Divergence of weighted square averages in $L^1$

TL;DR

This paper investigates the divergence of weighted ergodic averages along squares in $L^1$, showing that divergence occurs for a large set of frequencies, extending previous divergence results and highlighting differences from linear weights.

Contribution

It extends divergence results for quadratic ergodic averages in $L^1$ and demonstrates that convergence for linear weights does not hold at $p=1$.

Findings

The set of frequencies causing divergence is residual and includes rationals and dense Liouville numbers.

Divergence along squares in $L^1$ is more prevalent than previously known.

Convergence results for linear weights in $L^p$, $p>1$, do not extend to $p=1$.

Abstract

We study convergence of ergodic averages along squares with polynomial weights. For a given polynomial $P\in \mathbb{Z}[\cdot]$, consider the set of all $\theta\in[0,1)$ such that for every aperiodic system $(X,\mu, T)$ there is a function $f\in L^1(X,\mu)$ such that the weighted averages along squares $$ {\frac{1}{N}\sum_{n=1}^N} e(P(n)\theta)T^{n^2}f $$ diverge on a set with positive measure. We show that this set is residual and includes the rational numbers as well as a dense set of Liouville numbers. This on one hand extends the divergence result for squares in $L^1$ of the first author and Mauldin and on the other hand shows that the convergence result for linear weights for squares due to the second author and Krause in $L^p$, $p>1$ does not hold for $p=1$.

Equations

$$\frac{1}{N}\sum _{n=1}^{N}e(P(n)\theta )T^{n^2}f$$

$$\frac{1}{N}\sum _{n=1}^N{T}^n{f}_1\cdot T^{2n}{f}_2\cdots T^{kn}{f}_k$$

$$\frac{1}{N}\sum _{n=1}^N{T}^{k_n}f$$

$$\frac{1}{N}\sum _{n=1}^N{a}_n{T}^{n}f$$

$$\frac{1}{N}\sum _{n=1}^{N}e(n\theta )T^{n^2}f$$

$$\frac{1}{N}\sum _{n=1}^{N}e(n\theta )T^n{f}_1\cdot T^{2n}{f}_2$$

$$\frac{1}{N}\sum _{n=1}^N{a}_n{T}^{k_n}f$$

$$u={\int }_{[0,1]}X(x)d\lambda (x)=0.99^2\cdot M\cdot 2^{-M-1}>0.9\cdot M\cdot 2^{-M-1}.$$

$$\lambda \left\{x:\frac{1}{K}\sum _{h=1}^K{X}_h(x)\ge \frac{u}{2}\right\}>1-\delta .$$

$$\frac{1}{N_x}\sum _{k=1}^{N_x}g(T^{k^2}(x))>\sum _{h=1}^K{X}_h(x).$$

$$\underset{N\in [{\mathbf{N}}_1,{\mathbf{N}}_2]}{\max }\frac{1}{N}\sum _{n=1}^N(T^{n^2}f)(l)>C$$

$$\frac{1}{p}<\frac{\epsilon }{4}\text{ and }32\cdot C\cdot 4^{-p}<\frac{\epsilon }{2}.$$

$$\lambda (U_p^{\prime })>1-\delta \text{ for }{U}_p^{\prime }:=\left\{x:\frac{1}{K}\sum _{h=1}^K{X}_h(x)>\frac{0.9}{2}\cdot M\cdot 2^{-M-1}\right\}.$$

$$\frac{1}{N_x}\sum _{k=1}^{N_x}g(T^{k^2}x)>\sum _{h=1}^K{X}_h(x)$$

$$\frac{1}{N_x}\sum _{k=1}^{N_x}g(T^{k^2}x)>\sum _{h=1}^K{X}_h(x)>K\cdot \frac{0.9}{2}\cdot M_p\cdot 2^{-M_p-1}.$$

$${\tilde{U}}_p=\left\{x:\underset{N}{\sup }\frac{1}{N}\sum _{k=1}^{N}g(T^{k^2}x)>t_p\right\}$$

$$\frac{\int gd\lambda }{t_p}<\frac{K\cdot 2^{-M_p+2}}{K\cdot \frac{0.9}{2}\cdot M_p\cdot 2^{-M_p-1}}<\frac{32}{M_p}.$$

$${\varphi}_{\tau}(x)=\frac{1}{\tau}\sum_{j=0}^{\tau-1}{\varphi}\Big{(}x+\frac{j}{\tau}\Big{)}\text{ and }\lambda_{\tau}=\tau\cdot\lambda|_{[0,1/\tau)},$$

$$\int_{[0,1)}{\varphi}(x)d\lambda(x)=\int_{[0,1/\tau)}\sum_{j=0}^{\tau-1}{\varphi}\Big{(}x+\frac{j}{\tau}\Big{)}d\lambda(x)$$

$$={\int }_{[0,1/\tau )}{\phi }_{\tau }(x)\cdot \tau d\lambda (x)={\int }_{[0,1/\tau )}{\phi }_{\tau }(x)d{\lambda }_{\tau }(x).$$

$${\int }_{[0,1/\tau )}{\mu }_{\tau ,x}(\overline{E})d{\lambda }_{\tau }(x)={\int }_{[0,1)}{\chi }_{\overline{E}}(x)d\lambda (x)=\lambda (\overline{E})<\frac{\epsilon }{2}.$$

$$\lambda_{\tau}\Big{(}\Big{\{}x\in[0,1/\tau):\mu_{\tau,x}(\overline{E})\geq\varepsilon\Big{\}}\Big{)}<\frac{1}{2}.$$

