Multiple ergodic theorems for arithmetic sets
aa r X i v : . [ m a t h . D S ] N ov MULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS
NIKOS FRANTZIKINAKIS AND BERNARD HOST
Abstract.
We establish results with an arithmetic flavor that generalize the polyno-mial multidimensional Szemerédi theorem and related multiple recurrence and conver-gence results in ergodic theory. For instance, we show that in all these statements wecan restrict the implicit parameter n to those integers that have an even number of dis-tinct prime factors, or satisfy any other congruence condition. In order to obtain theserefinements we study the limiting behavior of some closely related multiple ergodicaverages with weights given by appropriately chosen multiplicative functions. Theseaverages are then analysed using a recent structural result for bounded multiplicativefunctions proved by the authors. Introduction and main results
Introduction.
The multi-dimensional Szemerédi theorem of H. Furstenberg andY. Katznelson [14], stated in ergodic terms, asserts that if T , . . . , T ℓ are commutingmeasure preserving transformations acting on the same probability space ( X, X , µ ) , thenfor every A ∈ X with µ ( A ) > there exists n ∈ N such that µ ( T − n A ∩ · · · ∩ T − nℓ A ) > . More recently, T. Tao [23] established mean convergence for some closely related multipleergodic averages by showing that for F , . . . , F ℓ ∈ L ∞ ( µ ) the averages N N X n =1 T n F · · · T nℓ F ℓ converge in the mean as N → ∞ . In this article we are interested in studying variantsof such statements where the parameter n is restricted to certain subsets of the integersof arithmetic nature. For instance, we are interested in knowing whether the previousresults remain true when we restrict the parameter n to those integers that have an even(or an odd) number of distinct prime factors. More generally, do they hold if we restrict n to those integers that have a mod b distinct prime factors for some a, b ∈ N ?We answer these questions affirmatively. In order to give some model results in this in-troductory section (extensions and related statements appear in Section 1.2) we introducesome notation. For a, b ∈ N we let S a,b consist of those n ∈ N whose number of distinctprime factors is congruent to a mod b . It can be shown that for every a ∈ { , . . . , b − } the set S a,b has density /b (see the second remark after Proposition 2.10). Theorem A.
Let T , . . . , T ℓ be commuting measure preserving transformations actingon the same probability space ( X, X , µ ) . Then for every A ∈ X with µ ( A ) > we have µ ( T − n A ∩ · · · ∩ T − nℓ A ) > for a set of n ∈ S a,b with positive lower density.We deduce from this ergodic statement, via the correspondence principle of H. Fursten-berg (see Section 1.3) that every set of integers with positive upper density containsarbitrarily long arithmetic progressions with common difference taken from the set S a,b ; Mathematics Subject Classification.
Primary: 37A45; Secondary: 05D10, 11B30, 11N37, 28D05.
Key words and phrases.
Multiple ergodic averages, multiple recurrence, multiplicative functions,higher degree uniformity. similar statements also hold for the multidimensional Szeméredi theorem, polynomialvariants of it (see Theorem 1.5), and for any shift of the sets S a,b . Theorem B.
Let T , . . . , T ℓ be commuting measure preserving transformations actingon the same probability space ( X, X , µ ) . Then for all F , . . . , F ℓ ∈ L ∞ ( µ ) , the averages(1) N X n ∈ S a,b ∩ [1 ,N ] T n F · · · T nℓ F ℓ converge in L ( µ ) . In fact, the limit is equal to lim N →∞ bN P Nn =1 T n F · · · T nℓ F ℓ .In order to analyze the averages (1) we do not use the theory of characteristic factors;even for averages of the form N P Nn =1 T n F · · · T nℓ F ℓ this theory is very intricate and notyet developed to an extent that facilitates our study. Instead, we proceed by comparingthe averages (1) with the averages bN P Nn =1 T n F · · · T nℓ F ℓ and show that the differenceconverges to in L ( µ ) . To do this, we work with some weighted multiple ergodic aver-ages with weights given by suitably chosen multiplicative functions. Then the assertedconvergence to is a consequence of the next statement. Theorem C.
Let f ∈ M conv be a multiplicative function (see definition in Section 1.2).If T , . . . , T ℓ are commuting measure preserving transformations acting on the same prob-ability space ( X, X , µ ) , then for all F , . . . , F ℓ ∈ L ∞ ( µ ) , the averages(2) N N X n =1 f ( n ) · T n F · · · T nℓ F ℓ converge in L ( µ ) . Furthermore, the limit is zero if f is aperiodic (see definition inSection 1.2)Let us briefly explain how we derive Theorem A and B from Theorem C. For b ∈ N we let ζ be a root of unity of order b and let f be the multiplicative function defined by f ( p k ) = ζ for all primes p and all k ∈ N . Note that(3) S a,b ( n ) = 1 b b − X j =0 ζ − aj ( f ( n )) j . It follows from Corollary 2.10 that for j = 1 , . . . , b − the multiplicative function f j isaperiodic. Combining this with (3) and Theorem C we get that the difference(4) N N X n =1 S a,b ( n ) · T n F · · · T nℓ F ℓ − bN N X n =1 T n F · · · T nℓ F ℓ converges to in L ( µ ) . Using this and the aforementioned convergence result of T. Taowe deduce Theorem B. Furthermore, since the difference (4) converges to in L ( µ ) ,letting F = . . . = F ℓ = A where µ ( A ) > , integrating over X , and using the multidi-mensional Szemerédi theorem of Furstenberg and Katznelson, we deduce Theorem A.The proof of Theorem C depends upon a deep structural result for multiplicativefunctions proved by the authors in [11]. Roughly speaking, it asserts that the generalmultiplicative function that is bounded by can be decomposed in two terms, one that isapproximately periodic and another that contributes negligibly to the averages (2). Theapproximately periodic component vanishes if the multiplicative function is aperiodic.In the general case, a careful analysis of the contribution of the structured componentallows us to conclude the proof of Theorem C.We note that if one is only interested in the weak convergence of the averages (2), analternate (and arguably simpler) approach is to use a decomposition result for multiplecorrelation sequences from [9]; we discuss this approach in more detail in Section 3.5. ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 3
Recurrence and convergence results.
Our main results cover a vastly moregeneral setting than the one described in the previous subsection. In order to facilitateexposition we introduce some definitions and notation. We start with some notions fromnumber theory related to multiplicative functions.
Definition.
