1503.08863.pdf

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arXiv:1503.08863v2 [math.DS] 2 Sep 2015

A GOOD UNIVERSAL WEIGHT FOR NONCONVENTIONAL ERGODIC AVERAGES IN NORM

IDRIS ASSANI AND RYO MOORE

Abstract. We will show that the sequence appearing in the double recurrence theorem is a good universal

weight for the Furstenberg averages. That is, given a system(X,F,µ,T)and bounded functions f1,f2∈L∞(µ),

there exists a set of full-measureXf1,f2 inXthat is independent of integersaandband a positive integerk

such that for allx∈Xf1,f2and for every other measure-preserving system(Y,G,ν,S), and each bounded and

measurable functiong1, . . . ,gk∈L∞(ν), the averages

1

N N

∑

n=1

f1(Tanx)f2(Tbnx)g1◦Sng2◦S2n· · ·gk◦Skn

converge inL2₍_ν₎_.

1. Introduction

1.1. Background.

1.1.1. Good universal weights. In some literatures (e.g. [3, Definitions 3.1-3.3]), the sequence (an)is called

a good universal weight for the pointwise ergodic theorem if for any probability measure preserving system

(Y,G,ν,S)and anyg∈L∞

(ν), the averages

(1) 1

N

N−1

∑

n=0

ang(Sny)

converge forν-a.e. y∈Y. Similarly, the sequence(an)is called agood universal weight for the mean ergodic

theoremif the averages in(1)converge inL2(ν).

In this paper, we will extend these classical notions of good universal weights to discuss the case where

the sequence(g◦Sn₎

n in(1)is replaced by other sequences of bounded and measurable functions(Xn)n.

Definition 1.1. We say(Xn)nis aprocessif for all nonnegative integers n≥0, Xnis a bounded and measurable

function on some probability measure space(Ω_,_S,P₎.

For instance, a sequence of bounded and measurable functions (Xn)n = (g◦Sn)n for any g ∈ L∞(ν)

on any probability measure-preserving system (Y,G,ν,S) is a process. Another process (Xn)n of our

interest is a product of multiple functions each iterated by different powers of a measure-preserving

transformation, such as

Xn(y) =g1(Sny)g2(S2ny)· · ·gk(Skny),

for any positive integerk, whereg1,g2, . . . ,gk∈L∞(ν)on any measure-preserving system(Y,G,ν,S).

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Definition 1.2. We denote by

M1= (

(an): sup N

1 N

N

∑

n=1

|an|<∞ )

.

• We say a sequence(an)∈ M1is agood universal weight for(Xn)n(a.e.) pointwiseif for any probability

measure space(Ω_,_S,P₎for which the process₍X_n₎_n is defined, the averages

1 N

N

∑

n=1

anXn(ω)

converge forP-a.e. _ω_∈Ω_.

• We say a sequence (an) ∈ M1 is a good universal weight for (Xn)n in norm if for any probability

measure space(Ω_,_S,P₎for which the process₍X_n₎_n is defined, the averages

1 N

N

∑

n=1

anXn(ω)

converge in L2₍_P₎

For example, if(an)n is a good universal weight for the process(Xn)n = (g◦Sn)n pointwise (resp. in

norm) for anyg∈L∞

(ν), where(Y,G,ν,S)is any probability measure-preserving system, then(an)nis a

good universal weight for the pointwise ergodic theorem (resp. the mean ergodic theorem) in the classical

sense.

1.1.2. History of the return times theorem. The studies of the return times theorem have shown that we

can randomly generate good universal weights. The basic principle of the return times theorem that

has been initially studied by A. Brunel in his Ph.D. thesis in 1966 [15] is as follows: Given a process

Xn(ω)converging in average (in norm or pointwise) and the characteristic function of a measurable set

with positive measure,1_A_{, do we still have the convergence of the averages along the subsequence given}

by the return times of Tnx to the set A? In other words, is the sequence (1_A₍Tnx₎₎_n _{a good universal}

weight (in norm or pointwise) for the averages of 1_A₍Tnx₎X_n₍_ω₎_{? In 1969, A. Brunel and M. Keane}

answered this question positively for a particular class of dynamical systems for both pointwise and norm

convergence [16]. Krengel’s book highlights some of the generalization of their work [22].

One of the important results in ergodic theory is the proof of return times theorem by J. Bourgain [12],

which was later simplified by J. Bourgain, H. Furstenberg, Y. Katznelson, and D. Ornstein (a.k.a. the

"BFKO" argument) [14]. This result strengthens Birkhoff’s pointwise ergodic theorem and generalizes the

above-mentioned results on return times.

Theorem 1.3(Bourgain’s Return Times Theorem). Let(X,F,µ,T)be a probability measure-preserving system and f ∈ L∞₍_µ₎_{. Then there exists a set X}

f ⊂ X of full measure such that for any other probability

measure-preserving system(Y,G,ν,S)and any g∈L∞₍_ν₎_,

1 N

N

∑

n=1

(3)

convergesν-almost everywhere for all x∈Xf.

While the set of full-measureXf depends on the function f and the transformationT, it is independent

of every other ergodic system. In terms of Definition 1.2, Bourgain has shown that forµ-a.e. x ∈ X, the

sequence an = f(Tnx)is a good universal weight for(Xn)n = (g◦Sn)n pointwise, whereg ∈ L∞(ν)for

any measure-preserving system(Y,G,ν,S).

1.1.3. Extensions of the return times theorem. Much of the background, historical development, and current

status of the return times can be found in the survey paper prepared by the first author and K. Presser [10].

Here, we will focus on discussing some of the developments on the return times theorem regarding mixing

of multiple recurrence and multi-term return times problems. Some new results that appeared since the

emergence of the survey paper are mentioned as well.