$${\int }_{[0,1/\tau )}{G}_{\tau }(x)d{\lambda }_{\tau }(x)={\int }_{[0,1)}G(x)d\lambda (x)<\frac{\epsilon }{2}.$$

$$\lambda_{\tau}\Big{(}\Big{\{}x\in[0,1/\tau):G_{\tau}(x)\geq\varepsilon\Big{\}}\Big{)}<\frac{1}{2}.$$

$$\underset{N\in [{\mathbf{N}}_1,{\mathbf{N}}_2]}{\max }\frac{1}{N}\sum _{n=1}^N(T^{n^2}f)(x)>C$$

$$(T^{\ast }f)(x):=\underset{n\in \mathbb{N}}{\sup }|(T_{n}f)(x)|\in [0,\infty ],x\in X,f\in L^1(X,\mu ).$$

$$\mu (T^{\ast }f>\lambda )\le \frac{C\Vert f{\Vert }_1}{\lambda }.$$

$$\mu \left(\underset{N\in \mathbb{N}}{\sup }\left|\frac{1}{N}\sum _{n=1}^N{a}_n{T}^{k_n}f\right|>\lambda \right)\le \frac{C}{\lambda }\Vert f{\Vert }_1\forall \lambda >0$$

$$\mu \left(\underset{1\le N\le M}{\sup }\left|\frac{1}{N}\sum _{n=1}^N{a}_n{\tilde{T}}^{k_n}f\right|>\lambda \right)\le \frac{C}{\lambda }\Vert f{\Vert }_1.$$

$$\mu \left(\underset{1\le N\le M}{\sup }\left|\frac{1}{N}\sum _{n=1}^N{a}_n{S}_l{T}^{k_n}{S}_l^{-1}f\right|>\lambda \right)\le \frac{C}{\lambda }\Vert f{\Vert }_1$$

Full text

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diverge on a set with positive measure. We show that this set is residual and includes the rational numbers as well as a dense set of Liouville numbers.

On one hand, this extends the divergence result for unweighted averages along squares in $L^{1}$ of the first author and Mauldin; on the other hand, it shows that the convergence result for linear weights for squares in $L^{p}$, $p>1$, due to Bourgain as well as the second author and Krause does not hold for $p=1$.

The first listed author was supported by the Hungarian National Foundation for Scientific Research Grant 124003.

1. Introduction

Originally motivated by physics, ergodic theory became an independent mathematical discipline in the 1930s with the appearence of the classical ergodic theorems due to von Neumann and Birkhoff. Since then, ergodic theorems have been extended and generalized in many directions and surprising connections to other areas of mathematics have been discovered. We mention three major directions: the multiple, the subsequential and the weighted ergodic theorems.

• •

Multiple ergodic theorems, dealing with averages of the form

[TABLE]

and introduced by Furstenberg in his celebrated ergodic theoretic proof of Szemerédi’s theorem, have deep connections to, e.g., combinatorics, harmonic analysis, number theory, group theory and form an active area of research, see Furstenberg [36, 37], Furstenberg, Katznelson [38], Bourgain [14], Host, Kra [42], Ziegler [64], Leibman [50], Bergelson, Leibman, Lesigne [9], Tao [59], Walsch [61], Donoso, Sun [26].

• •

Subsequential ergodic theorems concern averages

[TABLE]

for a subsequence $(k_{n})$ of $\mathbb{N}$ and are natural from the physical point of view. They have been studied by Furstenberg, Bourgain, Wierdl and others using methods from harmonic analysis and number theory, see Bourgain [13], Wierdl [62], Nair [54], Bellow, Losert [7], Rosenblatt, Wierdl [56], Akcoglu, Bellow, Jones, Losert, Reinhold-Larsson, Wierdl [1], Krause [47], Zorin-Kranich [65], Eisner [28].

• •

Weighted ergodic theorems concerning averages of the form

[TABLE]

with $(a_{n})\subset\mathbb{C}$ go back to the Wiener-Wintner theorem [63] and are connected to both additive number theory and the recent Sarnak conjecture, see Green, Tao [39], Sarnak [57], El Abdalaoui, Kulaga-Przymus, Lemańczyk, de la Rue [5].

The Wiener-Wintner theorem deals with pointwise convergence for the family of linear weights $(e(n\theta))$, $\theta\in\mathbb{R}$, where $e(x):=e^{2\pi ix}$, see Assani [2]. Lesigne [51, 52] extended it to polynomial weights of the form $(e(P(n)))_{n=1}^{\infty}$ for real polynomials $P$, see also Frantzikinakis [35] and Assani [2] and the generalization to nilsequences by Host, Kra [43] and Eisner, Zorin-Kranich [31]. For topological versions of the Wiener-Wintner theorem and its generalizations see Robinson [55], Assani [2, Chapter 2.6] and Fan [32].

For more weighted ergodic theorems see Bellow, Losert [7], Berend, Lin, Rosenblatt, Tempelman [8], Bourgain, Furstenberg, Katznelson, Ornstein [15], Lin, Olsen, Tempelman [53], Eisner, Lin [30], Fan [33]. For a different type of weighted ergodic theorems with arithmetic weights we refer to Cuny, Weber [25] and Buczolich [17].

A related question to weighted ergodic theorems is about the convergence of ergodic series $\sum_{n=0}^{\infty}a_{n}f(T^{n}x).$ We can refer to Cohen and Lin [22], Fan [34] and Cuny and Fan [24]. In Izumi [44] the question about the convergence of $\sum\frac{1}{n}f(T^{n}x)$ was first raised. Positive answers would be improvements of the ergodic theorems. But there are no such answers in general. Negative answers were given by Halmos [40], Dowker and Erdős [27], moreover by Kakutani and Petersen [46].