A function f : N → C is called multiplicative if f ( mn ) = f ( m ) f ( n ) whenever ( m, n ) = 1 . We let M := { f : N → C multiplicative such that | f ( n ) | ≤ for every n ∈ N } and M conv := n f ∈ M : lim N →∞ N N X n =1 f ( an + b ) exists for every a, b ∈ N o . We say that f ∈ M is aperiodic if lim N →∞ N P Nn =1 f ( an + b ) = 0 for every a, b ∈ N .If a multiplicative function takes real values, then a well known theorem of E. Wirs-ing [25] states that it has a mean value; furthermore it belongs to M conv . But there existcomplex valued multiplicative functions that do not have a mean value, for example if f ( n ) = n it for some t = 0 ; then N P Nn =1 f ( n ) = N it it + o N →∞ (1) . Lending terminologyfrom [18], it can be shown that f ∈ M conv unless f ( n ) “pretends” to be n it χ ( n ) for some t ∈ R and Dirichlet character χ . Necessary and sufficient conditions for checking when amultiplicative function belongs to the set M conv can be found in Theorem 2.9 below.Next we introduce some notions from ergodic theory. Definition. • A bounded sequence of complex numbers ( w ( n )) is a good universalweight for polynomial multiple mean convergence if for every ℓ, m ∈ N , probability space ( X, X , µ ) , invertible commuting measure preserving transformations T , . . . , T ℓ : X → X ,functions F , . . . , F m ∈ L ∞ ( µ ) , and polynomials p i,j : Z → Z , i = 1 , . . . ℓ , j = 1 , . . . , m ,the averages N N X n =1 w ( n ) · ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m converge in L ( µ ) , where Q ℓi =1 S i denotes the composition S ◦ · · · ◦ S ℓ . A set of integers S is a set of polynomial multiple mean convergence if the sequence ( S ( n )) is a gooduniversal weight for polynomial multiple mean convergence. • A set of integers S is set of polynomial multiple recurrence if for every ℓ, m ∈ N ,probability space ( X, X , µ ) , invertible commuting measure preserving transformations T , . . . , T ℓ : X → X , set A ∈ X with µ ( A ) > , and polynomials p i,j : Z → Z , i = 1 , . . . ℓ , j = 1 , . . . , m , with p i,j (0) = 0 , we have(5) µ (cid:0) ( ℓ Y i =1 T p i, ( n ) i ) A ∩ · · · ∩ ( ℓ Y i =1 T p i,m ( n ) i ) A (cid:1) > for a set of n ∈ S with positive lower density. Remark.
All the statements in this article refer to sets of integers S with positive density,hence there is no need to normalize the relevant averages. Furthermore, although we al-ways work under the assumption that the measure preserving transformations commute,with some additional work our arguments extend to the case where the transformationsgenerate a nilpotent group; we discuss this in more detail in Section 3.4.Our first result generalizes Theorem C from the introduction. ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 4
Theorem 1.1.
Let f ∈ M conv be a multiplicative function. Then the sequence ( f ( n )) isa good universal weight for polynomial multiple mean convergence. Furthermore, if f isaperiodic, then the corresponding weighted ergodic averages converge to in L ( µ ) . Remark.
Examples of periodic systems show that one does not have convergence if f / ∈ M conv . Nevertheless, following the method of [10] it is possible to show that for every f ∈ M there exist t ∈ R and a slowly varying sequence η ( n ) (meaning max x ≤ n ≤ x | η ( n ) − η ( x ) | → as x → ∞ ), where both t and η depend only on f , such that the correspondingweighted ergodic averages multiplied by N − it e ( − η ( n )) converge in the mean.If S is the set of square-free integers, applying Theorem 1.1 for the multiplicativefunction f := S we deduce that S is a set of polynomial multiple mean convergence. Next, we give generalizations of Theorems A and B from the introduction. They areconsequences of a result that we state next. A subset of the unit interval or the unit circleis called
Riemann-measurable if its indicator function is a Riemann integrable function.It is known that this condition is equivalent to having boundary of Lebesgue measure . Theorem 1.2.
Let f ∈ M be a multiplicative function taking values on the unit circle. (i) If for some k ∈ N the function f takes values on k -th roots of unity and K is anon-empty subset of its range, then f − ( K ) is a set of polynomial multiple meanconvergence. If in addition f j is aperiodic for j = 1 , . . . , k − , then any shift ofthe set f − ( K ) is a set of polynomial multiple recurrence. (ii) If f j is aperiodic for all j ∈ N and K is a Riemann-measurable subset of the unitcircle of positive measure, then any shift of the set f − ( K ) is a set of polynomialmultiple recurrence and polynomial multiple mean convergence.In fact, under the aperiodicity assumptions of part ( i ) or ( ii ) we get that the set f − ( K ) is Gowers uniform (see definition in Section 2.2). We denote by ω ( n ) the number of distinct prime factors of an integer n and by Ω( n ) the number of prime factors of n counted with multiplicity. We let S ω,A,b := { n ∈ N : ω ( n ) ≡ a mod b for some a ∈ A } and similarly we define S Ω ,A,b . Corollary 1.3.
For every b ∈ N and A ⊂ { , . . . , b − } non-empty, any shift of thesets S ω,A,b and S Ω ,A,b is a set of polynomial multiple recurrence and polynomial multiplemean convergence. In fact, all these sets are Gowers uniform. Remark.
As the polynomial a n b takes values in S Ω ,a,b , the multiple recurence property(5) for some n ∈ S Ω ,a,b can be inferred from the polynomial Szemerédi theorem. Thisargument does not apply for non-trivial shifts of the sets S Ω ,a,b and S ω,a,b . On the otherhand, see Section 3.6 for an alternate argument that can be used to prove “linear” multiplerecurrence statements by finding IP k -patterns within any shift of S ω,a,b and S Ω ,a,b .For α ∈ R and A ⊂ [0 , / we let S ω,A,α := { n ∈ N : k ω ( n ) α k ∈ A } , S Ω ,A,α := { n ∈ N : k Ω( n ) α k ∈ A } , where k x k := d ( x, Z ) for x ∈ R . Corollary 1.4.
For every irrational α and Riemann-measurable set A ⊂ [0 , / ofpositive measure, any shift of the sets S ω,A,α and S Ω ,A,α is a set of polynomial multiplerecurrence and mean convergence. In fact, all these sets are Gowers uniform. Remark.
Similar results hold if S ω,A,α and S Ω ,A,α are defined using fractional parts. This can also be deduced directly from Theorem 2.2 by approximating S in density by periodic sets. ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 5
Combinatorial implications.
We give some combinatorial implications of theprevious multiple recurrence results. We define the upper Banach density d ∗ ( E ) of a set E ⊂ Z ℓ as d ∗ ( E ) := lim sup | I |→∞ | E ∩ I || I | , where the lim sup is taken over all parallelepipeds I ⊂ Z ℓ whose side lengths tend to infinity. We use the following modification of thecorrespondence principle of H. Furstenberg (the proof can be found in [4]): Furstenberg Correspondence Principle ([13]) . Let ℓ ∈ N and E ⊂ Z ℓ . There exista probability space ( X, X , µ ) , invertible commuting measure preserving transformations T , . . . , T ℓ : X → X , and a set A ∈ X with µ ( A ) = d ∗ ( E ) , such that d ∗ (( E − ~n ) ∩ . . . ∩ ( E − ~n m )) ≥ µ (cid:0) ( ℓ Y i =1 T n i, i ) A ∩ . . . ∩ ( ℓ Y i =1 T n i,m i ) A (cid:1) for all m ∈ N and ~n j = ( n ,j , . . . , n ℓ,j ) ∈ Z ℓ for j = 1 , . . . , m .Using this result and Corollaries 1.3 and 1.4, we immediately deduce the following: Theorem 1.5.