Since the result of Bourgain emerged, the return times theorem has been extended in multiple direction.

One way is to find a new universal weight in which the return-times averages converge. For instance, the

first author shows in Proposition 5.3 of [3] that if (X,F,µ,T) is a weakly-mixing, standard uniquely

ergodic system with Lebesgue spectrum, and f ∈ C(X), then(f(Tnx))is a good universal weight for the

pointwise ergodic theoremfor all x∈X. Recently, P. Zorin-Kranich announced the extension of Bourgain’s

return times theorem by showing that the double recurrence sequence is a good universal weight for the

pointwise ergodic theorem forµ-a.e. x∈ X[28].

The return times theorem has also been extended to averages with more than two terms. One example

of such is the multiterm return times theorem that was obtained by D. Rudolph in 1998 [25], which answers

one of the questions raised by the first author in 1991. Rudolph’s proof utilized the method of joinings and

fully generic sequences, while the method of factor decomposition was absent, which was one of the key

tools in the BFKO argument of the return times theorem. Later, the first author and K. Presser identified

characteristic factors for the multiterm return times theorem [8,9]. Furthermore, P. Zorin-Kranich provided

a different proof of the multiterm return times theorem based on these factor structures, and showed that

multiterm return times averages can be extended to Wiener-Wintner type averages with nilsequences [27].

Also, T. Eisner [17] showed the convergence of Wiener-Wintner type averages for multiterm return times

theorem with linear sequences.

In another direction, the return times theorem has been extended by mixing weights from the a.e.

multiple recurrence and the multiterm return times theorem. This idea was introduced by the first author

in 1998, in which he proved the following:

Theorem 1.4( [2, Theorem 3]). Let(X,F,µ,T)be a weakly mixing dynamical system such that for all positive integers H, for all f1,f2, . . .fH ∈L∞(µ), for all(b1,b2, . . . ,bH)∈ZHwhere bi distinct and not equal to zero, the

sequence

1 N

N

∑

n=1

H

∏

i=1

fi(Tbinx) !

converges a.e. to

∏

H i=1 Z

(4)

Then there exists a set of full measure X′_{for any other weakly mixing system}₍_Y1_,_G

1,S1,ν1)and any g1∈ L∞(ν1), there exists a set of full measure Yg1 in Y1 such that if y1 ∈ Yg1, then . . . for any other weakly mixing system (Yk−1,Gk−1,Sk−1,νk−1) and any gk−1 ∈ L∞(νk−1) there exists a set of full measure Ygk−1 in Yk−1 such that if

yk−1∈Ygk−1, then for any other weakly mixing system(Yk,Gk,Sk,νk), the sequence

ξn(x,y1,y2, . . . ,yk) = H

∏

i=1

fi(Tbinx) ! _k

∏

j=1

gj(Snjyj) !

is a good universal weight for the pointwise ergodic theorem forνk-a.e. yk∈Yk.

For instance, if (X,F,µ,T) is a weakly mixing system for which the restriction of T to its Pinsker

algebra has singular spectrum, then the hypothesis of the theorem above holds. This result was proven by

the first author in 1998 [1].

In terms of Definition 1.2, Theorem 1.4 says that fork =1, there exists a set of full-measureYg1 ⊂ Y1

such that for all y1 ∈ Y1, the sequence(∏H_i₌₁f_i(Tbin_x₎₎_n _{is a good universal weight for}_µ_-a.e. _x _∈ _X _for

the process Xn(z) = Xn[y1,g1,S1](z) = g1(S₁ny1)h(Rnz) pointwise, for any measure-preserving system (Z,Z,η,R)and a functionh∈L∞₍_η₎_.

In 2009, B. Host and B. Kra showed in [21] that given an ergodic system (X,F,µ,T)and f ∈ L∞₍_µ₎_,

the sequence(f(Tn_x₎₎_{is a good universal weight for}_µ_-a.e. _x _∈_X_{for the convergence in}_L2_{-norm of the} Furstenberg averages, i.e. they have shown that there exists a set of full-measure X′ ⊂ Xsuch that for anyx ∈ X′ _{and any other measure-preserving system}₍_Y,_G,_ν_,_S₎ _{with functions} _{g1, . . . ,}_g

k ∈ L∞(ν), the

averages

(2) 1

N

∑

n=1

f(Tnx) k

∏

i=1

gi◦Sin,

converge inL2₍_ν₎_{. In particular, if} _f ₌₁

Afor some measurable setA∈ F, then they have shown that the

averages of the sequence(∏k_i₌₁g_i(Sin_y₎₎

n along the subsequence of the return times ofTnx to the set A

converge in L2(ν)-norm. In the language of Definition 1.2, forµ-a.e. x ∈ X, the sequence (f(Tn_x₎₎_{is a} good universal weight for(Xn)n in norm, where(Xn)nis a process of the form

(3) (Xn)n= k

∏

i=1

gi◦Sin !

n

for anyg1, . . . ,gk ∈L∞(ν)on any m.p.s.(Y,G,ν,S), for anyk≥1.

This result extends their earlier work in [20], where they proved the result for f =1_X_{. To show this result,}

they used the machinery of nilsequences (see [11, 21] for more background on nilsequences); they showed

that if a bounded sequence (an)n ∈ ℓ∞has a property that the Cesaro averages ofanbn converge for any

k-step nilsequence(bn)n, then(an)n is a good universal weight fork-term multiple recurrent averages in

the L2_{-norm. Then the convergence of the averages in} ₍₂₎ _{follows from the fact that there exists a set} of full-measure X′ _{so that for any} _x _∈ _X′ _{and any nilsequence} ₍_b_n₎_n_{, the Cesaro averages of} _f₍_Tn_x₎_b

n

(5)

and P. Zorin-Kranich, the generalized Wiener-Wintner theorem was extended to any measure-preserving

system (with not necessarily ergodic transformation) with uniform counterpart, and used this to extend

the result to a case with polynomial actions [18].

