∈ I a ( k ) i ( Rn ) β i (cid:13)(cid:13)(cid:13) , where the sum ranges over those i ∈ { , . . . , ℓ } that satisfy k i = k , and the β i ’s are real numbers,not all of them zero (we used here that χ is non-trivial and the rational independence of the˜ β i,j ’s), that belong to the finite set B = ℓ [ i =1 n k ! s X j =1 l i,j ˜ β i,j : | l i,j | ≤ M o . Combining this estimate with (30) gives(32) (cid:13)(cid:13)(cid:13) X i ∈ I a ( k ) i ( Rn ) β i (cid:13)(cid:13)(cid:13) ≤ MR k for some β i ∈ B .We are now ready to estimate the average E ≤ n ≤ N | A R,n | . Given ε > E ≤ n ≤ N | A R,n | = E ≤ n ≤ N ( S ,R,ε ( n ) · | A R,n | ) + E ≤ n ≤ N ( S ,R,ε ( n ) · | A R,n | ) = Σ ,R,N,ε + Σ ,R,N,ε , where S ,R,ε = n n ∈ N : (cid:13)(cid:13)(cid:13) X i ∈ I a ( k ) i ( Rx ) β i (cid:13)(cid:13)(cid:13) ≤ ε for some β i ∈ B, not all of them 0 o , S ,R,ε = N \ S ,R,ε . We estimate Σ ,R,N,ε . Using Lemma 2.1 and our assumptions, we conclude that log t ≺ a ( k ) i ( t ) ≺ t for i ∈ I . Furthermore, since the functions a i ( t ) for i ∈ I have different growthrates and belong to the same Hardy field, we deduce that the functions a ( k ) i ( t ) for i ∈ I havedifferent growth rates. It follows thatlog t ≺ b R ( t ) = X i ∈ I a ( k ) i ( Rt ) β i ≺ t. Since b R ∈ H and log t ≺ b R ( t ) ≺ t , we get (e.g. using Theorem 3.1) that for every R ∈ N thesequence ( b R ( n ) Z ) n ∈ N is equidistributed in T . Hence, | ≤ n ≤ N : k b R ( n ) k ≤ ε | N = 2 ε + o N →∞ ; R (1) . It follows that | Σ ,R,N,ε | ≤ k F k ∞ ε + o N →∞ ; R (1) . We estimate Σ ,R,N,ε . Notice that for n ∈ S ,R,ε we have (cid:13)(cid:13)(cid:13)P i ∈ I a ( k ) i ( Rn ) β i (cid:13)(cid:13)(cid:13) ≥ ε . Asa result, if R is large enough, then (32) fails, and as a consequence the finite sequence( b p ,R,n ( r )1 x , . . . , b p ℓ,R,n ( r ) ℓ x ℓ ) ≤ r ≤ R is δ -equidistributed in X . Hence, if R is large enough, then | A R,n | ≤ δ for every n ∈ S ,R,ε . Therefore, for every N ∈ N we have | Σ ,R,N,ε | ≤ δ + o R →∞ (1) . Putting the previous estimates together we find E ≤ n ≤ N | A R,n | ≤ k F k ∞ ε + δ + o N →∞ ; R (1) + o R →∞ (1) . etting N → ∞ , then R → ∞ , and then ε, δ → 0, we deduce thatlim R →∞ lim sup N →∞ E ≤ n ≤ N | A R,n | = 0 , or equivalently thatlim R →∞ lim sup N →∞ E ≤ n ≤ N (cid:12)(cid:12)(cid:12) E ≤ r ≤ R F ( b [ a ( Rn + r )]1 Γ , . . . , b [ a ℓ ( Rn + r )] ℓ Γ) (cid:12)(cid:12)(cid:12) = 0 . Combining this with (27) gives (26), completing the proof. (cid:3) Random sequences of sub-exponential growth In this section we shall prove Theorem 1.4. In what follows, when we introduce a nilpotentLie group G or a nilmanifold X , we assume that it comes equipped with a Mal’cev basis andthe corresponding (right invariant) metric d G or d X that was introduced in [23]. When thereis no danger of confusion we are going to denote d G or d X with d . We denote by B M the ballin G of radius M , that is B M = { g ∈ G : d ( g, id G ) ≤ M } . A reduction. We start with some initial maneuvers that will allow us to reduce Theo-rem 1.4 to a more convenient statement.We remind the reader of our setup. We are given a sequence ( X n ( ω )) n ∈ N of 0 − P ( { ω ∈ Ω : X n ( ω ) = 1 } ) = σ n , where ( σ n ) n ∈ N is a decreasingsequence of real numbers satisfying lim n →∞ nσ n = ∞ . Our objective is to show that almostsurely the averages(33) 1 N N X n =1 F ( b a n ( ω ) Γ)converge as N → ∞ for every nilmanifold X = G/ Γ, function F ∈ C ( X ), and element b ∈ G .We caution the reader