Let ℓ, m ∈ N , ~q , . . . , ~q m : Z → Z ℓ be polynomials with ~q i (0) = ~ for i = 1 , . . . , m , and let E ⊂ Z ℓ with d ∗ ( E ) > . Then the set (cid:8) n ∈ N : d ∗ (cid:0) ( E − ~q ( n )) ∩ . . . ∩ ( E − ~q m ( n )) (cid:1) > (cid:9) intersects any shift of the sets S ω,A,b , S Ω ,A,b , S ω,A,α , S Ω ,A,α (we assume that the set A satisfies the assumptions of Corollaries 1.3 and 1.4) on a set of positive lower density. Pointwise convergence.
Variants of the previous mean convergence results thatdeal with pointwise convergence of multiple ergodic averages are, for the most part,completely open. The situation is only clear for the single term ergodic averages(6) N N X n =1 f ( n ) · F ( T n x ) where F ∈ L ∞ ( µ ) . If f is the Möbius or the Liouville function, then it is shown in [1,Proposition 3.1] that these averages converge pointwise to . This is done by combiningthe spectral theorem with some classical quantitative bounds of H. Davenport [7] foraverages of the form N P Nn =1 f ( n ) e( nt ) ; note though that such bounds do not hold forgeneral aperiodic multiplicative functions.For more general f ∈ M conv we can treat pointwise convergence of the averages (6)as follows: If F is orthogonal to the Kronecker factor of the system, then for every f ∈ M the averages (6) converge pointwise to . We can establish this by combining anorthogonality criterion of I. Kátai [21] with a result of J. Bourgain [3]; the former impliesthat the averages (6) converge to zero if N P nn =1 F ( T an x ) · F ( T bn x ) → for every a, b ∈ N with a = b and the latter confirms this property pointwise almost everywhere when F isorthogonal to the Kronecker factor of the system. On the other hand, suppose that F is an eigenfunction with eigenvalue e ( α ) for some α ∈ R . If α is irrational, then using aresult of H. Daboussi [5, 6] we deduce that for all f ∈ M the averages (6) converge to pointwise. If α is rational, then they converge for all f ∈ M conv . Furthermore, in eithercase, the averages (6) converge to if f is aperiodic. Combining the above and usingan approximation argument, we get that if f ∈ M conv , then the averages (6) convergepointwise, and they converge to if f is aperiodic. We deduce from this that all the sets S ω,A,b , S Ω ,A,b , S ω,A,α , S Ω ,A,α defined in Section 1.2 are good for pointwise convergenceof single ergodic averages and under the obvious non-degeneracy assumptions for the set A we get that for ergodic systems the normalized averages converge to R F dµ for all F ∈ L ∞ ( µ ) . Furthermore, an approximation argument allows to extend these results toall F ∈ L ( µ ) .We record here a related open problem regarding multiple ergodic averages with arith-metic weights (perhaps the simplest of this type). ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 6
Problem.
Let f ∈ M conv be a multiplicative function. Is it true that for every measurepreserving system ( X, X , µ, T ) , and every F, G ∈ L ∞ ( µ ) , the averages (7) N N X n =1 f ( n ) · F ( T n x ) · G ( T n x ) converge pointwise? Do they converge to if f is aperiodic? When f = 1 the averages (7) converge pointwise by a result of J. Bourgain [3]. Ingeneral, the problem is open even when f is the Liouville function; that is, it is not knownwhether the averages N P Nn =1 S ( n ) · F ( T n x ) · G ( T n x ) converge pointwise when S isthe set of integers that have an even number of prime factors counted with multiplicity.1.5. Notation and conventions.
For reader’s convenience, we gather here some nota-tion that we use throughout the article. We denote by N the set of positive integers andby P the set of prime numbers. For N ∈ N we let Z N := Z /N Z and [ N ] := { , . . . , N } .We let e( t ) := e πit . With o N →∞ (1) we denote a quantity that converges to when N → ∞ and all other implicit parameters are fixed. Given transformations T i : X → X , i = 1 , . . . , ℓ , with Q ℓi =1 T i we denote the composition T ◦ · · · ◦ T ℓ . We use the letter f to denote a multiplicative function. A Dirichlet character, denoted by χ , is a completelymultiplicative function that is periodic and satisfies χ (1) = 1 .1.6. Acknowledgement.
We would like to thank M. Lemanczyk for helpful remarks.2.
Main ingredients
Multiple recurrence and convergence results.
In order to prove our mainresults we will use some well known multiple recurrence and convergence results in ergodictheory. The first is the polynomial Szemerédi theorem stated in ergodic terms.
Theorem 2.1 (Bergelson, Leibman [4]) . The set of positive integers is a set of polynomialmultiple recurrence.
The second is a mean convergence result for multiple ergodic averages.
Theorem 2.2 (Walsh [24]) . The set of positive integers is a set of polynomial multiplemean convergence.
Gowers norms and estimates.
We recall the definition of the U s -Gowers unifor-mity norms from [16]. Definition (Gowers norms on Z N [16]) . Let N ∈ N and a : Z N → C . For s ∈ N the Gowers U s ( Z N ) -norm k a k U s ( Z N ) of a is defined inductively as follows: For every t ∈ Z N we write a t ( n ) := a ( n + t ) . We let k a k U ( Z N ) := (cid:12)(cid:12)(cid:12) N X n ∈ Z N a ( n ) (cid:12)(cid:12)(cid:12) and for every s ∈ N we let k a k U s +1 ( Z N ) := (cid:16) N X t ∈ Z N k a · a t k s U s ( Z N ) (cid:17) / s +1 . If a : N → C is an infinite sequence, then by k a k U s ( Z N ) we denote the U s ( Z N ) -norm ofthe restriction of a to the interval [ N ] , thought of as a function on Z N .The following uniformity estimates will be used to analyze the limiting behavior ofmultiple ergodic averages. ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 7
Lemma 2.3 (Uniformity estimates [12, Lemma 3.5]) . Let ℓ, m ∈ N , ( X, X , µ ) be a prob-ability space, T , . . . , T ℓ : X → X be invertible commuting measure preserving transfor-mations, F , . . . , F m ∈ L ∞ ( µ ) be functions bounded by , and p i,j : Z → Z , i ∈ { , . . . , ℓ } , j ∈ { , . . . , m } , be polynomials. Let w : N → C be a sequence of complex numbers thatis bounded by . Then there exists s ∈ N , depending only on the maximum degree of thepolynomials p i,j and the integers ℓ and m , such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N X n =1 w ( n ) · ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) ≪ (cid:13)(cid:13) [ N ] · w (cid:13)(cid:13) U s ( Z sN ) + o N (1) . Furthermore, the implicit constant and the o N (1) term depend only on the integer s . We also need the following result which follows from Lemma A.1 and A.2 in [11].
Lemma 2.4.
Let s ≥ be an integer and ε, κ > . Then there exist δ > and N ∈ N ,such that for all integers N, e N with N ≤ N ≤ e N ≤ κN , every interval J ⊂ [ N ] , and f : Z e N → C with | f | ≤ , the following implication holds:if k f k U s ( Z e N ) ≤ δ, then k J · f k U s ( Z N ) ≤ ε. Gowers uniform sets.