Surprisingly, pointwise convergence of the simplest mixture of the subsequential and weighted ergodic averages, namely polynomial averages with polynomial weights, is still open in general. Pointwise convergence of the unweighted square averages $\frac{1}{N}\sum_{n=1}^{N}T^{n^{2}}f$ was proved by Bourgain [10, 11, 13] for $L^{p}$, $p>1$, answering a question of Bellow and Furstenberg, see also Krause [47], whereas divergence in $L^{1}$ was shown by Buczolich, Mauldin [19], extended by LaVictoire [60] to all monomials. The averages along squares with linear weights

[TABLE]

were treated by Bourgain [12, 13] and Eisner, Krause [29] where pointwise convergence in $L^{p}$, $p>1$, was shown for every $\theta$, partially uniformly in $\theta$. Note that a Wiener-Wintner type result for the weighted double averages

[TABLE]

for bounded functions was obtained by Assani, Duncan, Moore [3] from Bourgain’s unweighted double recurrence result [14]. See also Assani, Moore [4] for an extension to polynomial weights.

In this paper we study convergence of the averages (1.1) for $f\in L^{1}$ and show that for many $\theta$ the weight $(e(\theta n))$ is $L^{1}$-universally bad, extending the mentioned above convergence results in $L^{p}$, $p>1$, by Bourgain and Eisner, Krause as well as the divergence result of unweighted averages by Buczolich, Mauldin in $L^{1}$. More precisely, we treat polynomial weights $(e(P(n)\theta))$ with $P\in\mathbb{Z}[\cdot]$.

Definition 1.1**.**

Let $(a_{n})\subset\mathbb{C}$ and $(k_{n})$ be a subsequence of $\mathbb{N}$. We say that the pair $((a_{n}),(k_{n}))$ is $L^{1}$-universally bad if for every ergodic invertible system $(X,\mu,T)$ on a nonatomic standard probability space there is $f\in L^{1}(X,\mu)$ such that the weighted averages along $(k_{n})$

[TABLE]

diverge on a set of positive measure.

Some authors use aperiodic instead of ergodic in the definition of universally bad sequences, see for example [2], [6] and [7]. One can show analogously to the unweighted case that the two versions of the definition are equivalent. In this paper we use the ergodic version and this way we follow for example [19], [56] and [60].

Thus, using the terminology of Definition 1.1, Buczolich and Mauldin [19] showed that $((1),n^{2})$ is $L^{1}$-universally bad. This result was slightly generalized in [60] by LaVictoire to show that $((1),n^{d})$ is $L^{1}$-universally bad and in [16] by Buczolich to show that for any polynomial $q(n)$ of degree two with integer coefficients the sequence $((1),q(n))$ is $L^{1}$-universally bad. Both these generalizations were based on the original argument of [19]. It is still an open question whether for a ”general” polynomial $q(n)$ of degree at least three with integer coefficients the sequence $((1),q(n))$ is $L^{1}$-universally bad.

In this paper we restrict our attention to the base case $k_{n}=n^{2}$. For the above mentioned slightly generalized cases which depend on the argument of [19] similar results to Theorem 1.2 can be obtained. The case of general polynomials $q(n)$ of degree at least three with integer coefficients is a challenging unsolved problem.

Our main result is the following.

Theorem 1.2**.**

Let $P\in\mathbb{Z}[\cdot]$ be a polynomial and let $\mathcal{M}$ be the set of all $\theta\in[0,1)$ such that the pair $(e(\theta P(n)),(n^{2}))$ is $L^{1}$-universally bad. Then the following assertions hold.

1. (a)

$\mathbb{Q}\cap[0,1)\subset\mathcal{M}$. 2. (b)

$\mathcal{M}\setminus\mathbb{Q}$* contains a dense set of Liouville numbers.* 3. (c)

$\mathcal{M}$* is a dense $G_{\delta}$ subset of $[0,1]$ and therefore residual.*

2. Main tool

Next we need to recall some definitions and theorems, and tweak some arguments from [19]. We start with the definition of $M-0.99$ distributed random variables.

Definition 2.1**.**

For a positive integer $M$ we say that a function or a random variable, $X:[0,1)\to\mathbb{R}$ is M$$-$$0.99* distributed* if $X(x)\in\{0,0.99,0.99\cdot\frac{1}{2},...,0.99\cdot 2^{-M+1}\}$, and $\lambda(\{x\in[0,1):X(x)=0.99\cdot 2^{-l}\})=0.99\cdot 2^{-M+l-1}$ for $l=0,...,M-1$, where $\lambda$ denotes the Lebesgue measure.

An easy calculation shows that

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On the probability space $([0,1),\lambda)$ we can consider pairwise independent $M-0.99$-distributed random variables $X_{h}$ for $h=1,...,K$ for a sufficiently large $K$. Assume that $u$ denotes the mean of these variables.

By the weak law of large numbers $\lambda\left\{x:\left|\frac{1}{K}\sum_{h=1}^{K}X_{h}(x)-u\right|\geq\frac{u}{2}\right\}\to 0$ as $K\to\infty$. Given $\delta>0$ we can select $K$ so large that

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With slight change of notation we recall Theorem 8 from [19].