that the set of probability 1 for which the averages (33) converge hasto be independent of the nilmanifold X = G/ Γ, the function F ∈ C ( X ), and the element b ∈ G .On the other hand, since up to isomorphism there exist countably many nilmanifolds X (seefor example [12]), and since the space C ( X ) is separable, it suffices to prove that for every fixednilmanifold X = G/ Γ and F ∈ C ( X ) the averages (33) converge almost surely for every b ∈ G .Furthermore, since G is a countable union of balls, it suffices to verify the previous statementwith B M in place of G for every M > A ( N, ω ) N X n =1 X n ( ω ) F ( b n Γ)where A ( N, ω ) = |{ n ∈ { , . . . , N } : X n ( ω ) = 1 }| . Since the expectation of X n is σ n , by thestrong law of large numbers we almost surely have that A ( N, ω ) /w ( N ) → , where w ( N ) = P Nn =1 σ n . It therefore suffices to work with the averages1 w ( N ) N X n =1 X n ( ω ) F ( b n Γ) . e shall establish convergence of these averages by comparing them with the averages1 w ( N ) N X n =1 σ n F ( b n Γ) . Notice that these last averages can be compared with the averages1 N N X n =1 F ( b n Γ)which, as we have mentioned repeatedly before, are known to be convergent.Up to this point we have reduced matters to showing that for every nilmanifold X = G/ Γ, F ∈ C ( X ), and M > 0, we almost surely have(34) lim N →∞ w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) = 0for every b ∈ B M .Next we show that we can impose a few extra assumptions on the nilmanifold X , and thefunction F ∈ C ( X ). Using the lifting argument of Section 2.2 we see that every sequence( F ( b n Γ)) n ∈ N can be represented in the form ( ˜ F (˜ b n ˜Γ)) n ∈ N for some nilmanifold ˜ X = ˜ G/ ˜Γ, with˜ G connected and simply connected, ˜ F ∈ C ( ˜ X ), and ˜ b ∈ ˜ G . Therefore, when proving (34) wecan assume that the nilmanifold X has the form G/ Γ, where the group G is connected andsimply connected. Furthermore, since the set Lip( X ), of Lipschitz functions F : X → C , isdense in C ( X ) in the uniform topology, an easy approximation argument shows that it sufficesto prove (34) for F ∈ Lip( X ).Summarizing, we have reduced Theorem 1.4 to proving: Theorem 6.1. Let ( X n ( ω )) n ∈ N be a sequence of − valued independent random variableswith P ( { ω ∈ Ω : X n ( ω ) = 1 } ) = σ n , where ( σ n ) n ∈ N is a decreasing sequence of real numberssatisfying lim n →∞ nσ n = ∞ . Let X = G/ Γ be a nilmanifold, with G connected and simplyconnected, F ∈ Lip ( X ) , and M > .Then almost surely we have lim N →∞ max b ∈ B M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 where w ( N ) = P Nn =1 σ n .Remark. We shall not use the fact that the convergence to zero is uniform; only the indepen-dence of the set of full measure on the set B M will be used.To prove Theorem 6.1 we are going to extend an argument used by Bourgain in [10] (wherethe case X = T was covered). A more detailed version of this argument can be found in [38].Since several steps of [38] carry over verbatim to our case we are only going to spell out thedetails of the genuinely new steps.6.2. A key ingredient. In this subsection we shall prove the following key result: Proposition 6.2. Let X = G/ Γ be a nilmanifold, with G connected and simply connected, and M be a positive real number. hen there exists k = k ( G, M ) ∈ N with the following property: for every N ∈ N , there exists B N,M ⊂ B M with | B N,M | = N k , and such that for every F ∈ Lip ( X ) with k F k Lip ( X ) ≤ , andsequence of real numbers ( c n ) n ∈ N with norm bounded by , we have (35) max b ∈ B M (cid:12)(cid:12)(cid:12) N X n =1 c n F ( b n Γ) (cid:12)(cid:12)(cid:12) = max b ∈ B N,M (cid:12)(cid:12)(cid:12) N X n =1 c n F ( b n Γ) (cid:12)(cid:12)(cid:12) + o N →∞ (1) . The proof of this result ultimately relies on the fact that multiplication on a nilpotentLie group is given by polynomial mappings. To make this precise we shall use a convenientcoordinate system, the proof of its existence can be found in [23] (for example).For every connected and simply connected Lie group G there exist a non-negative integer m (we call m the dimension of G ) and a continuous isomorphism φ from ( G, · ) to ( R m , · ) withmultiplication defined as follows: If u = ( u , . . . , u m ) and v = ( v , . . . , v m ), then for i = 1 , . . . , m the i -th coordinate of u · v has the form u i + v i + P i ( u , . . . , u i − , v , . . . , v i − )where P i : R i − × R i − → R is a polynomial of degree at most i . It follows that the i -thcoordinate of u n has the form nu i + Q i ( u , . . . , u i − , n )where Q i : R i − × R → R is a polynomial.We shall use the following result (Lemma A.4 in [23]): Lemma 6.3 ( Green & Tao [23] ). Let G be a connected and simply connected nilpotent Liegroup of dimension m .Then there exists k = k ( G ) ∈ N such that for every K > we have K − k | u − v | ≤ d ( g, h ) ≤ K k | u − v | , for every g, h ∈ G , and u = φ ( g ) , v = φ ( h ) ∈ R m that satisfy | u | , | v | ≤ K , where | · | denotesthe sup-norm in R m . Using this, we are going to show: Lemma 6.4. Let G be a connected and simply connected nilpotent Lie group.Then there exists k = k ( G ) ∈ N such that for every M > we have d ( g n , h n ) ≪ G,M n k d ( g, h ) for every n ∈ N and g, h ∈ B M .Proof. We first establish the corresponding estimate in “coordinates”. Suppose that the dimen-sion of G is m . Let φ ( g ) = u = ( u , . . . , u m ) and φ ( h ) = v = ( v , . . . , v m ) satisfy | u | , | v | ≤ K .Using the multiplication formula in local coordinates we deduce that | ( u n ) i − ( v n ) i | ≤ i X j =1 | ( u j − v j ) | | R j ( u , . . . , u j − , v , . . . , v j − , n ) | for some polynomials R j : R j − × R j − × R → R of degree of degree depending only on G . Ifwe consider R j as a polynomial of a single variable n , then its coefficients depend polynomially n the parameters u i , v i (which are bounded by K ) and the structure constants of the Mal’cevbasis of G . Hence, | R j ( u, v, n ) | ≪ G,K n l j for some l j = l j ( G ) ∈ N . It follows that | ( u n ) i − ( v n ) i | ≪ G,K n k i X j =1 | ( u j − v j ) | for some k = k ( G ). As a consequence(36) | u n − v n | ≪ G,K n k | u − v | for some k = k ( G ).To finish the proof, we use (36) to deduce an analogous estimate for elements of G withthe metric d . We argue as follows. First, using Lemma 6.3 we conclude that if g ∈ B M , then | u | ≪ G,M 1. As a result, (36) gives that(37) | u n | ≪ G,M n k for every g ∈ B M . Next, notice that by Lemma 6.3 there exists k = k ( G ) ∈ N such that forevery K > K − k | u − v | ≤ d ( g, h ) ≤ K k | u − v | for every g, h ∈ G that satisfy | u | , | v | ≤ K (remember that u = φ ( g ) , v = φ ( h ) ∈ R m ).Combining the estimates (36), (37), and (38) we get d ( g n , h n ) ≪ G,M n k k | u n − v n | ≪ G,M n k + k k | u − v | ≪ G,M n k + k k d ( g, h ) . This establishes the advertised estimate with k = k + k k . (cid:3) Proof of Proposition 6.2. By Lemma 6.4 we get that there