We introduce here the notion of a Gowers uniform subsetof the integers that was used repeatedly in the statements of our main results.
Definition.
We say that a set of positive integers S is Gowers uniform if there exists apositive constant c such that lim N →∞ k S − c k U s ( Z N ) = 0 for every s ∈ N . Remark.
If such a constant exists, then applying the defining property for s = 1 givesthat c is the density of the set S .If S is a Gowers uniform set, then applying Lemma 2.3 for the weight w ( n ) = S ( n ) − c , n ∈ N , and combining the definition of Gowers uniformity with Lemma 2.4, we deducethat for V n := ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m , we have | S ∩ [ N ] | X n ∈ S ∩ [ N ] V n − N N X n =1 V n → L ( µ ) . Using this, the recurrence result of Theorem 2.1, and the convergence result of Theo-rem 2.2, we deduce the following:
Proposition 2.5.
Suppose that the set S ⊂ N is Gowers uniform. Then any shift of S is a set of polynomial multiple recurrence and polynomial multiple mean convergence. Structure theorem for multiplicative functions and aperiodicity.
Next westate a structural result from [11] that is crucial for our study. We first introduce somenotation from [11, Section 3]. Given f : N → C and N ∈ N we let f N := f · [ N ] and whenever appropriate we consider f N as a function in Z N . The Fourier transform c f N of f N is defined by c f N ( ξ ) := 1 N N X n =1 f ( n ) e (cid:0) − n ξN (cid:1) for ξ ∈ Z N . ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 8
By a kernel on Z N we mean a non-negative function on Z N with average . For everyprime number N and θ > , in [11] we defined two positive integers Q = Q ( θ ) and V = V ( θ ) , and for N > QV , a function φ N,θ : Z N → C given by the formula φ N,θ := X ξ ∈ Ξ N,θ (cid:0) − (cid:13)(cid:13)(cid:13) QξN (cid:13)(cid:13)(cid:13)
NQV (cid:1) e (cid:0) n ξN (cid:1) where(8) Ξ N,θ := n ξ ∈ Z N : (cid:13)(cid:13)(cid:13) QξN (cid:13)(cid:13)(cid:13) < QVN o . Then for every ξ ∈ Z N we have(9) d φ N,θ ( ξ ) = − (cid:13)(cid:13)(cid:13) QξN (cid:13)(cid:13)(cid:13)
NQV if ξ ∈ Ξ N,θ ;0 otherwise. Theorem 2.6 (Structure theorem for multiplicative functions [11, Theorem 8.1]) . Let s ∈ N and ε > . Then there exist a real number θ > and N ∈ N , depending on s and ε only, such that for every prime N ≥ N , every f ∈ M admits the decomposition f ( n ) = f N, st ( n ) + f N, un ( n ) , for every n ∈ [ N ] , where f N, st , f N, un : [ N ] → C are bounded by and respectively and satisfy: (i) f N, st = f N ∗ φ N,θ where φ N,θ is the kernel on Z N defined previously and theconvolution product is defined in Z N ; (ii) k f N, un k U s ( Z N ) ≤ ε . Remark.
In [11] this result is stated with f multiplied by a certain cut-off. The cut-offis not needed for our purposes and exactly the same argument proves the current version.We think of f N, st and f N, un as the structured and uniform component of f respectively.From this point on we assume that N > QV . When convenient we identify Z N with the set { , . . . , N − } and we denote by ( a, b ) mod N the set that consists of those ξ ∈ Z N such that ξ + kN ∈ ( a, b ) for some k ∈ Z . Note that ξ ∈ Ξ N,θ if and only if thereexists p ∈ Z such that ξ − pQ N ∈ ( − V, V ) mod N . Hence, Ξ N,θ = Q − [ p =0 (cid:0) pQ N − V, pQ N + V (cid:1) mod N, We may choose to include or omit the endpoints of each interval (if they are integers),since for these values the Fourier transform of φ N,θ is . Hence, we can assume that(10) Ξ N,θ = Q − [ p =0 Ξ N,θ,p where for p = 0 , . . . , Q − we have Ξ N,θ,p := (cid:8)(cid:4) pQ N (cid:5) + j mod N : − V < j ≤ V (cid:9) . Note that for fixed
N > QV and θ > the sets Ξ N,θ,p , p = 0 , . . . , Q − , are disjoint,each of cardinality V , hence | Ξ N,θ | = 2 QV . Furthermore, if N ≡ Q , then(11) Ξ N,θ,p = (cid:8) pQ ( N −
1) + j mod N : − V < j ≤ V (cid:9) . Restricting N to a specific congruence class mod Q is needed in the proof of Lemma 3.3.We will also use the following consequence of Theorem 2.6; it can be derived bycombining Theorem 2.4 and Lemma A.1 in [11]. ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 9
Theorem 2.7 (Aperiodic multiplicative functions [11]) . Let f ∈ M be an aperiodicmultiplicative function and for N ∈ N let I N be a subinterval of [ N ] . Then lim N →∞ k I N · f k U s ( Z N ) = 0 for every s ∈ N . Halász’s theorem and consequences.
To facilitate exposition, we define thedistance between two multiplicative functions as in [18]:
Definition. If f, g ∈ M we let D : M × M → [0 , ∞ ] be given by D ( f, g ) = X p ∈ P p (cid:0) − Re (cid:0) f ( p ) g ( p ) (cid:1)(cid:1) Remark.
Note that if | f | = | g | = 1 , then D ( f, g ) = P p ∈ P p | f ( p ) − g ( p ) | . It can be shown (see [18]) that D satisfies the triangle inequality D ( f, g ) ≤ D ( f, h ) + D ( h, g ) . Also for all f , f , g , g ∈ M we have (see [17, Lemma 3.1])(12) D ( f f , g g ) ≤ D ( f , g ) + D ( f , g ) . We will also use that if f ∈ M is such that for some c in the unit circle we have f ( p ) = c for all primes p , then D ( f, n it ) = ∞ for every t = 0 . In particular we have D (1 , n it ) = ∞ for every t = 0 . Using this and the triangle inequality, one deduces that for f ∈ M wehave D ( f, n it ) < ∞ for at most one value of t ∈ R . We will use the following celebratedresult of G. Halász: Theorem 2.8 (Halász [19]) . A multiplicative function f ∈ M has mean value zero ifand only if for every t ∈ R we either have D ( f, n it ) = ∞ or f (2 k ) = − ikt for all k ∈ N . Remark.
Since f is aperiodic if and only if for every Dirichlet character χ the multi-plicative function f · χ has mean value zero, this result also gives necessary and sufficientconditions for aperiodicity.Another consequence of the mean value theorem of Halász (see for example [8, Theo-rem 6.3]) is the following result that gives easy to check necessary and sufficient conditionsfor a multiplicative function to have a mean value (not necessarily zero). Theorem 2.9.
Let f ∈ M . Then f has a mean value if and only if we either have (i) D ( f, n it ) = ∞ for every t ∈ R , or (ii) P p ∈ P p (1 − f ( p )) converges, or (iii) For some t ∈ R we have D ( f, n it ) < ∞ and f (2 k ) = − ikt for all k ∈ N . Remark.