Theorem 2.2**.**

Given $\delta>0$, $M$ and $K$ there exist $\tau_{0}\in{\mathbb{N}},$ which defines a translation on $[0,1)$ by $T(x)=x+\frac{1}{\tau_{0}}$ modulo $1$, $E_{\delta}{\subset}[0,1)$ with $\lambda(E_{\delta})<\delta$, a measurable function $g:[0,1)\to[0,+{\infty})$ with $\int_{[0,1)}gd\lambda<K\cdot 2^{-M+2},$ and pairwise independent M$$-$$0.99-distributed random variables $X_{h},$ $h=1,...,K$ defined on $([0,1),\lambda)$ such that for all $x\in[0,1){\setminus}E_{\delta}$ there exists $N_{x}$ satisfying

[TABLE]

This theorem is highly non-trival and its proof in [19] is quite technical. In [18] there is some heuristic outline of this argument. One of the key elements of this argument is coming from number theory and is related to the randomness of quadratic residues (see [48] and [49]). Another version of the argument of [19] can be found in [60].

Our main tool will be the following corollary of Theorem 2.2. Here and later, we denote by $\mathbb{Z}_{d}$ the cyclic group $\{0,\ldots,d-1\}$. On $\mathbb{Z}_{d}$ we denote by $T$ the shift transformation $Tx=x+1\mod d$.

Corollary 2.3**.**

For every ${\mathbf{N}}_{1}\in\mathbb{N}$ and every $\varepsilon,C>0$ there exist an arbitrarily large $\tau\in\mathbb{N}$, a set $E\subset\mathbb{Z}_{\tau}$ with proportion less than $\varepsilon$ in $\mathbb{Z}_{\tau}$, a positive bounded function $f$ on $\mathbb{Z}_{\tau}$ with $\int f\,d\mu<\varepsilon$ ($\mu$ being the normalised counting measure), and ${\mathbf{N}}_{2}>{\mathbf{N}}_{1}$ such that the inequality

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holds for every $l\in\mathbb{Z}_{\tau}\setminus E$.

Verification of Corollary 2.3 based on methods of [19].

We can now modify slightly the proof of Theorem 1 on p. 1528 of [19]. We select $p$ such that

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We let $M_{p}=4^{p}$, $\delta=\frac{1}{p}$ and select $K$ such that for $M_{p}-0.99$-distributed random variables $X_{h}$, $h=1,...,K$ we have (2.2) satisfied and hence for this $K$ by (2.1) we have

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By Theorem 2.2 used with $\delta=\frac{1}{p},$ $M_{p}$ and $K$ there exist $\tau_{0}\in\mathbb{N},$ $E_{1/p}\subset[0,1)$ and a periodic transformation $T:[0,1)\to[0,1),$ $T(x)=x+\frac{1}{\tau_{0}}$ modulo $1,$ $g:[0,1)\to[0,+\infty)$, $X_{h}$ pairwise independent $M_{p}-0.99$-distributed random variables defined on $[0,1)$ such that $\lambda(E_{1/p})<\frac{1}{p}$ and for all $x\in[0,1)\setminus E_{1/p}$ there exists $N_{x}$ such that

[TABLE]

and $\int_{[0,1)}fd\lambda<K\cdot 2^{-M_{p}+2}.$

One can observe that on p. 1527 of [19] at the beginning of the proof of Theorem 8 one can choose $A={\mathbf{N}}_{1}$ instead of $A=1$ and this implies that $N_{x}\geq{\mathbf{N}}_{1}$ holds in (2.3). By using a slightly larger exceptional set ${\widetilde{E}_{1/p}}\supset{{E_{1/p}}}$ still satisfying $\lambda(\widetilde{E}_{1/p})<1/p$ we can select ${\mathbf{N}}_{2}$ such that ${\mathbf{N}}_{1}\leq N_{x}\leq{\mathbf{N}}_{2}$ holds for any $x\not\in\widetilde{E}_{1/p}$. One can also observe that any integer multiple of $\tau_{0}$ could also be used, so $\tau_{0}$ can be arbitrarily large.

Put $U_{p}=U_{p}^{\prime}\setminus\widetilde{E}_{1/p}.$ Then $\lambda(U_{p})>1-\frac{2}{p}$ and for $x\in U_{p}$ there exists $N_{x}\in[{\mathbf{N}}_{1},{\mathbf{N}}_{2}]$ such that

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Thus letting $t_{p}=K\cdot\frac{0.9}{2}\cdot M_{p}\cdot 2^{-M_{p}-1}$, and

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we have $U_{p}\subset\widetilde{U}_{p}$ and hence $\lambda(\widetilde{U}_{p})>1-\frac{2}{p}.$ On the other hand,

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Now set $G=\frac{C}{t_{p}}\cdot g.$ Then (2.5) and the above inequality imply that $\int Gd\lambda<\varepsilon/2.$ We also put $\overline{E}=[0,1)\setminus{\widetilde{U}}_{p}$. Then $\lambda(\overline{E})<\varepsilon/2$.

Next we need a sort of a transference argument to move the above results onto the integers.

Given ${\varphi}:[0,1)\to\mathbb{R}$, $x\in[0,1/\tau)$ put

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where $\lambda|_{[0,1/\tau)}$ is the restriction of the Lebesgue measure onto $[0,1/\tau)$.

By this notation we have

[TABLE]

[TABLE]

For $x\in[0,1/\tau)$ and a measurable set $A\subset[0,1)$ let \mu_{\tau,x}(A):=\chi_{A,\tau}(x)=\frac{1}{\tau}\sum_{j=0}^{\tau-1}\chi_{A}\Big{(}x+\frac{j}{\tau}\Big{)}. Then using (2.8) with ${\varphi}(x)=\chi_{\overline{E}}(x)$ we obtain that

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This implies that

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Using (2.8) with ${\varphi}(x)=G(x)\geq 0$ we obtain similarly

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and this implies that

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Since $\lambda_{\tau}([0,1/\tau))=1$ by (2.9) and (2.10) we can select an $x^{*}\in[0,1/\tau)$ such that $\mu_{\tau,x^{*}}(\overline{E})<\varepsilon$ and $G_{\tau}(x^{*})<\varepsilon.$

Now we can define $f:\mathbb{Z}\to[0,+\infty)$, periodic by $\tau$ such that $f(l)=G(x^{*}+\{l/\tau\})$ where $\{.\}$ denotes fractional part, this also defines a function on $\mathbb{Z}_{\tau}$ which, for ease of notation is also denoted by $f$.