1.2. The main theorem. In this paper, we will prove the following:

Theorem 1.5 (The main result). Let (X,F,µ,T) be a probability measure-preserving system, with functions f1,f2 ∈ L∞

(µ). Then forµ-a.e. x ∈ X, the sequence un = (f1(Tanx)f2(Tbnx))n is a good universal weight for

a k-term Furstenberg averages in norm for any positive integer k. More precisely, there exists a set of full-measure Xf1,f2 ⊂ X such that for any x ∈ Xf1,f2, a,b ∈ Z and any positive integer k ≥ 1, and any other probability

measure-preserving system(Y,G,ν,S)with g1, . . . ,gk∈L∞(ν), the averages

1 N

N

∑

n=1

f1(Tanx)f2(Tbnx) k

∏

i=1

gi◦Sin

converge in L2₍_ν₎_.

In particular, if f1 =1_A_and f2₌1_B_{for some measurable sets} A,B _{∈ F} _{with positive measures, then}

we see that the averages of the sequence (∏k_i₌₁g_i(Sin_y₎₎_n _{along the subsequence of the return times of}

Tanx to the set A and Tbnx to the set B converge in L2₍_ν₎_{-norm. This theorem mixes the weights from} the a.e. double recurrent convergence result and the norm convergence of the multiple recurrent theorem.

In terms of Definition 1.2, we show that for µ-a.e. x ∈ X, the sequence (f1(Tan_x₎_f2₍_Tbn_x₎₎

n is a good

universal weight for the process(Xn)n of the form in(3)in norm. Note that this theorem generalizes the

result obtained by B. Host and B. Kra, since ifa=1 and f2=1_X_{, then the averages in the theorem become}

the averages seen in(2).

The Cesaro averages of the sequence (f1(Tan_x₎_f2₍_Tbn_x₎₎_n _{is known to converge for}_µ_-a.e. _x _∈ _X _by

Bourgain’s double recurrence theorem [13]. It was recently extended to a Wiener-Wintner result [5], and

further to a polynomial Wiener-Wintner result [7]. Note that the case k = 1 of the main result follows

immediately from this Wiener-Wintner result. In fact, we will show that this is the key step required to

establish the "base case" of our inductive argument in the proof.

In the proof, we will assume that the systems(X,F,µ,T)and(Y,G,ν,S)are ergodic, since we can apply

the ergodic decomposition to show that the result holds for general measure-preserving systems. To prove

the main result fork≥2, we will first decompose the functions f1and f2into an appropriate characteristic

factor of(X,F,µ,T), and treat the cases when either f1or f2belongs to the orthogonal complement of this

factor, or the case when both of them belong to the factor. For the first case, we will prove it by induction

onk. We will show that the casek = 2 follows from the fact that the theorem holds for the casek = 1;

to do so, we will show that the L2₍_ν₎_{-norm limit of the averages can be controlled by the limit of the} double recurrence Wiener-Wintner averages. We will also show that the casek=3 follows from the case

(6)

when either one of g1, . . . ,gk belongs to the orthogonal complement of this factor, and when all of them

belong to the factor separately. For the first sub-case, we will control the norm limit of the averages with

a seminorm that characterizes this factor, and uses this to show that the norm averages converge to 0. For

the second sub-case, we will use the structure of nilmanifolds and Leibman’s convergence result [24] to

prove the claim.

The factors we use are the Host-Kra-Ziegler factors [20, 26]. Throughout this paper, we denote Zk(T)

to be thek-th Host-Kra-Ziegler factor of (X,T), which is characterized by the k+1-th Gowers-Host-Kra

seminorm|||·|||_k₊₁[19, 20]. Using the language of these factors, Theorem 1.5 can be shown by proving the

following:

Theorem 1.6. Let(X,F,µ,T)be an ergodic system, and f1,f2∈L∞

(µ)such thatkfikL∞_(ν)≤1for both i=1, 2.

Fix a positive integer k≥1. Then the following statements are true.

(a) Suppose either f1,f2 ∈ Zk+1(T)⊥. Then there exists a set of full-measure X˜k ⊂ X such that for any

x ∈ X, any other measure-preserving system˜ (Y,G,ν,S), and functions g1,g2, . . . ,gk ∈ L∞(ν) where

gj

_L∞_(ν)≤1for each1≤ j≤k, the averages

(4) 1

N

∑

n=1

∏

j=1

gj◦Sjn

converge to0in L2₍_ν₎_.

(b) For any f1,f2∈L∞₍_µ₎_{, there exists a set of full-measure}_X_ˆ

ksuch that for any x∈X and any other ergodicˆ

system(Y,G,ν,S), and functions g1, . . . ,gk∈L∞(ν)with one of them belonging toZk(S)⊥, the averages

in(4)converge to0in L2(ν).

(c) Suppose both f1,f2∈ Zk+1(T). Then there exists a set of full-measure X′k ∈X such that for any x ∈ X′,

and for any other ergodic system (Y,G,ν,S)and functions g1, . . . ,gk ∈ L∞(ν)∩ Zk(S), the averages in (4)converge in L2₍_ν₎_.