Since f ∈ M conv if and only if for every Dirichlet character χ the multiplicativefunction f · χ has a mean value, this result also gives necessary and sufficient conditionsfor a multiplicative function to be in M conv .We deduce from the previous results the following criterion that will be used in theproof of Theorem 1.2 and the proof of Corollary 1.3 and 1.4: Proposition 2.10.
Let f ∈ M . (i) If for some k ∈ N , f takes values on the k -th roots of unity, then f ∈ M conv . (ii) If α ∈ R is not an integer and f ( p ) = e ( α ) for all p ∈ P , then f is aperiodic. Remarks. • Sharper results can be obtained using a theorem of R. Hall [20] and theargument in [17, Corollary 2]. For instance, it can be shown that if f ( p ) takes values ina finite subset of the unit disc for all p ∈ P , then f ∈ M conv , and if in addition f ( p ) = 1 for all p ∈ P , then f is aperiodic. • By taking averages in (3) and using that Part ( ii ) of the previous result impliesaperiodicity of f j for j = 1 , . . . , b − , we deduce that d ( S a,b ) = b . ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 10
Proof.
We prove ( i ) . It suffices to show that for every Dirichlet character χ the multi-plicative function f · χ has a mean value. Note that χ takes values on roots of unity offixed order for all but finitely many primes (on which it is ). Hence, it suffices to showthat if for some m ∈ N a multiplicative function g takes values on the m -th roots of unityfor all but finitely many primes, then g has a mean value. So let g be such a multiplicativefunction. If D ( g, n it ) = ∞ for every t ∈ R , then we are done by Theorem 2.9. Supposethat there exists t ∈ R such that D ( g, n it ) < ∞ . Using that D (1 , n it ) = ∞ for every t = 0 we have by (12) that m D ( g, n it ) ≥ D ( g m , n imt ) = D (1 , n imt ) + O (1) = ∞ , for every t = 0 , where the lower bound follows from (12). Hence, t = 0 , which implies that X p ∈ P p (cid:0) − Re( g ( p )) (cid:1) < ∞ . Since g takes finitely many values on the unit disc, there exists c > such that for all p ∈ P we either have g ( p ) = 1 or − Re( g ( p )) ≥ c . Hence, X p ∈ P ,g ( p ) =1 p < ∞ . Since | − g ( p ) | ≤ for all p ∈ P , we deduce that X p ∈ P | − g ( p ) | p < ∞ . Theorem 2.9 again gives that g has a mean value, completing the proof of ( i ) .We prove ( ii ) . Using the remark following Theorem 2.8, it suffices to show that forevery t ∈ R and Dirichlet character χ we have D ( f · χ, n it ) = ∞ . So let χ be a Dirichletcharacter. Then there exists m ∈ N such that ( χ ( p )) m = 1 for all but a finite number ofprimes p . Then using (12) we get m D ( f · χ, n it ) ≥ D ( f m · χ m , n imt ) = D ( f m , n imt ) + O (1) = ∞ , for every t = 0 , where the last distance is infinite since f m is constant on primes and mt = 0 . It remainsto show that D ( f · χ,
1) = ∞ . Suppose that χ has period d . Since χ (1) = 1 , we have χ ( n ) = 1 whenever n ≡ d , and since f ( p ) = e ( α ) for all p ∈ P , we have D ( f · χ, ≥ (1 − cos(2 πα )) · X p ∈ P ∩ ( d Z +1) p = ∞ , where we used that (1 − cos(2 πα )) = 0 because α / ∈ Z and the divergence of the lastseries follows from Dirichlet’s theorem. This completes the proof of ( ii ) . (cid:3) Proof of main results
Proof of Theorem 1.1.
We start with a few elementary lemmas.
Lemma 3.1.
Let ( V n ) be a bounded sequence of elements of a normed space such that lim N →∞ N N X n =1 V n = V. For N ∈ N let e N > N be integers such that the limit β := lim N →∞ N e N exists. Then forevery α ∈ R we have N N X n =1 e (cid:0) n α e N (cid:1) V n = c · V where c = β R β e (cid:0) αy (cid:1) dy if β = 0 and c = 1 if β = 0 . ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 11
Proof.
For n ∈ N let S n := n X k =1 (cid:0) V k − V ) . Our assumption gives that S n /n → as n → ∞ . Using partial summation we get thatthe modulus of the average N N X n =1 e (cid:0) n α e N (cid:1) (cid:0) V n − V ) is at most N (cid:0) N − X n =2 k S n k (cid:12)(cid:12) e (cid:0) ( n + 1) α e N (cid:1) − e (cid:0) n α e N (cid:1)(cid:12)(cid:12) + k S N k (cid:1) + o N →∞ (1) . Let ε > . Since S n /n → as n → ∞ , we have | S n | ≤ εn for every sufficiently large n ,and thus the last expression is bounded by N (cid:0) N − X n =2 εn | πα | e N + εN (cid:1) + o N →∞ (1) ≤ ( β | πα | + 1) ε + o N →∞ (1) . Since ε is arbitrary, we get that lim N →∞ N N X n =1 e (cid:0) n α e N (cid:1) (cid:0) V n − V (cid:1) = 0 . Note also that if β > , then lim N →∞ N N X n =1 e (cid:0) n α e N (cid:1) = 1 β Z β e (cid:0) αy (cid:1) dy and the previous limit is if β = 0 . Combining the above we get the asserted claim. (cid:3) Next we show that the discrete Fourier transform of elements of M conv along certain“major arc” frequencies converges. Lemma 3.2.
Let f ∈ M conv . Let Q ∈ N , p, ξ ′ ∈ Z , and ξ N = pQ N + ξ ′ Q , N ∈ N . Then the averages (13) N N X n =1 f ( n ) e (cid:0) − n ξ N N (cid:1) converge.Proof. Notice first that the expression in (13) is equal to Q Q X r =1 e (cid:0) − r pQ (cid:1) ⌊ N/Q ⌋ ⌊ N/Q ⌋ X n =1 f ( Qn + r ) e (cid:0) − ( Qn + r ) ξ ′ QN (cid:1) + o N →∞ (1) . Hence, it suffices to show that for every fixed
Q, ξ ′ , and r ∈ [ Q ] , the averages ⌊ N/Q ⌋ ⌊ N/Q ⌋ X n =1 f ( Qn + r ) e (cid:0) − ( Qn + r ) ξ ′ QN (cid:1) converge. Since e( − rξ ′ / ( QN )) → as N → ∞ , it suffices to show that the averages(14) ⌊ N/Q ⌋ ⌊ N/Q ⌋ X n =1 f ( Qn + r ) e (cid:0) − n ξ ′ N (cid:1) ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 12 converge. Since f ∈ M conv , we have that the averages N N X n =1 f ( Qn + r ) converge and using Lemma 3.1 for a ( n ) := f ( Qn + r ) , N replaced by ⌊ N/Q ⌋ , and e N replaced by N , we deduce the needed convergence for the averages (14). This completesthe proof. (cid:3) Next we analyze the asymptotic behavior of the relevant ergodic averages with weightsgiven by the structured components (defined by Theorem 2.6) of an element of M conv . Lemma 3.3.