To define the exceptional set $E$ we say that $l\in\mathbb{Z}$ belongs to $E$ iff $x^{*}+\{l/\tau\}\in\overline{E}$. Then $f$ and $E$ are both periodic by $\tau$, $\int fd\mu<\varepsilon$, $\mu(E)<\varepsilon$, the definition of $G$ and (2.7) imply (2.4). ∎

{comment}

We will use the following corollary of results in Buczolich, Mauldin [19].

Theorem 2.4** (Buczolich, Mauldin).**

For every ${\mathbf{N}}_{1}\in\mathbb{N}$ and every $\varepsilon,C>0$ there exist an arbitrarily large $\tau\in\mathbb{N}$, a periodic transformation $T:[0,1)\to[0,1)$ given by $Tx=x+\frac{1}{\tau}\mod 1$, a set $E\subset[0,1)$ with $\lambda(E)<\varepsilon$, a function $f\in L^{\infty}(X,\mu)$ with $f\geq 0$ and $\int f\leq\varepsilon$, and ${\mathbf{N}}_{2}>{\mathbf{N}}_{1}$ such that the inequality

[TABLE]

holds for every $x\in[0,1)\setminus E$.

3. Weighted Conze Principle

Definition 3.1**.**

Let $(X,\mu)$ be a probability space and let $(T_{N})$ be a sequence of bounded linear operators on $L^{1}(X,\mu)$. Define the corresponding maximal operator $T^{*}$ by

[TABLE]

We say that $(T_{N})$ satisfies a weak $(1,1)$ maximal inequality if there exists a constant $C>0$ such that for every $f\in L^{1}(X,\mu)$ and every $\lambda>0$

[TABLE]

The following is a corollary of Sawyer’s variation of Stein’s principle, see [58, Corollary 1.1].

Lemma 3.2** (Sawyer).**

Let $(X,\mu,\mathbf{T})$ be an ergodic measure-preserving dynamical system and let $(T_{N})$ be a sequence of bounded linear operators on $L^{1}(X,\mu)$ commuting with the Koopman operator $\mathbf{T}$. Assume that $(T_{N})$ does not satisfy a weak $(1,1)$ maximal inequality. Then there exists a function $f\in L^{1}(X,\mu)$ such that $T^{*}f=\infty$ a.e., and, in particular, $(T_{N}f)$ diverges a.e. (Moreover, the set of such functions is residual in the Baire category sense.)

We will need the following variation of Conze’s principle.

Theorem 3.3** (Weighted Conze principle).**

Let $(a_{n})\subset\mathbb{C}$ be bounded and $(k_{n})$ be a subsequence of $\mathbb{N}$. Let $C\leq\infty$ be minimal such that for every system $(X,\mu,T)$ and every $f\in L^{1}(X,\mu)$

[TABLE]

*holds. Then $C<\infty$ if and only if there exists an ergodic invertible system $(X,\mu,T)$ on a nonatomic standard probability space such that for every $f\in L^{1}(X,\mu)$, the weighted averages (1.2) converge a.e. Equivalently, $C=\infty$ if and only if $((a_{n}),(k_{n}))$ is $L^{1}$-universally bad. *

The proof is an adaptation of the argument in Rosenblatt, Wierdl [56, Proof of Theorem 5.9] which is based on the original work by Conze [23].

Proof.

The “only if” part is trivial. To show the “if” part, assume that for an ergodic invertible system $(X,\mu,T)$ on a nonatomic standard probability space and every $f\in L^{1}(X,\mu)$, the weighted averages (1.2) converge a.e. By Lemma 3.2, there is $C>0$ such that (3.2) holds for every $f\in L^{1}(X,\mu)$.

Since all nonatomic standard probability spaces are isomorphic, it suffices to show that for every (invertible) transformation $\tilde{T}$ on $(X,\mu)$, (3.2) holds for the same constant $C$ and every $f\in L^{1}(X,\mu)$. Take such $\tilde{T}$. By the Halmos conjugacy lemma, there exists a sequence $(S_{l})$ of invertible transformations such that $\lim_{l\to\infty}S_{l}TS_{l}^{-1}=\tilde{T}$ in the weak topology. Then by a standard approximation argument, the Koopman operators on $L^{1}(X,\mu)$ (which we denote by the same letter) satisfy $\lim_{l\to\infty}S_{l}TS_{l}^{-1}=\tilde{T}$ in the strong operator topology. Thus $\lim_{l\to\infty}S_{l}T^{n}S_{l}^{-1}=\tilde{T}^{n}$ in the strong operator topology for every $n\in\mathbb{N}$.