Proof that Theorem1.6implies Theorem1.5. Fix a positive integerk≥1. Let f_i′= fi−E(fi|Zk+1)fori=1, 2. We rewrite the averages in(4)as follows:

1 N

N

∑

n=1

∏

j=1

gj◦Sjn

(5)

= 1

N

∑

n=1

E₍f1|Z_k₊₁₎₍Tanx₎E₍f2|Z_k₊₁₎₍Tbn₎

k

∏

j=1

gj◦Sjn

+ 1

N

N−1

∑

n=0

E₍f1|Z_k₊₁₎₍Tanx₎f₂′₍Tbnx₎

k

∏

j=1

gj◦Sjn

+ 1

N

∑

n=1

f′

1(Tanx)E(f2|Zk+1)(Tbnx)

k

∏

j=1

(7)

+ 1

N

∑

n=1

f′

1(Tanx)f2′(Tbnx)

k

∏

j=1

gj◦Sjn.

We know that, by Theorem 1.6(a), there exists a universal set of full-measure ˜Xksuch that for all x∈ X˜k,

the last three averages of the right hand side of(5)converge to 0 inL2₍_ν₎_{. And by Theorem 1.6(b-c), the} first averages also converge inL2₍_ν₎_{for all}_x_∈_X_ˆ

k∩Xk′. So if we set

Xf1,f2,k=X˜k∩Xˆk∩X

′

k,

thenXf1,f2,kis a set full-measure that only depends on f1, f2, the transformationT, and the positive integer

k, since it is a finite intersection of the sets of full-measure, each only depending on the functions f1, f2, and the transformationT. Thus, for any x ∈ Xf1,f2,k, a,b ∈ Z, and any other ergodic system(Y,G,ν,S)

with functionsg1, . . . ,gk∈L∞(ν), the averages in(4)converge inL2(ν). This implies that the set

Xf1,f2 = ∞ \

k=1 Xf1,f2,k

is a set of full-measure that only depends on the functions f1, f2, and the transformation T, and this is

indeed the desired universal set for Theorem 1.5.

1.3. Organization of the paper. In §2, we will prove (a) of Theorem 1.6, which treats the case where either f1or f2belongs to the orthogonal complement of the appropriate factor of(X,T). The case where f1and f2 both belong to the appropriate factor is discussed in §3, where we first look at the case where either

one of the functionsg1, . . . ,gk belongs to the orthogonal complement of the appropriate factor of (Y,S)

(which corresponds to (b) of Theorem 1.6), and the case all of them belong to the appropriate factor (which

corresponds to (c) of Theorem 1.6).

1.4. Acknowledgment. We thank the anonymous referee for his/her comments.

2. The case where either f1or f2belongs toZ_k₊₁(T)⊥(Proof of(a)ofTheorem1.6)

The idea of the proof is as follows: We will first prove the statement for the casek=2. We first identify

the set of full-measure for which the averages in (4)converges to 0; the fact that this is indeed a set of

full-measure can be shown by using Fatou’s lemma and the following inequality obtained in [5]:

(6)

Z

lim sup

N→∞

sup

t∈R

1 N

N−1

∑

n=0

f1(Tanx)f2(Tbnx)e2πint

2

dµ(x).a,bmin i=1,2|||fi|||

2 3.

The key observation of the proof is the fact thatS is a measure-preserving transformation allows us to

bound the L2(ν)-norm of the averages by the double recurrence Wiener-Wintner averages; to do so, we apply van der Corput’s lemma [23], Hölder’s inequality, and the spectral theorem. This allows us to show

(8)

Then we will proceed for the casek=3 to demonstrate that the claim can be proven inductively for the

casek>_{2. Again we start by identifying the set of full-measure. To show that the averages converge to 0}

on this set, we rely on the result obtained for the casek=2.

Before we prove this part of the theorem, we will prove this for the case wherek=2, 3 to demonstrate

the inductive step for simple cases. For the casek=2, we would like to show that

(7) lim sup

N→∞ 1 N N

∑

n=1

f1(Tan_x₎_f2₍_Tbn_x₎_g1_◦_Sn_g2_◦_S2n 2

L2_(ν) =0.

Consider a set

(8) ˜ X2=     

x∈X: lim inf

H→∞   1 H H

∑

h=1 lim sup

N→∞

sup

t∈R

1 N

N−h

∑

n=1

f1·f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)e2πint 2  1/2 =0     

First we show that ˜X2is a set of full-measure. To do so, we apply Fatou’s lemma and the inequality(6)to

obtain

Z

lim inf

H→∞   1 H H

∑

h=1 lim sup

N→∞

sup

t∈R

1 N

N−h

∑

n=1

f1· f1◦Tah₍_Tan_x₎_f2_·_f2_◦_Tbh₍_Tbn_x₎_e2πint 2  1/2 dµ

≤lim inf

H→∞   1 H H

∑

h=1 Z lim sup

N→∞

sup

t∈R

1 N

N−h

∑

n=1

f1·f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)e2πint 2 dµ   1/2

.a,b min i=1,2lim infH→∞

1 H H

∑

h=1

fi·fi◦T h

₃≤min_i

=1,2 lim infH→∞

1 H H

∑

h=1

fi·fi◦T h 8 3 !1/8 =min

i=1,2|||fi||| 2 4.