Let θ > , Q ∈ N , and f ∈ M conv . For N ∈ N let e N > N be a primethat satisfies e N ≡ Q and suppose that the limit β := lim N →∞ N e N exists. Let f e N, st := f e N ∗ φ e N,θ where φ e N,θ is defined by (9) and the convolution product is defined in Z e N . Then for every probability space ( X, X , µ ) , invertible commuting measure preservingtransformations T , . . . , T ℓ : X → X , functions F , . . . , F m ∈ L ∞ ( µ ) , and polynomials p i,j : Z → Z , i = 1 , . . . ℓ , j = 1 , . . . , m , the averages (15) N N X n =1 f e N, st ( n ) · ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m converge in L ( µ ) .Proof. By the definition of f e N, st and φ e N,θ we have that f e N, st ( n ) = X ξ ∈ Ξ e N,θ c f e N ( ξ ) d φ e N,θ ( ξ )e (cid:0) n ξ e N (cid:1) , n ∈ [ e N ] , where Ξ e N,θ is defined in (8). Since e N ≡ Q , it follows from (10) and (11) that for e N > QV if ξ ∈ Ξ e N,θ , then ξ can be uniquely represented as ξ = pQ e N + ξ ′ Q for some p ∈ { , . . . , Q − } and ξ ′ ∈ Ξ ′ p,θ where for p = 0 , , . . . , Q − we have Ξ ′ p,θ := (cid:8) − p + jQ : − V < j ≤ V (cid:9) . Hence, it suffices to show that the averages (15) satisfy the asserted asymptotic whenthe (finite) sequence ( f e N, st ( n )) n ∈ [ N ] in (15) is replaced by the sequence c f e N (cid:0) pQ e N + ξ ′ Q (cid:1) · d φ e N,θ (cid:0) pQ e N + ξ ′ Q (cid:1) · e (cid:0) n ( pQ + ξ ′ Q e N ) (cid:1) , n ∈ [ N ] , for all p ∈ { , . . . , Q − } and ξ ′ ∈ Ξ ′ p,θ .Recall that if f : Z e N → C and ξ ∈ Z e N , then b f ( ξ ) = e N P n ∈ Z e N f ( n ) e (cid:0) − n ξ e N (cid:1) . ByLemma 3.2 the limit lim N →∞ c f e N (cid:0) pQ e N + ξ ′ Q (cid:1) exists and it follows from (9) that d φ e N,θ (cid:0) pQ e N + ξ ′ Q (cid:1) = 1 − ξ ′ QV , whenever e N ≥ QV.
As a consequence, both terms can be factored out from the averaging operation.
ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 13
It remains to deal with the term e (cid:0) n ( pQ + ξ ′ Q e N ) (cid:1) . Let V n := ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m , n ∈ N . By Theorem 2.2 we have that the averages N N X n =1 e (cid:0) n pQ (cid:1) V n converge in L ( µ ) . Using Lemma 3.1 we deduce that for all p, ξ ′ ∈ Z , Q ∈ N , the averages N N X n =1 e (cid:0) n ( pQ + ξ ′ Q e N ) (cid:1) V n converge in L ( µ ) . This completes the proof. (cid:3) We are ready now to prove Theorem 1.1.
Proof of Theorem 1.1.
Let ε > . Without loss of generality we can assume that allfunctions are bounded by . We let s ≥ be the integer and C s be the implicit constantdefined in Lemma 2.3. We apply Lemma 2.4 for ε/ (2 C s ) in place of ε and for κ := 2 . Weget that there exist δ > and N ∈ N such that for all integers N, e N with N ≤ sN ≤ e N ≤ sN and f : Z e N → C with | f | ≤ , the following implication holds:(16) if k f k U s ( Z e N ) ≤ δ, then (cid:13)(cid:13) [ N ] · f (cid:13)(cid:13) U s ( Z sN ) ≤ ε C s . We use the structural result of Theorem 2.6 for this δ in place of ε and for the previouslydefined s . We get that there exists θ = θ ( δ, s ) > such that for all large enough N ∈ N ,if e N denotes the smallest prime such that e N > sN and e N ≡ Q ( Q was introducedin Section 2.4 and depends only on θ ), then we have the decomposition(17) f ( n ) = f e N, st ( n ) + f e N, un ( n ) , n ∈ [ e N ] , where f e N, st = f e N ∗ φ e N,θ ( φ e N,θ is defined by (9)) and (cid:13)(cid:13)(cid:13) f e N, un (cid:13)(cid:13)(cid:13) U s ( Z e N ) ≤ δ. It follows from (16) that for all large enough N we have(18) (cid:13)(cid:13)(cid:13) [ N ] · f e N, un (cid:13)(cid:13)(cid:13) U s ( Z sN ) ≤ ε C s . Note also that the prime number theorem on arithmetic progressions implies that lim N →∞ N e N = 1 s . For ℓ, m ∈ N let ( X, X , µ ) be a probability space, T , . . . , T ℓ : X → X be invertiblecommuting measure preserving transformations, F , . . . , F m ∈ L ∞ ( µ ) , p i,j : Z → Z bepolynomials where i = 1 , . . . ℓ , j = 1 , . . . , m . Let V n := ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m , n ∈ N . If N ∈ N , for a given a N : [ N ] → C we define A N ( a N ) := 1 N N X n =1 a N ( n ) · V n . ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 14
Since f e N, un is bounded by and the functions F i are bounded by , it follows fromLemma 2.3 and (18) that lim sup N →∞ (cid:13)(cid:13)(cid:13) A N ( f e N, un ) (cid:13)(cid:13)(cid:13) L ( µ ) ≤ ε. Hence, lim sup N →∞ (cid:13)(cid:13)(cid:13) A N ( f ) − A N ( f e N, st ) (cid:13)(cid:13)(cid:13) L ( µ ) ≤ ε. Furthermore, by Lemma 3.3 we have that the averages A N ( f e N, st ) converge in L ( µ ) as N → ∞ . Combining the above we deduce that the sequence ( A N ( f )) is Cauchy in L ( µ ) and hence it converges in L ( µ ) . Therefore, the sequence ( f ( n )) is agood universal weight for polynomial multiple mean convergence.Finally, we prove the last claim of Theorem 1.1. If the multiplicative function f isaperiodic, then by Theorem 2.7 we have that lim N →∞ (cid:13)(cid:13) [ N ] · f (cid:13)(cid:13) U s ( Z sN ) = 0 for every s ≥ . Using Lemma 2.3 we deduce that the averages A N ( f ) converge to in L ( µ ) .This verifies the asserted convergence. (cid:3) Proof of Theorem 1.2.
