Fluctuations of ergodic averages for amenable group actions

TL;DR

This paper establishes exponential decay estimates for the probability of fluctuations in ergodic averages for amenable group actions, extending previous results to a broader class of groups and F{46}lner sequences.

Contribution

It introduces a universal estimate for fluctuations in ergodic averages along specific F{46}lner sequences for amenable groups, generalizing prior work.

Findings

Provides exponential decay bounds for fluctuation probabilities.

Shows any countable amenable group admits suitable F{46}lner sequences.

Extends classical results from 46}lner sequences in 46}Z^d to general amenable groups.

Abstract

We show that for any countable amenable group action, along F{\o}lner sequences that have for any $c>1$ a two sided $c$-tempered tail, one have universal estimate for the probability that there are $n$ fluctuations in the ergodic averages of $L^{\infty}$ functions, and this estimate gives exponential decay in $n$. Any two-sided F{\o}lner sequence can be thinned out to satisfy the above property, and in particular, any countable amenble group admits such a sequence. This extends results of S. Kalikow and B. Weiss for $\mathbb{Z}^{d}$ actions and of N. Moriakov for actions of groups with polynomial growth.

Equations

$$D_N=D_{(F_n),f,\alpha ,\beta ,N}=\{x:\text{Anf(x) fluctuates across (α,β) at least N times}\}$$

$$\mu (D_N)\le c_1{c}_0^N(\forall N)$$

$$\left|\bigcup _{i<n}{F}_i^{-1}{F}_n\right|\le c\left|F_n\right|,$$

$$\left|\bigcup _{i<n}{F}_n{F}_i^{-1}\right|\le c\left|F_n\right|,$$

$$\mu (D_N)\le c_1{c}_0^N(\forall N)$$

$${\varphi }_l=1_{(2l{\mathbb{N}}_0+[0,l-1])}$$

$$A_{n+1}=\{\begin{array}{ll}[0,\frac{2}{\lambda }{M}_n]\cup \left((2l{\mathbb{N}}_0+[0,l-1])\cap [0,\frac{2+\lambda }{\lambda }{M}_n]\right)& \text{n+1 is odd}\\ [0,\frac{2}{\lambda }{M}_n]\cup \left((2l{\mathbb{N}}_0+[l,2l-1])\cap [0,\frac{2+\lambda }{\lambda }{M}_n-2l+1]\right)& \text{n+1 is even}\end{array}$$

$$A_n-\bigcup _{i=0}^{n-1}{A}_i\subset A_n-[0,M_{n-1}]\subset [-M_{n-1},M_n]$$

$$|A_n-\bigcup _{i=0}^{n-1}{A}_i|\le M_n+M_{n-1}\le \frac{2+\lambda }{\lambda }{M}_{n-1}+M_{n-1}\le (1+\lambda )|A_n|.$$

$$\frac{1}{|A_n|}\sum _{z\in A_n}{\varphi }_l(z+2lk+i)$$

$$\ge \frac{1}{\left|A_n\right|}\cdot \left[l\cdot \left(\frac{\frac{2}{\lambda }{M}_{n-1}}{2l}-1\right)+\frac{3}{4}l\cdot \left(\frac{M_{n-1}}{2l}-1\right)\right]$$

$$\ge \frac{\left(\frac{1}{\lambda }+\frac{3}{8}\right)M_{n-1}-2l}{\left(\frac{2}{\lambda }+\frac{1}{2}\right)M_{n-1}+l}$$

$$\ge \frac{1}{2}+\frac{\lambda }{4(4+\lambda )}-\frac{4}{l}$$

$$\frac{1}{|A_n|}\sum _{z\in A_n}{\varphi }_l(z+2lk+i)$$

$$\le \frac{\left(\frac{1}{\lambda }+\frac{1}{8}\right)M_{n-1}+2l}{\left(\frac{2}{\lambda }+\frac{1}{2}\right)M_{n-1}-l}$$