Since either f1or f2belongs toZ3(T)⊥, either|||f1|||₄or|||f2|||₄equals zero. This shows thatµ(_X2˜ _{) =}_1. Now we claim(7)holds for allx∈X2. In fact, we show that for any 1˜ ≤H< N, we have

1 N N

∑

n=1

f1(Tanx)f2(Tbnx)g1◦Sng2◦S2n 2

L2_(ν)

.a,b _H1 +   1 H H

∑

h=1 sup

t∈R

1 N

N−h

∑

n=1

f1·f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)e2πint 2  1/2 (9)

To do so, we proceed with van der Corput’s lemma; using the fact thatSis a measure preserving

trans-formation, we obtain

1 N N

∑

n=1

L2_(ν)

≤ 2 H + 4 H H

∑

h=1 Z ₁ N

N−h

∑

n=1

(9)

= 2 H + 4 H H

∑

h=1 Z ₁ N

N−h

∑

n=1

f1· f1◦Tah(Tanx)f2·f2◦Tbh(Tbnx)(g1·g1◦Sh)(g2·g2◦S2h₎_◦_Sn_d_ν ≤ 2 H + 4 H H

∑

h=1 1 N N

∑

n=1

f1·f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)(g2·g2◦S2h₎_◦_Sn _L₂ (ν)

(by Hölder’s inequality)

≤ 2 H +   16 H H

∑

h=1 1 N N

∑

n=1

f1· f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)(g2·g2◦S2h₎_◦_Sn 2

L2_(ν) 



1/2 ,

where the last inequality follows from the Cauchy-Schwarz inequality. We apply the spectral theorem to

the square of theL2(ν)-norm in the last line to obtain

1 N N

∑

n=1

f1· f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)(g2·g2◦S2h₎ 2

L2_(ν)

= Z 1 N N

∑

n=1

f1·f1◦Tah(Tanx)f2·f2◦Tbh(Tbnx)e2πint 2

dσ_g₂_·_g₂_◦_S2h(t)

≤sup

t∈R

1 N N

∑

n=1

f1· f1◦Tah(Tanx)f2· f2◦Tbh(Tbnx)e2πint 2 ,

which tells us that(9)holds. Thus, ifx ∈ _X˜2_{, and we let} _N_→_∞_{(and consequently} _H_→_∞_{) in}₍₉₎_{, we} obtain

lim sup

N→∞ 1 N N

∑

n=1

L2_(ν)

.a,b 

lim inf H→∞

1 H H

∑

h=1 lim sup

N→∞

sup

t∈R

1 N

N−h

∑

n=1

f1· f1◦Tah(Tanx)f2·f2◦Tbh(Tbnx)e2πint 2  1/2 =0.

This proves the case for k = 2. Now we show that the holds for the case k = 3 using the fact that the

convergence to 0 holds fork=2. We letF1,h1 = f1·f1◦T

ah1 _andF

2,h1 = f2· f2◦T

bh1_{. Then we set}

˜ X3=

(

x∈X: lim inf

H1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

lim sup

N→∞

sup

t∈R

1 N N

∑

n=1

F1,h1·F1,h1◦T

ah2₍Tanx₎F2,_h

1·F2,h1◦T

bh2₍Tbnx₎e2πint 2  1/4 =0      .

We first show that ˜X3is a set of full-measure. To see that, we apply Fatou’s lemma twice to interchange

the integral and the lim inf’s, Hölder’s inequality, the inequality(6), and the Cauchy-Schwarz inequality

multiple times to obtain

Z

lim inf

H1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

(10)

lim sup

N→∞

sup

t∈R

1 N N

∑

n=1

F1,h1·F1,h1◦T

1·F2,h1◦T

bh2₍Tbnx₎e2πint 2 dµ(x)





1/4

≤lim inf

H1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

Z

lim sup

N→∞

sup

t∈R

1 N N

∑

n=1

F1,h1·F1,h1◦T

1·F2,h1◦T

bh2₍Tbnx₎e2πint 2 dµ(x)





1/4

.a1,a2 lim inf_H 1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

min i=1,2

Fi,h1·Fi,h1◦T a_ih2

2 3 !1/4

(wherea1=a,a2=b)

≤lim inf

H1→∞   1 H1 H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

min i=1,2

Fi,h1·Fi,h1◦T a_ih2

8 3

!1/4 

1/4

1 H1

H1

∑

h1=1

min i=1,2

fi· fi◦T a_ih1

4 4 !1/4

≤lim inf

H1→∞

1 H1

H1

∑

h1=1

min i=1,2

fi·fi◦T a_ih1

16 4 !1/16

.a1,a2 _imin

=1,2|||fi||| 2 5,

and since either f1or f2belongs toZ4(T)⊥, either|||f1|||₅or|||f2|||₅equals zero. Hence, we know that ˜X3 is a set of full-measure.

Now we will show that the averages converge to 0 whenx ∈X3. To do so, we wish to show that˜

1 N N

∑

n=1

f1(Tanx)f2(Tbnx)

3

∏

j=1

gj◦Sjn 2

L2_(ν)

.a,b _H11 + _H11 H1−1

∑

h1=0

2 H2+   16 H2

H2−1

∑

h2=0

sup

t∈R

1 N N

∑

n=1

F1,h1·F1,h1◦T

1·F2,h1◦T

bh2₍Tbnx₎e2πint 2      1/4

Indeed, we apply van der Corput’s lemma and the Cauchy-Schwarz inequality to show that

1 N N

∑

n=1

3

∏

j=1

gj◦Sjn 2

L2_(ν)

≤ 2 H1+ 4 H1 H

∑

h=1 Z ₁ N

N−h1

∑

n=1

f1· f1◦Tah1₍Tanx₎f2_· f2_◦Tbh1₍Tbnx₎

3

∏

j=1

(gj·gj◦Sjh1)◦Sjndν = 2 H1+ 4 H1 H1

∑

h1=1

Z ₁ N

N−h1

∑

n=1

f1· f1◦Tah(Tanx)f2·f2◦Tbh1₍Tbnx₎

3

∏

j=1

(11)

≤ 2 H1+   16 H1 H1

∑

h1=1

1 N

N−h1

∑

n=1

f1·f1◦Tah1₍Tanx₎f2_· f2_◦Tbh1₍Tbnx₎

3

∏

j=2

(gj·gj◦Sjh1)◦S(j−1)n 2

L2_(ν) 