Proof of part ( i ) of Theorem 1.2. Recall that the range of f is contained in a set of theform R = { , ζ, . . . , ζ k − } where ζ is a root of unity of order k . Then(19) f − ( K ) ( n ) = X z ∈ K k k − X j =0 z − j ( f ( n )) j . We establish the first claim of part ( i ) . For ℓ, m ∈ N let ( X, X , µ ) be a probabilityspace, T , . . . , T ℓ : X → X be invertible commuting measure preserving transformations, F , . . . , F m ∈ L ∞ ( µ ) , p i,j : Z → Z be polynomials where i = 1 , . . . ℓ , j = 1 , . . . , m . Using(19) we see that in order to verify the asserted convergence it suffices to show that for V n := ( ℓ Y i =1 T p i, ( n ) i ) F · . . . · ( ℓ Y i =1 T p i,m ( n ) i ) F m , n ∈ N , the averages N N X n =1 ( f ( n )) j V n converge in L ( µ ) for j = 0 , . . . , k − . This follows from the first part of Theorem 1.1and the fact that under the stated assumptions on the range of f we have f j ∈ M conv for j = 0 , . . . , k − by part ( i ) of Proposition 2.10.We establish now the second claim in part ( i ) . Suppose that f j is aperiodic for j = 1 , . . . , k − . By Proposition 2.5 it suffices to show that the set f − ( K ) is Gowersuniform. Let s ≥ be an integer. We claim that lim N →∞ (cid:13)(cid:13)(cid:13)(cid:13) f − ( K ) − | K | k (cid:13)(cid:13)(cid:13)(cid:13) U s ( Z N ) = 0 . Using (19) and the triangle inequality for the U s -norms, we see that in order to verifythe claim it suffices to show that for j = 1 , . . . , k − we have lim N →∞ (cid:13)(cid:13) f j (cid:13)(cid:13) U s ( Z N ) = 0 for every s ∈ N . Since f j is by assumption aperiodic for j = 1 , . . . , k − , this followsfrom Theorem 2.7. (cid:3) ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 15
Proof of part ( ii ) of Theorem 1.2. We claim that if F is a Riemann integrable functionon T with integral zero, then (cid:0) ( F ◦ f )( n ) (cid:1) is a Gowers uniform sequence, meaning,(20) lim N →∞ k F ◦ f k U s ( Z N ) = 0 , for every s ∈ N . Applying this for F := K − m T ( K ) , and using that m T ( K ) > , we deduce that the set f − ( K ) is Gowers uniform; hence by Proposition 2.5 it is a set of polynomial multiplerecurrence and mean convergence.We verify now (20). Without loss of generality we can assume that k F k ∞ ≤ / . Let s ∈ N and ε > .We first claim that the sequence ( f ( n )) is equidistributed on the unit circle. Indeed,using Weyl’s equidistribution criterion it suffices to show that for every non-zero j ∈ Z we have lim N →∞ N N X n =1 ( f ( n )) j = 0 . This follows at once since by assumption f j is aperiodic for j ∈ N , hence it has average . Taking complex conjugates we get a similar property for all negative j as well.Since F is Riemann integrable, bounded by / , and has zero mean, there exists atrigonometric polynomial P on T , bounded by , with zero constant term, such that k F − P k L ( m T ) ≤ (cid:0) ε (cid:1) s . Since ( f ( n )) is equidistributed in T and the function F − P is Riemann integrable, wededuce that lim N →∞ N N X n =1 | F ( f ( n )) − P ( f ( n )) | = k F − P k L ( m T ) ≤ (cid:0) ε (cid:1) s . Using this, the fact that | F − P | is bounded by , and the estimate k a k s U s ( Z N ) ≤ k a k s − L ∞ ( Z N ) k a k L ( Z N ) which can be easily proved using the inductive definition of the norms U s ( Z N ) , we deducethat(21) lim sup N →∞ k F ◦ f − P ◦ f k U s ( Z N ) ≤ ε. We know by assumption that f j is aperiodic for all j ∈ N and taking complex conju-gates we get that f j is aperiodic for all non-zero j ∈ Z . Hence, Theorem 2.7 gives that lim N →∞ (cid:13)(cid:13) f j (cid:13)(cid:13) U s ( Z N ) = 0 for all non-zero j ∈ Z . Since the trigonometric polynomial P haszero constant term, it follows by the triangle inequality that lim N →∞ k P ◦ f k U s ( Z N ) = 0 .From this and (21) we deduce that lim sup N →∞ k F ◦ f k U s ( Z N ) ≤ ε. Since ε > is arbitrary, we get lim N →∞ k F ◦ f k U s ( Z N ) = 0 and the proof is complete. (cid:3) Proof of Corollaries 1.3 and 1.4.
Proof of Corollary 1.3.
Let b ≥ be an integer, ζ be a root of unity of order b , and a ∈ { , . . . , b − } .In order to deal with the set S ω,A,b we define the multiplicative function f by f ( p k ) = ζ for all k ∈ N and primes p . Using part ( ii ) of Proposition 2.10 we deduce that f j isaperiodic for j = 1 , . . . , b − . Applying Theorem 1.2 for this multiplicative function andfor K := { ζ a : a ∈ A } we deduce the asserted claims for the set S ω,A,b . ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 16
In a similar fashion we prove the asserted claims for the set S Ω ,A,b ; the only differenceis that we apply Theorem 1.2 for the multiplicative function f defined by f ( p k ) = ζ k ,for all k ∈ N and primes p . (cid:3) Proof of Corollary 1.4.
In order to deal with the set S Ω ,A,α we define the multiplicativefunction f by f ( p k ) = e( α ) for all k ∈ N and primes p . Using part ( ii ) of Proposi-tion 2.10 we deduce that f j is aperiodic for all j ∈ N . Applying Theorem 1.2 for thismultiplicative function and for K := { e( t ) : t ∈ ( − A ) ∪ A } we deduce the asserted claimsfor the set S Ω ,A,α .In a similar fashion, we prove the asserted claims for the set S Ω ,A,α ; the only differenceis that we apply Theorem 1.2 for the multiplicative function f defined by f ( p k ) = e( kα ) for all k ∈ N and primes p . (cid:3) Extension to nilpotent groups.
Essentially the same arguments used in the pre-vious subsections can be replicated in order to extend the main results of this article tothe case where the transformations T , . . . , T ℓ generate a nilpotent group. The only extradifficulty that we do not address here is to prove a variant of the uniformity estimatesof Lemma 2.3 that deals with this more general setup. This requires a non-trivial mod-ification of the PET induction argument used in [12, Lemma 3.5] along the lines of theargument used to prove [24, Theorem 4.2]. Assuming these estimates, substituting theconvergence result of Theorem 2.2 with its nilpotent version (again due to M. Walsh),and the multiple recurrence result of Theorem 2.1 with a result of S. Leibman [22], therest of the arguments carries without any change.3.5. An alternate approach for weak convergence.
If one is satisfied with analyzingweak convergence of the multiple ergodic averages in our main results (which suffices forproving multiple recurrence), then an alternate way to proceed is as follows: Using themain result from [9] we get that sequences of the form C ( n ) = R F · T n F · · · T nℓ F ℓ dµ , n ∈ N , can be decomposed in two terms, one that is an ℓ -step nilsequence ( N ( n )) and another that contributes negligibly in evaluating weighted averages of the form N P Nn =1 f ( n ) C ( n ) . This reduces matters to analyzing the limiting behavior of aver-ages of the form N P Nn =1 f ( n ) N ( n ) , a task that has been carried out in [11]. Thisway one can prove a version of Theorem 1.1 and related corollaries that deal with weakconvergence, avoiding the full strength of the main structural result in [11].3.6. An alternate approach for recurrence.