Let now $f\in L^{1}(X,\mu)$, $\lambda>0$ and $M\in\mathbb{N}$. By monotonicity it suffices to show that

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Since $(a_{n})$ is bounded, $\frac{1}{N}\sum_{n=1}^{N}a_{n}S_{l}T^{k_{n}}S_{l}^{-1}f$ converges to $\frac{1}{N}\sum_{n=1}^{N}a_{n}\tilde{T}^{k_{n}}f$ in $L^{1}(X,\mu)$ for every $N\in\mathbb{N}$, and the same of course holds for $\sup_{1\leq N\leq M}$ of the absolute value. Since for every sequence $(g_{n})\subset L^{1}(X,\mu)$ converging in norm to $g\in L^{1}(X,\mu)$ one has $\lim_{n\to\infty}\mu(x:g_{n}(x)>\lambda)=\mu(x:g(x)>\lambda)$, it suffices to show that

[TABLE]

or equivalently, by the measure-preserving property of $S_{l}$,

[TABLE]

for $g:=S_{l}^{-1}f$. But this holds by the definition of the constant $C$ and monotonicity. ∎

4. Proof of the $G_{\delta}$ property

We first prove that $\mathcal{M}$ is a $G_{\delta}$ set. The denseness follows from Theorem 1.2 a) or b) and will be proven in the following sections.

Proof of Theorem 1.2 c) assuming Theorem 1.2 a) or b).

Observe that by Theorem 3.3, $\theta\in\mathcal{M}$ if and only if for every $C\in\mathbb{Q}$ there exist a system $(X,\mu,T)$, a function $f\in L^{1}(X,\mu)$ with $\|f\|_{1}=1$ and $\lambda>0$ such that

[TABLE]

Since $\displaystyle\frac{1}{N}\sum_{n=1}^{N}e(P(n)\theta)T^{n^{2}}f(x)$ appears at many places in this proof we introduce the notation

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By monotonicity of the sets $\{x\in X:\max_{N\leq k}|\sigma_{N}^{(\theta)}f(x)|>\lambda\},$ (4.1) is equivalent to the existence of $k\in\mathbb{N}$ such that

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Thus we obtain

[TABLE]

where under the union sign $(X,\mu,T)$ denotes an arbitrary measure-preserving system, $f\in L^{1}(X,\mu)$ an arbitrary function with $\|f\|_{1}=1$, $\lambda>0$ an arbitrary real number and $k$ an arbitrary natural number. It remains to show that each of the sets on the right is open, and for that it suffices to show that for given $(X,\mu,T)$, $f$, $\lambda$ and $k$, the $1$-periodic function

[TABLE]

is continuous.

Let $\theta\in\mathbb{R}$, $(\theta_{j})_{j=1}^{\infty}\subset\mathbb{R}$ with $\lim_{j\to\infty}\theta_{j}=\theta$ and let $\varepsilon>0$. Using the elementary estimate $|e(x)-e(y)|=|e^{2\pi ix}-e^{2\pi iy}|\leq 2\pi|x-y|$, by

[TABLE]

we obtain for every $j\in\mathbb{N}$

[TABLE]

Since $T$ is $\mu$-preserving and $\|f\|_{1}=1$, the last summand on the right hand side equals

[TABLE]

Therefore we have for every $\varepsilon>0$

[TABLE]

implying, by letting $\varepsilon\to 0$,

[TABLE]

Analogously one shows $g(\theta)\geq\limsup_{j\to\infty}g(\theta_{j})$, implying the continuity of $g$ and completing the proof.

∎

5. Reduction

We first reduce Theorem 1.2 a) and b) to the following.

Theorem 5.1**.**

Let $P\in\mathbb{Z}[\cdot]$ be a polynomial with $P(0)=0$. For every rational number $\frac{p}{q}\in[0,1)$ and every $k\in\mathbb{N}$ there exist $r>0$, a system $(X,\mu,T)$ and a positive function $f\in L^{\infty}(X,\mu)$ satisfying

[TABLE]

for every $\theta\in[\frac{p}{q}-r,\frac{p}{q}+r]$.

The system $(X,\mu,T)$ will be a shift on $\mathbb{Z}_{\tau q^{2}}$ with suitable $\tau$ and $q$. We now show that Theorem 5.1 implies Theorem 1.2.

Proof of Theorem 1.2 a) and b) based on Theorem 5.1.

Assume that Theorem 5.1 holds and let $P\in\mathbb{Z}[\cdot]$. Since multiplication by a non-zero constant does not affect divergence, we can assume without loss of generality that $P(0)=0$.

Let $\frac{p_{1}}{q_{1}}\in[0,1)$ be arbitrary. The claim for $k:=1$ implies the existence of an arbitrarily small $r_{1}>0$ such that there exist a system $(X_{1},\mu_{1},T_{1})$ and a positive $f_{1}\in L^{\infty}(X_{1},\mu_{1})$ such that

[TABLE]

holds for every $\theta\in[\frac{p_{1}}{q_{1}}-r_{1},\frac{p_{1}}{q_{1}}+r_{1}]$. Take now an arbitrary rational number $\frac{p_{2}}{q_{2}}\in(\frac{p_{1}}{q_{1}}-r_{1},\frac{p_{1}}{q_{1}}+r_{1})$. The claim for $k:=2$ implies the existence of an arbitrarily small $r_{2}$, a system $(X_{2},\mu_{2},T_{2})$ and a positive function $f_{2}\in L^{\infty}(X_{2},\mu_{2})$ such that

[TABLE]

holds for every $\theta\in[\frac{p_{2}}{q_{2}}-r_{2},\frac{p_{2}}{q_{2}}+r_{2}]\subset[\frac{p_{1}}{q_{1}}-r_{1},\frac{p_{1}}{q_{1}}+r_{1}]$. Repeating the procedure we get a nested sequence of rapidly decreasing intervals $[\frac{p_{k}}{q_{k}}-r_{k},\frac{p_{k}}{q_{k}}+r_{k}]$ such that

[TABLE]

satisfies the following property: For every $k\in\mathbb{N}$ there exist a system $(X,\mu,T)$ and a positive function $f_{k}\in L^{\infty}(X,\mu)$ with property (5.1) satisfied with $f_{k}$ instead of $f$. Since $1-\frac{1}{k}\geq\frac{1}{2}$ for $k\geq 2$ by the weighted Conze principle (Theorem 3.3), $((e(\theta P(n))),(n^{2}))$ is $L^{1}$-univerally bad. Note that we have some freedom in the above construction of $\theta$ by choosing each $\frac{p_{k}}{q_{k}}$ and by taking $r_{k}$ as small as we wish.