1/2

We can now apply the inequality(9)and the Cauchy-Schwarz inequality to obtain

1 N N

∑

n=1

3

∏

j=1

gj◦Sjn 2

L2_(ν)

._a_,_b 1

H1+ 1 H1

H1

∑

h1=1

1 H2+   1 H2 H2

∑

h2=1

sup

t∈R

1 N

N−h1−h2

∑

n=1

F1,h1·F1,h1◦T

ah2₍_Tan_x₎_F2,

h1·F2,h1◦T

bh2₍_Tbn_x₎_e2πint 2      1/4

Therefore, we have shown that the averages converge to 0 inL2(ν)whenx∈_X3.˜

One of the key observations in showing that the case k=2 implies the casek =3 was the use of the

inequality(9). We will show that this can be done fork≥ 4. For the following lemma, we will use the

following notations for our convenience: We shall denotea1=aanda2=b. Let~_h

l= (h1,h2, . . . ,hl)∈Nl.

With this notation, we define the following functions recursively:

F_1,~_h₍₁₎= f1·f1◦Ta1h1, F_2,~_h₍₁₎= f2·f2◦Ta2h1,

F_1,~_h₍₂₎=F_1,~_h₍₁₎·F_1,~_h₍₁₎◦Ta1h2, F_2,~_h₍₂₎=F_2,~_h₍₁₎·F_2,~_h₍₁₎◦Ta2h2,

· · ·, · · ·,

F_1,~_h₍_k₋₁₎=F_1,~_h₍_k₋₂₎·F_1,~_h₍_k₋₂₎◦Ta1hk−1, F_2,~_h₍_k₋₁₎=F_2,~_h₍_k₋₂₎·F_2,~_h₍_k₋₂₎◦Ta2hk−1.

Lemma 2.1. Let everything as in (a) of Theorem1.6. Then for each positive integer k≥2, we have

lim sup

N→∞ 1 N N

∑

n=1

f1(Ta1nx₎f2₍Ta2nx₎ k

∏

i=1

gi◦Sin 2

L2_(ν)

(10)

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

· · ·

lim inf

H_k−1→∞

1 Hk−1

H_k−1

∑

h_k−1=1

lim sup

N→∞

sup

t∈R

1 N N

∑

n=1

F_1,~_h₍₁₎(Ta1nx)F_2,~_h₍₁₎(Ta2nx)e2πint

2 

2−(k−1)

Proof. We will show this by using induction. The base casek=2 has been treated by the estimate(9)after we let N → ∞_and _H _→ ∞_{. Now suppose the estimate holds when we have}_k₋_{1 terms. By applying}

the van der Corput’s lemma and the Cauchy-Schwarz inequality, the left hand side of the estimate(10)is

bounded above by the universal constant depending ona1anda2times

lim inf

H1→∞   1 H1 H1

∑

h1=1

lim sup

N→∞ 1 N N

∑

n=1

F_1,~_h₍_k₋₁₎(Ta1nx)F_2,~_h₍_k₋₁₎(Ta2nx)

k

∏

i=2

(gi·gi◦Sih1)◦S(i−1)n 2

L2_(ν) 



(12)

and we can apply the inductive hypothesis on this lim sup of the square of theL2_{-norm and the}

Cauchy-Schwarz inequality to obtain the desired estimate.

Proof of Theorem1.6(a). The set ˜X1 can be obtained from the double recurrence Wiener-Wintner result [5]

by applying the spectral theorem. Fork≥2, we consider a set

˜ Xk =

(

x∈X: lim inf

H1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

· · ·

lim inf

H_k→∞

1 Hk−1

H_k

∑

h_k=1

lim sup

N→∞

sup

t∈R

1 N

N

∑

n=1

F_1,~_h₍_k₋₁₎(Ta1nx)F_2,~_h₍_k₋₁₎(Ta2nx)e2πint

2 

2−(k−1)

=0

  

 

.

We will show that this set is the desired set of full-measure. To show thatµ(_X˜_k_{) =}_{1, we will show that}

the integral

Z

lim inf

H1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

· · · (11)

lim inf

Hk−1→∞

1 Hk−1

H_k−1

∑

hk−1=1

lim sup

N→∞

sup

t∈R

1 N

N

∑

n=1

2 

2−(k−1)

dµ=0,

which would show that the averages inside the integral equals zero forµ-a.e. x∈Xsince the averages are

nonnegative. To do so, we apply Fatou’s lemma and Hölder’s inequality to show that the integral above

is bounded above by

lim inf

H1→∞

1 H1

H1

∑

h1=1

lim inf

H2→∞

1 H2

H2

∑

h2=1

· · ·

lim inf

H_k−1→∞

1 Hk−1

Hk−1

∑

h_k−1=1

Z

lim sup

N→∞

sup

t∈R

1 N

N

∑

n=1

2 dµ





2−(k−1)

.

Note that the integral above is bounded above by min

i=1,2

Fi,~h(k−1)

2

3 by the estimate (6). By applying lim infH_j → ∞ for eachi = 1, 2, . . . ,k−1, we conclude that the integral on the left hand side of (11)is

bounded above by the minimum of |||f1|||2

k+2or |||f2|||2k+2. Since either f1 or f2 belongs toZk+1(T)⊥, we know that either|||f1|||_k₊₂=0 or|||f2|||_k₊₂=0. Thus,(11)holds, which implies that ˜Xkis indeed a set of

full-measure.