We mention here an alternate way toprove “linear” multiple recurrence results for shifts of the sets S ω,A,b , S Ω ,A,b . After modi-fying the argument below along the lines of the proof of part ( ii ) of Theorem 1.2 we getsimilar results for the sets S ω,A,α , S Ω ,A,α . Definition. An IP k -set of integers is a set of the form (cid:8) a i + · · · + a i ℓ : 1 ≤ ℓ ≤ k, i < i < · · · < i ℓ (cid:9) where a , . . . , a k are distinct positive integers.For example, an IP -set has the form { m, n, r, m + n, m + r, n + r, m + n + r } with m, n, r ∈ N distinct. Proposition 3.4.
Let d, ℓ ∈ N and L , . . . , L ℓ : N d → N be pairwise independent linearforms. Let b, c ∈ N , a ∈ { , . . . , b − } , and S a,b,c be either S ω,a,b + c or S Ω ,a,b + c . Thenthe set (22) (cid:8) m ∈ N d : L ( m ) , . . . , L ℓ ( m ) ∈ S a,b,c (cid:9) has density b − ℓ . Therefore, the set S a,b,c contains IP k -sets of integers for every k ∈ N . Remark.
Similar results hold for affine linear forms and all sets defined in Theorem 1.2.
ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 17
Proof.
Note first that for n > c we have(23) S a,b,c ( n ) = 1 b b − X j =0 ζ − aj ( f ( n − c )) j where f is the multiplicative function and ζ is the b -th root of unity defined in the proofof Corollary 1.3. Note also that the density of the set in (22) is equal to(24) lim N →∞ N d X m ∈ [ N ] d ℓ Y i =1 S a,b,c ( L i ( m )); the existence of the limit will be established momentarily. By [10, Theorem 1.1], if atleast one of the functions f , . . . , f ℓ ∈ M is aperiodic, then(25) lim N →∞ N d X m ∈ [ N ] d ℓ Y j =1 f j ( L j ( m )) = 0 . Using that f j is aperiodic for j = 1 , . . . , b − , in conjunction with (23) and (25), wededuce that the limit in (24) exits and is equal to b − ℓ . (cid:3) Theorem 3.5 (Furstenberg, Katznelson [15, Theorem 10.1]) . Let T , . . . , T ℓ be commut-ing measure preserving transformations acting on the same probability space ( X, X , µ ) .Then for every A ∈ X with µ ( A ) > there exists k ∈ N such that set { n ∈ N : µ ( T − n A ∩ · · · ∩ T − nℓ A ) > } intersects non-trivially every IP k -set of integers. Combining Proposition 3.4 with Theorem 3.5 we get that the set of return times inTheorem 3.5 intersects non-trivially each of the sets S ω,a,b,c and S Ω ,a,b,c . Unfortunately,a polynomial extension of Theorem 3.5 is not yet available and we cannot get the fullstrength of the recurrence results of Corollaries 1.3 and 1.4 using such methods. References [1] E. el Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk, T. de la Rue. The Chowla and the Sarnakconjectures from ergodic theory point of view. Preprint. arXiv:1410.1673[2] A. Balog, A. Granville, K. Soundararajan. Multiplicative functions in arithmetic progressions.
Annales mathématiques du Québec (2013), 3–30.[3] J. Bourgain. Double recurrence and almost sure convergence. J. Reine Angew. Math. (1990),140–161.[4] V. Bergelson, A. Leibman. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems.
J. Amer. Math. Soc. (1996), 725–753.[5] H. Daboussi. Fonctions multiplicatives presque périodiques B. D’après un travail commun avecHubert Delange. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974),pp. 321–324. Asterisque (1975), 321–324.[6] H. Daboussi, H. Delange. Quelques proprietes des functions multiplicatives de module au plusegal 1.
C. R. Acad. Sci. Paris Ser. A (1974), 657–660.[7] H. Davenport. On some infinite series involving arithmetical functions II.
Quart. J. Math. Oxf. (1937), 313–320.[8] P. Elliott. Probabilistic Number Theory I. Springer-Verlag, New York, Heidelberg, Berlin (1979).[9] N. Frantzikinakis. Multiple correlation sequences and nilsequences. Inventiones Math. (2015),875–892.[10] N. Frantzikinakis, B. Host. Asymptotics for multilinear averages of multiplicative functions.Preprint. arXiv:1502.02646[11] N. Frantzikinakis, B. Host. Higher order Fourier analysis of multiplicative functions and applica-tions. Preprint. arXiv:1403.0945[12] N. Frantzikinakis, B. Host, B. Kra. The polynomial multidimensional Szemerédi theorem alongshifted primes.
Israel J. Math. (2013), 331–348.[13] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmeticprogressions.
J. Analyse Math. (1977), 204–256. ULTIPLE ERGODIC THEOREMS FOR ARITHMETIC SETS 18 [14] H. Furstenberg, Y. Katznelson. An ergodic Szemerédi theorem for commuting transformations.
J. Analyse Math. (1979), 275–291.[15] H. Furstenberg, Y. Katznelson. An ergodic Szemerédi theorem for IP-systems and combinatorialtheory. J. Analyse Math. (1985), 117–168.[16] T. Gowers. A new proof of Szemerédi’s theorem. Geom. Funct. Anal. (2001), 465–588.[17] A. Granville, K. Soundararajan. Large character sums: pretentious characters and the Pólya-Vinogradov theorem. J. Amer. Math. Soc. (2007), no. 2, 357–384.[18] A. Granville, K. Soundararajan. Multiplicative Number Theory: The pretentious approach. Bookmanuscript in preparation.[19] G. Halász. Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad.Sci. Hung. (1968), 365–403.[20] R. Hall. A sharp inequality of Halász type for the mean-value of a multiplicative function. Math-ematika (1995), 144–157.[21] I. Kátai. A remark on a theorem of H. Daboussi. Acta Math. Hungar. (1986), 223–225.[22] A. Leibman. Multiple recurrence theorem for measure preserving actions of a nilpotent group. Geom. Funct. Anal. (1998), 853–931.[23] T. Tao. Norm convergence of multiple ergodic averages for commuting transformations. ErgodicTheory Dynam. Systems (2008), no. 2, 657–688.[24] M. Walsh. Norm convergence of nilpotent ergodic averages. Annals of Mathematics (2012),no. 3, 1667–1688.[25] E. Wirsing. Das asymptotische Verhalten von Summen uber multiplikative Funktionen, II.
ActaMath. Acad. Sci. Hung. (1967), 411–467.(Nikos Frantzikinakis) University of Crete, Department of mathematics, Voutes Univer-sity Campus, Heraklion 71003, Greece
E-mail address : [email protected] (Bernard Host) Université Paris-Est Marne-la-Vallée, Laboratoire d’analyse et de math-ématiques appliquées, UMR CNRS 8050, 5 Bd Descartes, 77454 Marne la Vallée Cedex,France
E-mail address ::