exists k = k ( G ) such that(39) d G ( g n , h n ) ≪ G,M n k d G ( g, h )for every g, h ∈ B M . For every K ∈ N there exist K m points that form an 1 /K -net for the set[0 , m with the sup-norm. Combining this with Lemma 6.3 we get that there exists k = k ( G )with the following property: for every N ∈ N there exists an 1 /N k +2 net of B M consisting of N k points.Let B N,M be any such 1 /N k +2 -net of B M . By construction, | B N,M | = N k for some k that depends only on G . Furthermore, for every b ∈ B M there exists b N ∈ B N,M such that d G ( b, b N ) ≤ /N k +2 . It follows from (39) thatmax ≤ n ≤ N d G ( b n , b nN ) ≪ G,M N k d G ( b, b N ) ≤ /N . Therefore, max ≤ n ≤ N d X ( b n Γ , b nN Γ) ≪ G,M /N . Using this, we deduce (35) (with o N →∞ (1) = k c n k ∞ k F k Lip( X ) /N ≤ /N ), completing theproof. (cid:3) .3. Proof of Theorem 6.1. We give a sketch of the proof of Theorem 6.1. The missingdetails can be extracted from [38].Without loss of generality we can assume that k F k Lip( X ) ≤ k = k ( G, M ) ∈ N and a subset B N,M of B M with | B N,M | = N k such that(40)max b ∈ B M (cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12) = max b ∈ B N,M (cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12) + o N →∞ (1) . Since the cardinality of B N,M is a power of N that depends only on G and M , it followsthat | B N,M | / log N is bounded by some constant that depends only on G and M . Hence, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) max b ∈ B N,M (cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L log N (Ω) ≪ G,M (41) max b ∈ B N,M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L log N (Ω) . Furthermore, arguing exactly as in [38] (pages 40-41), it can be shown that for every sequenceof complex numbers ( c n ) n ∈ N with k c n k ∞ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) w ( N ) N X n =1 ( X n ( ω ) − σ n ) c n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L log N (Ω) ≪ s log Nw ( N ) . Combining (40), (41), and (42) (with c n = F ( b n Γ)), gives(43) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) max b ∈ B M (cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L log N (Ω) ≪ G,M s log Nw ( N ) + o N →∞ (1) . Next we make use of the following simple lemma: Lemma 6.5. Let ( Y k ) k ∈ N be a sequence of bounded, complex-valued random variables on aprobability space (Ω , Σ , P ) .Then almost surely we have lim sup k →∞ | Y k ( ω ) |k Y k k L log k (Ω) ≤ e where e is the Euler number. Combining this lemma with (43), we conclude that for every nilmanifold X = G/ Γ, with G connected and simply connected, and F ∈ Lip( X ) with k F k Lip( X ) ≤ 1, there exists a setΩ F,G,M of probability 1, such that for every ω ∈ Ω F,G,M we havemax b ∈ B M (cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12) ≪ ω,G,M s log Nw ( N ) + o N →∞ (1) . ince by assumption log N/w ( N ) → 0, we get that for every nilmanifold X , F ∈ Lip( X ), and M > 0, we almost surely havelim N →∞ max b ∈ B M (cid:12)(cid:12)(cid:12) w ( N ) N X n =1 ( X n ( ω ) − σ n ) F ( b n Γ) (cid:12)(cid:12)(cid:12) = 0 . This completes the proof of Theorem 6.1, and finishes the proof of Theorem 1.4. References [1] L. Auslander, L. Green, F. Hahn. Flows on homogeneous spaces. With the assistance of L. Markus andW. Massey, and an appendix by L. Greenberg, Annals of Mathematics Studies , , Princeton UniversityPress, Princeton, N.J. (1963).[2] V. Bergelson, B. Host, B. 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