Now we need to show that ifx ∈ _X˜_k_{, then the averages in}₍₄₎_{converge to 0 in}_L2₍_ν₎_{. But this follows} immediately from Lemma 2.1, since ifx∈_X˜_k_{, the right hand side of}₍₁₀₎_{is 0.}

3. When both f1and f2are inZ_k₊₁(T)(Proof of(b)and(c)ofTheorem1.6)

3.1. When one of the functionsg1,g2, . . .gk belongs to Zk(S)⊥. We first consider the case where either

(13)

by a seminorm on L∞₍_ν₎_{. We remark here that B. Host and B. Kra have obtained an estimate sharper}

than the one we provide, using the tools of nilsequences (cf. [21, Corollary 7.3]). However, the less sharp

estimate that we provide here is sufficient to prove our claim. We will also achieve this estimate without

the machinery of nilsequences.

We prove this for the case(Y,G,ν,S)is an ergodic system, and the general case holds by applying an

ergodic decomposition on(Y,S).

Proposition 3.1 (See also: [21, Corollary 7.3]). Let (Y,G,ν,S) is an ergodic system, (an)n ∈ ℓ∞ such that

|an| ≤1for each n, and functions g1, . . . ,gk∈ L∞(ν). Then

(12) lim sup

N→∞ 1 N

N−1

∑

n=0 an k

∏

i=1

gi◦Sin 2

L2_(ν)

≤2k+1 min

1≤i≤ki· |||gi|||

2

k+1.

Proof. We proceed by induction on k. For the case k = 1, we apply van der Corput’s lemma and the

Cauchy-Schwarz inequality to obtain

lim sup

N→∞ 1 N

N−1

∑

n=0

ang1◦Sn 2

L2_(ν)

≤lim inf

H→∞

2 H+

4 H

H−1

∑

h=0

lim sup

N→∞ 1 N

N−h

∑

n=0

anan+h !

Z

g1·g1◦Shdν

!

≤lim inf

H→∞ 



2 H+4

1 H

H−1

∑

h=0 Z

g1·g1◦Shdν

2!1/2

=4|||g1|||2₂,

which proves the base case.

Now suppose the statement holds fork=l−1; i.e. we assume that for any(bn)n∈ℓ∞andG1, . . . ,Gl−1∈ L∞

(ν), we have

(13) lim sup

N→∞ 1 N

N−1

∑

n=0

bn l−1

∏

i=1

Gi◦Sin 2

L2_(ν)

≤2l+1 min

1≤i≤li· |||Gi|||

2

l

To prove this for the case k = l, we again apply van der Corput’s lemma and the Cauchy-Schwarz

inequality to obtain

lim sup

N→∞ 1 N

N−1

∑

n=0 an l

∏

i=1

gi◦Sin 2

L2_(ν)

≤lim inf

H→∞

4 H

H−1

∑

h=0

lim sup

N→∞ Z ₁ N

N−1

∑

n=0

anan+h l

∏

i=1

gi·gi◦Sih

◦S(i−1)n_d_ν

≤lim inf

H→∞

4 H

H−1

∑

h=0

lim sup

N→∞ 1 N

N−1

∑

n=0

anan+h l

∏

i=2

gi·gi◦Sih

◦S(i−1)n _L₂_(ν)

(14)

By settingbn =anan+handGi= gi−1·gi−1◦S(i−1)hfor eachh, we can apply the inequality(13)to show that

lim sup

N→∞

1 N

N−1

∑

n=0

an l

∏

i=1

gi◦Sin

2

L2_(ν)

≤4 min 2≤i≤llim infH→∞

2l+1 H

H−1

∑

h=0

gi·gi◦S ih

_l

≤2l+3 min

2≤i≤li·lim infH→∞

1 H

H

∑

h=0

gi·gi◦S ih

2l

l !2−l

=2l+3 min

2≤i≤li· |||gi|||

2

l+1.

To keep|||g1|||2_l₊₁, we compute

lim sup

N→∞

1 N

N−1

∑

n=0

an l

∏

i=1

gi◦Sin

2

L2_(ν)

≤lim inf

H→∞

4 H

H−1

∑

h=0

lim sup

N→∞

Z ₁ N

N−1

∑

n=0

anan+h

∏

1≤i≤l,i6=j

gi·gi◦Sih

◦S(i−j)n_d_ν

≤22l+3 min

1≤i≤l,i6=ji· |||gi|||

2

l+1

for a fixedj. When these estimates are combined, we have

(14) lim sup

N→∞

1 N

N−1

∑

n=0

an l

∏

i=1

gi◦Sin

2

L2_(ν)

≤2l+3 min

1≤i≤li· |||gi|||

2

l+1,

which completes the proof.

With this estimate, Theorem 1.6(b) can be proven immediately.

Proof of (b) of Theorem1.6. Set an = f1(Tanx)f2(Tbnx)in Proposition 3.1. Since f1,f2 ∈ L∞, there exists a

set of full-measure ˆXk for which the sequence an ∈ ℓ∞. Because one of the functions g1, . . . ,gk belongs

toZk(S)⊥, we must have|||gi|||k =0 for one of them. Hence, we know from(12)that the averages must

converge to 0 inL2₍_ν₎_.

3.2. When all of the functionsg1, . . . ,gkbelong toZk(S). Here we use Leibman’s pointwise convergence

result on nilmanifold to show that the averages converges if all the functions belong to the appropriate

Host-Kra-Ziegler factors.

Theorem 3.2(Leibman, [24]). Let p(n)be a polynomial sequence in a nilpotent Lie group G. For any x∈ X =

G/Γ_{for some discrete compact subgroup}Γ_{, F}_{∈ C}₍_X₎_,

lim

N→∞

1 N

N

∑

n=1

F(p(n)x)

exists.