To show (a), by taking in the above construction $p_{k}:=p_{1}$ and $q_{k}:=q_{1}$ for every $k\in\mathbb{N}$, we have $\theta:=\frac{p_{1}}{q_{1}}$ which was arbitrary, and (a) follows.

To show (b), take in the above construction $\frac{p_{k}}{q_{k}}$ being all different and decrease $r_{k}$ such that $r_{k}<\frac{1}{q_{k}^{k}}$ and $r_{k}<|\frac{p_{k}}{q_{k}}-\frac{p_{k-1}}{q_{k-1}}|$ hold. Then $\theta\notin\{\frac{p_{1}}{q_{1}},\frac{p_{2}}{q_{2}},\ldots\}$ and $\theta$ is Liouville, therefore irrational.

Thus Theorem 1.2 follows. ∎

6. Proof of Theorem 5.1

Proof of Theorem 5.1.

Let $P\in\mathbb{Z}[\cdot]$ satisfy $P(0)=0$, let $\frac{p}{q}\in[0,1)$ be arbitrary and $k\in\mathbb{N}$. We will define bounded positive functions $f_{1},\ldots,f_{q}$ and then $f:=\sum_{l=1}^{q}f_{l}$ on $\mathbb{Z}_{\tau q^{2}}$ for some large $\tau\in\mathbb{N}$ with the normalised counting measure and the right shift. Denote $N_{0}:=1$.

Construction of ${N}_{1}$, $\tau_{1}$ and $f_{1}$ on $\mathbb{Z}_{\tau_{1}q^{2}}$.

Let $0<\varepsilon_{1}\leq 1$ and $C_{1}>0$ to be chosen later. By Corollary 2.3, used with ${\mathbf{N}}_{1}:=1$, there exist $\tau_{1}\in\mathbb{N}$ with $\tau_{1}\gg q$, the right shift transformation $T$ modulo $\tau_{1}$ on $\mathbb{Z}_{\tau_{1}}$ with the normalised counting measure, a set $\tilde{E}_{1}\subset\mathbb{Z}_{\tau_{1}}$ with proportion less than $\varepsilon_{1}$ in $\mathbb{Z}_{\tau_{1}}$, a function $\tilde{f}_{1}$ on $\mathbb{Z}_{\tau_{1}}$ with $\tilde{f}_{1}\geq 0$ and $\int_{\mathbb{Z}_{\tau_{1}}}\tilde{f}_{1}\leq\varepsilon_{1}$, and $N_{1}:={\mathbf{N}}_{2}>1={\mathbf{N}}_{1}$ such that the inequality

[TABLE]

holds for every $l\in\mathbb{Z}_{\tau_{1}}\setminus\tilde{E}_{1}$.

Consider $\mathbb{Z}_{\tau_{1}q^{2}}$ together with the normalised counting measure and the right translation which we denote by $T$ again. Consider further the function $f_{1}$ on $\mathbb{Z}_{\tau_{1}q^{2}}$ given by

[TABLE]

The support of this function is contained in

[TABLE]

and $\int_{\mathbb{Z}_{\tau_{1}q^{2}}}f_{1}\leq\frac{\varepsilon_{1}}{q^{2}}$ holds. Observe

[TABLE]

Moreover, for every $n\leq qN_{1}$ and every $\theta\in[\frac{p}{q}-r,\frac{p}{q}+r]$ observe

[TABLE]

By choosing $r<\frac{1}{4\pi\sup_{[1,qN_{1}]}|P|}$ we have by $P(0)=0$

[TABLE]

Define $E_{1}:=\{mq^{2}:\,m\in\tilde{E_{1}}\}$ and consider now $l=mq^{2}\in U_{1}\setminus E_{1}$. The inequalities (6.1), (6.2) and (6.3) imply

[TABLE]

Construction of $N_{2}$ and $f_{2}$ on $\mathbb{Z}_{\tau_{2}q^{2}}$.

Let $\varepsilon_{2}\leq\varepsilon_{1}$ and $C_{2}\geq C_{1}$ to be chosen later. By Corollary 2.3, there exist $\tau_{2}\in\mathbb{N}$ with $\tau_{2}\gg q$, the right shift transformation $T$ modulo $\tau_{2}$ on $\mathbb{Z}_{\tau_{2}}$ with the normalised counting measure, a set $\tilde{E}_{2}\subset\mathbb{Z}_{\tau_{2}}$ with proportion less than $\varepsilon_{2}$ in $\mathbb{Z}_{\tau_{2}}$, a function $\tilde{f}_{2}$ on $\mathbb{Z}_{\tau_{2}}$ with $\tilde{f}_{2}\geq 0$ and $\int_{\mathbb{Z}_{\tau_{2}}}\tilde{f}_{2}\leq\varepsilon_{2}$, and $N_{2}={\mathbf{N}}_{2}>N_{1}={\mathbf{N}}_{1}$ such that the inequality

[TABLE]

holds for every $l\in\mathbb{Z}_{\tau_{2}}\setminus\tilde{E}_{2}$.