(15)

Proof of (c) of Theorem1.6. With appropriate factors maps, we assume (X,F,µ,T) and (Y,G,ν,S) to be nilsystems, i.e. X=G1/Γ_{1, and}_Y₌_G2/Γ_{2, where}_G1_{is a}₍_k₊₁₎_{-step nilpotent Lie group,}_G2_{is a}_k-step

nilpotent Lie group , andΓ₁_andΓ₂_{are discrete co-compact subgroups of}_G1_and_{G2, respectively. In this}

proof, we will assume that f1,f2 ∈ C(X), and g1, . . . ,gk ∈ C(Y). By taking the product ofX2andYk, we

would have another nilmanifold:

X2_×_Yk_{= (}_G1/_Γ

1)2×(G2/Γ2)k∼= (G12×G2k)/(Γ21×Γ2k).

Letτ ∈ G1such that the action ofτ on an element ofXis determined to beτ·x = Tx. Similarly, we

defineσ∈G2so thatσ·y=Sy. We define a polynomial sequence ponX2×Yk_{as follows:}

p(n) = (τan,τbn,σn,σ2n, . . . ,σkn).

Clearly, p(n)∈G2

1×G2kfor alln∈Z, and it acts onX2×Ykin a way that p(n)·(x1,x2,y1, . . . ,yk) = (Tanx1,Tbnx2,Sny1, . . . ,Sknyk).

Define a continuous functionF∈ C(X2_×_Yk₎_{such that}

F(x1,x2,y1, . . . ,yk) = f1(x1)f2(x2) k

∏

j=1

gj(yj).

Theorem 3.2 tells us that the averages

1 N

N

∑

n=1

F(p(n)·(x1,x2,y1, . . . ,yk)) = _N1 N

∑

n=1

f1(Tanx1)f2(Tbnx2) k

∏

j=1

gj(Sjnyj)

converge for all(x1,x2,y1, . . . ,yk)∈X2×Yk. So in particular, if the averages were taken a point(x,x,y, . . . ,y)∈

X2_×_Yk_{for any}_x_∈_X_and_y_∈_{Y, the desired convergence result holds.}

By a standard approximation argument, we can extend this result for the case f1,f2∈L∞₍_µ₎_{∩ Z} k+1(T)

and g1, . . . ,gk ∈ L∞(ν)∩ Zk(S). In this process, we neglect a null-set for which the averages may not

converge, which allows us to obtain a set of full-measureX′

k⊂Xthat satisfies (c) of Theorem 1.6.

Remarks

(1) Recently, the first author announced in [4] that the Wiener-Wintner result obtained in [5] can be

extended to a nilsequence Wiener-Wintner result, providing a positive answer to the question

raised by B. Weiss in 2014 Ergodic Theory Workshop at UNC Chapel Hill. A similar result was

also recently announced by P. Zorin-Kranich [29].

(2) Recently, we have extended Theorem 1.5 so that the sequence an = f1(Tanx)f2(Tbnx) isµ-a.e. a

good universal weight for multiple recurrent averages with commuting transformations. More

(16)

Theorem ( [6, Theorem 1.1]). Let (X,F,µ,T) be a measure-preserving system, and suppose f1,f2 ∈ L∞₍_µ₎_{. Then there exists a set of full-measure X}

f1,f2 such that for any x ∈ Xf1,f2, for any a,b ∈ Zand

any positive integer k ≥ 1, for any other measure-preserving system with k commuting transformations

(Y,G,ν,S1,S2, . . .Sk), and for any g1,g2, . . .gk∈L∞(ν), the averages

1 N

N−1

∑

n=0

∏

i=1

gi◦Sni converge in L2(ν).

In other words, in terms of Definition 1.2, we have shown that for µ-a.e. x ∈ X, the sequence

(f1(Tan_x₎_f2₍_Tbn_x₎₎_n_{is a good universal weight for the process}₍_X_n₎_n_{for norm convergence, where}

(Xn)nis of the form

Xn = k

∏

i=1

gi◦Sni,

whereg1,g2, . . . ,gk∈L∞(ν)for any measure-preserving system with commuting transformations

(Y,G,ν,S1,S2, . . . ,Sk), for any positive integerk≥1.

References

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[2] I. Assani. Multiple return times theorems for weakly mixing systems.Ann. Inst. Henri Poincaré, Probabilités et Statistiques, 36:153– 165, 2000.

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[5] I. Assani, D. Duncan, and R. Moore. Pointwise characteristic factors for Wiener-Wintner double recurrence theorem.Ergod. Th. and Dynam. Sys., 2015. Available on CJO 2015 doi:10.1017/etds.2014.99.

[6] I. Assani and R. Moore. A good universal weight for multiple recurrence averages with commuting transformations in norm. Preprint, available on arXiv:1506.05370, June 2015.

[7] I. Assani and R. Moore. Extension of Wiener-Wintner double recurrence theorem to polynomials. Available on http://www.unc.edu/math/Faculty/assani/WWDR_poly_final_abSept19.pdf, submitted, September 2014.

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[10] I. Assani and K. Presser. Survey of the return times theorem. InErgodic Theory and Dynamical Systems: Proceedings of the Ergodic Theory Workshops at University of North Carolina at Chapel Hill, 2011-2012, De Gruyter Proc. Math., pages 19–58. De Gruyter, Berlin, 2014.

[11] V. Bergelson, B. Host, and B. Kra with appendix by I Ruzsa. Multiple recurrence and nilsequences.Invent. Math., 160:261–303, 2005.

[12] J. Bourgain. Return time sequences of dynamical systems. Unpublished preprint, 1988.

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Department of_Mathematics_{, T}he_University of_North_Carolina at_Chapel_Hill_{, C}hapel_Hill_{, NC 27599} E-mail address:[email protected]

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Department ofMathematics, TheUniversity ofNorthCarolina atChapelHill, ChapelHill, NC 27599 E-mail address:[email protected]