Stretch $\tilde{f}_{2}$ to $\mathbb{Z}_{q^{2}\tau_{2}}$ as follows. Define the function $f_{2}$ on $\mathbb{Z}_{\tau_{2}q^{2}}$ given by

[TABLE]

The support of this function is contained in

[TABLE]

and $\int_{\mathbb{Z}_{\tau_{2}q^{2}}}f_{2}\leq\frac{\varepsilon_{2}}{q^{2}}$ holds.

Observe

[TABLE]

Moreover, for every $n\leq qN_{2}$ and every $\theta\in[\frac{p}{q}-r,\frac{p}{q}+r]$ observe

[TABLE]

By choosing $r<\frac{1}{4\pi\sup_{[1,qN_{2}]}|P|}$ we have by $P(0)=0$

[TABLE]

Define now $E_{2}:=\{mq^{2}+1:\,m\in\tilde{E_{2}}\}$ and consider $l=mq^{2}+1\in U_{2}\setminus E_{2}$. The inequalities (6.4), (6.5) and (6.6) imply

[TABLE]

Now, choosing $C_{2}>c\|f_{1}\|_{\infty}$ with $c>0$ to be chosen later, we see that

[TABLE]

Construction of $\tau$ and $f$

In such a fashion we construct for every $j\in\{1,\ldots,q^{2}\}$ and for $\varepsilon_{j}\leq\varepsilon_{j-1},C_{j}>0$ to be chosen later (where $\varepsilon_{0}:=1$) an integer $\tau_{j}\gg q$, a set $E_{j}\subset\mathbb{Z}_{\tau_{j}q^{2}}$ with proportion less than $\varepsilon_{j}$ in $Z_{\tau_{j}q^{2}}$, a natural number $N_{j}>N_{j-1}$ and positive bounded functions $f_{j}$ on $\mathbb{Z}_{\tau_{j}q^{2}}$ with $\int f_{j}\leq\varepsilon_{j}$ and supported on

[TABLE]

such that for every $l\in U_{j}\setminus E_{j}$ and every $\theta\in[\frac{p}{q}-r,\frac{p}{q}+r]$ with $r<\frac{1}{4\pi\sup_{[1,qN_{j}]}|P|}$

[TABLE]

Moreover, we choose $C_{j}$ large enough to satisfy

[TABLE]

We now consider $\tau:=\tau_{1}\cdot\ldots\cdot\tau_{q^{2}}$ and extend the functions $f_{j}$ and the sets $E_{j}$ periodically to $\mathbb{Z}_{\tau q^{2}}$. (We use the same notation for these extensions.) These sets and functions have the unchanged proportion in $\mathbb{Z}_{\tau q^{2}}$ and unchanged integrals, respectively. Moreover, (6.7) holds for every $l\in\mathbb{Z}_{\tau q^{2}}\setminus E_{j}$ and (6.8) is still true. We denote by $\mu$ the normalised counting measure on $\mathbb{Z}_{\tau q^{2}}$.

Define now $f:=f_{1}+\ldots+f_{q^{2}}$. We have by the monotonicity of $\varepsilon_{j}$

[TABLE]

by choosing $\varepsilon_{1}\leq\frac{1}{q^{2}}$. Define further $E:=E_{1}\cup\ldots\cup E_{q^{2}}$. Note that the proportion of $E$ in $\mathbb{Z}_{\tau q^{2}}$ is less than $\sum_{j=1}^{q^{2}}\varepsilon_{j}\leq q^{2}\varepsilon_{1}\leq\frac{1}{2k}$ by choosing $\varepsilon_{1}\leq\frac{1}{2kq^{2}}$.

Take $\theta\in[\frac{p}{q}-r,\frac{p}{q}+r]$ with $r<\frac{1}{4\pi\sup_{[1,qN_{q^{2}}]}|P|}$ and $l\in\mathbb{Z}_{\tau q^{2}}\setminus E$. Then $l\in U_{j}$ for some $j\in\{1,\ldots,q^{2}\}$. Let $N$ satisfy $qN_{j-1}\leq N\leq qN_{j}$ and decompose

[TABLE]

Observe that by (6.7), since $l\in U_{j}\setminus E_{j}$,

[TABLE]

Moreover, by (6.8),

[TABLE]

if we choose $c\geq 8q^{3}$.

Finally, by the triangle inequality and $\mu(f_{m}>1)\leq\int f_{m}\leq\varepsilon_{m}$ for each $m$,

[TABLE]

by choosing $\varepsilon_{j+1}\leq{1}/{(2kq^{5}N_{j})}$.

Denote now $F_{j}:=\{l\in U_{j}:\,\sup_{qN_{j-1}\leq N\leq qN_{j}}|III_{N}|>q^{2}\}$ and $F:=\cup_{j=1}^{q^{2}}F_{j}$. By the above, $\mu(F)\leq\sum_{j=1}^{q^{2}}\frac{1}{2kq^{2}}=\frac{1}{2k}$. Thus if we assume in addition that $l\notin F$, the triangle inequality (6), (6.10) and (6) lead to

[TABLE]

if we choose $C_{j}\geq 4kq+q^{2}$. Therefore, for every $l\in\mathbb{Z}_{\tau q^{2}}\setminus(E\cup F)$

[TABLE]

Since $\mu(E\cup F)\leq 2\frac{1}{2k}=\frac{1}{k}$, the proof is complete. ∎

7. Further questions

There are many open questions related to our results. We just mention two here.

Is every Liouville number universally $L^{1}$-bad?

• 2)

Is there an $L^{1}$-good number?

8. Acknowledgement

We thank the referee of this paper for making valuable comments and pointing out some useful references.