$$\le \frac{1}{2}-\frac{\lambda }{4(4+\lambda )}+\frac{4}{l}$$

$$l_{k+1}=\max (A_{2l_m}^{l_m,2l_m}).$$

$$D_N$$

$$D_{N,M}$$

$$h>\frac{(L+\max A_{2l_m}^{l_m,2l_m})}{ϵ/4}.$$

$$\mu \left(x\in B:\forall N\le N^{\prime },\sum _{i=0}^{h-1}{1}_{D_{N,n^{\prime }}^f}(T^{i}x)>\mu (D_{N,n^{\prime }}^f)-ϵ/4\right)>(1-ϵ/4)\mu (B)$$

$$\hat{f}(x)=\{\begin{array}{ll}{\varphi }_{l_m}(i)& x\in T^i{B}^{\prime },0\le i\le h-1\\ f(x)& x\in X\backslash {\bigcup }_{i=0}^{h-1}{T}^i{B}^{\prime }\end{array}$$

$$T^{2k{l}_m+i}{B}^{\prime }\subset D_{l_m}^{\hat{f}}\subset D_{N^{\prime \prime }}^{\hat{f}}$$

$$\left(\frac{l_m}{4}-1\right)\cdot \left(\frac{h}{2l_m}-1\right)/h>\frac{1}{8}-\frac{l_m}{h}-\frac{1}{l_m}$$

$$\mu \left(D_{N^{\prime \prime }}^{\hat{f}}\right)\ge \frac{1}{9}\mu \left(\bigcup _{n=0}^{h-L}{T}^n{B}^{\prime }\right)=\frac{1}{9}(0.99\delta )>\frac{1}{10}\delta$$

$$\bigcup _{n=0}^{h-L}{T}^n\left(B\backslash B^{\prime }\right)\subset \left\{x:{\left(f(T^{i}x)\right)}_{i=0}^{L-1}={\left(\hat{f}(T^{i}x)\right)}_{i=0}^{L-1}\right\}$$

$$\mu \left({\left(f(T^{i}x)\right)}_{i=0}^{L-1}={\left(\hat{f}(T^{i}x)\right)}_{i=0}^{L-1}\right)$$

$$>1-ϵ/4-ϵ/4-0.99\delta$$

$$>1-\delta .$$

$$\bigcup _{n=0}^{h-L}{T}^n\left(B\backslash B^{\prime }\right)\cap D_{N,n^{\prime }}^f\subset D_{N,n^{\prime }}^{\hat{f}}$$

$$\mu \left(\bigcup _{n=0}^{h-L}{T}^n\left(B\backslash B^{\prime }\right)\cap D_{N,n^{\prime }}^f\right)\ge \mu \left(D_{N,n^{\prime }}^f\right)\mu \left(\bigcup _{n=0}^{h-L}{T}^n\left(B\backslash B^{\prime }\right)\right)-\frac{3}{4}ϵ$$

$$\mu \left(D_N^{\hat{f}}\right)$$

$$\ge \mu \left(D_{N,n^{\prime }}^f\right)\mu \left(\bigcup _{n=0}^{h-L}{T}^n\left(B\backslash B^{\prime }\right)\right)-\frac{3}{4}ϵ+\frac{1}{9}\mu \left(\bigcup _{n=0}^{h-L}{T}^n{B}^{\prime }\right)$$

$$\ge \min \left\{\mu \left(D_{N,n^{\prime }}^f\right),\frac{1}{9}\right\}\mu \left(\bigcup _{n=0}^{h-L}{T}^{n}B\right)-\frac{3}{4}ϵ$$

Full text

1 Introduction

A real valued sequence is said to fluctuate $N$ times across a gap $(\alpha,\beta)$, if there are integers $n_{1}<n_{2}<...<n_{2N}$ s.t. for odd $i$, $a_{n_{i}}\leq\alpha$, and for even $i$, $a_{n_{i}}\geq\beta$. Let $(X,\mu,\mathscr{B},(T_{g})_{g\in G})$ be a measure preserving action of a countable amenable group $G$, and fix some (left) F lner sequence $(F_{n})$ in $G$. For any $N,$ we define the set $D_{N}$ by:

[TABLE]

where $\mathsf{A}_{n}f=\frac{1}{|F_{n}|}\sum_{g\in F_{n}}f\circ T_{g}$ denotes the sequence of ergodic averages of a function $f$ on $X$ along $(F_{n})$. In [1] it was shown that for $G=\mathbb{Z}^{d}$ and $F_{n}=[-n,n]^{d}$, the following holds:

Theorem 1.1**.**

For any $0<\alpha<\beta$, there are constants $0<c_{0}<1$ and $c_{1}>0$, s.t. for every m.p.s. $(X,\mathscr{B},\mu,\{T_{g}\}_{g\in G})$ and every measurable $f\geq 0$, one has:

[TABLE]

.

In [3] this result was extended to measure preserving actions of groups of polynomial growth, where the fixed F lner sequence is taken to be balls of increasing radii, that is, $F_{n}=S^{n}$ where $S$ is a finite symmetric set of generators which contains the unit.

The aim of this paper is to extend these results to general actions of amenable groups. In this context, the notion of temperedness is of importance: A sequence $(F_{n})$ is left $c$-tempered if for all $n$,

[TABLE]

right $c$-tempered if for all $n$,

[TABLE]

and $c$-bi-tempered if it is both left and right $c$-tempered. In this paper, a sequence that for any $c>1$ has some tail which is $c$-bi-tempered, will be called strongly tempered. Notice that any two-sided F lner sequence can be thinned out to be strongly tempered.

The class of tempered F lner sequences is the most general class of sequences which are known to satisfy the pointwise ergodic theorem [2, 5]. That is, the averages along any (left) tempered F lner sequence of any integrble function converges a.e. Consequently, if the fixed F lner sequence $\left(F_{n}\right)$ is tempered, then for any $\alpha<\beta$ and any integrable function $f$, the measure of $D_{N}=D_{(F_{n}),f,\alpha,\beta,N}$ decreases to zero as $N\rightarrow\infty$. Thus, along such sequences, one might hope to have some control on the rate of $\mu(D_{N})$, as in Theorem 1.1:

Question**.**

Does every amenable group have a F lner sequence that satisfies (in some sense) Theorem 1.1? Can one find for any F lner sequence a subsequence with this property?

Our main result is the following theorem and its corollary, which says that one can successfully bound the rate of decrease of $\mu(D_{N})$ in any amenable group, provided that $f$ is bounded, and that the averages are taken along strongly tempered F lner sequences.

Theorem 1.2**.**

For any $\alpha<\beta$ and $S>0$, there exist $\lambda>0$ and $0<c_{0}<1$, s.t. for any $(1+\lambda)$-bi-tempered F lner sequence $\left(F_{n}\right)$, any m.p.s. $(X,\mu,\mathscr{B},(T_{g})_{g\in G})$ and any $f\in L_{\mu}^{\infty}(X)$ with $||f||_{\infty}\leq S$, one has

[TABLE]

for some $c_{1}>0$ which depends only on the sequence $\left(F_{n}\right)$ (and neither on the m.p.s. nor on the function $f$).

If $\left(F_{n}\right)$ is strongly tempered, then for any gap $(\alpha,\beta)\subset\mathbb{R}$ and any $S>0$, some tail of the sequence, say $\left(F_{n}\right)_{n>n_{0}}$, satisfies the hypothesis of Theorem 1.2, while the first $n_{0}$ elements of $\left(F_{n}\right)$ atribute at most $O(n_{0})$ fluctuations. Thus, enlarging $c_{1}$ depending on that $n_{0}$, we get:

Corollary 1.3**.**

Let $\left(F_{N}\right)$ be a strongly tempered F lner sequence. For any $\alpha<\beta$ and $S>0$, there exist $0<c_{0}<1$ and $c_{1}>0$, s.t. for any m.p.s. $(X,\mu,\mathscr{B},(T_{g})_{g\in G})$ and any $f\in L_{\mu}^{\infty}(X)$ with $||f||_{\infty}\leq S$, one has

[TABLE]

As the proof of Theorem 1.2 indicates, the bi-temperedness condition could be slightly relaxed, and was chosen for the clarity of presentation. In addition, the dependency of $c_{1}$ on the sequence $\left(F_{n}\right)$ could be replaced by restricting the theorem to sequences with some certain properties. For example, assuming $e\in F_{1}$ would be enough for determine $c_{1}$, regardless of what $\left(F_{n}\right)$ is.

In contrast to Theorem 1.2, we show that the temperedness property (with any fixed $c>1$) alone, isn’t enough to bound the rate of decrease of $\mu(D_{N})$ for any given gap $(\alpha,\beta)$. More precisely, we show that in any measure preaserving $\mathbb{Z}$-action $(X,\mu,\mathscr{B},\{T^{n}\}_{n\in\mathbb{Z}})$ one has the following:

Theorem 1.4**.**

Let $(X,\mathscr{B},\mu,\{T^{n}\}_{n\in\mathbb{Z}})$ be a m.p.s. and let $\omega(n)\searrow 0$ be any sequence which decreases to 0. For any $\lambda>0$, there are some $\alpha<\beta$, a bounded function $0\leq f\leq 1$ and a $(1+\lambda)$-tempered F lner sequence $\left(F_{n}\right)$, for which $\mu(D_{(F_{n}),f,\alpha,\beta,N})>\omega(N)$ for all but finitely many $N$.

Although this shows that the requirement for $\left(F_{n}\right)$ to have a left $(1+\lambda)$-tempered tail for any $\lambda$ is essential for Corollary 1.3 to take place, it is not clear whether the other requirements are. More generally, the following question remains open:

Question 1.5**.**

Does every left F lner sequence in a countable amenable group $G$ have a subsequence which satisfies the conclusion of Corollary 1.3?

Acknowledgement*.*

I would like to thank my advisor Michael Hochman, for suggesting me the problem studied in this paper, and for many helpful discussions.

2 Proof of Theorem 1.4

Proof of Theorem 1.4: Let $\lambda>0$. We first construct finite sequences of subsets of $\mathbb{Z}$, which have good fluctuation and invariance properties, and then concatenate such sequences to get the whole sequence $\left(F_{n}\right)$ in question. Fix $l,N\in\mathbb{N}$, and let $\phi_{l}:\mathbb{N}_{0}\rightarrow\{0,1\}$ be the indicator function

[TABLE]

(here $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$.) We define a sequence of subsets $\left(A_{n}\right)_{n=1}^{2N}=\left(A_{n}^{l,N}\right)_{n=1}^{2N}$ recursively:

[TABLE]

where $M_{0}=l^{2}$, and $M_{n}=\max(A_{n})$ for $n>0$. This sequence has the following properties:

(a) $\left(A_{n}\right)_{n=1}^{2N}$ is $(1+\lambda)$-tempered: For any $n$,

[TABLE]

thus

[TABLE]

(b) $A_{n}$ is $([-\sqrt{l},\sqrt{l}],2/\sqrt{l})$-invariant for all $n$; that is, for any $b\in[-\sqrt{l},\sqrt{l}]$, one have $\frac{|(b+A_{n})\triangle A_{n}|}{|A_{n}|}\leq 2/\sqrt{l}$: This follows immidiately from the fact that $A_{n}$ is a union of segments, the first one of size at list $l^{2}$, and all but the last one of size at least $l$.

(c) Assuming $l$ large enough, there are some $0<\alpha<\beta$ s.t. for any $0\leq i\leq l/4$, and any $k$, averaging $\phi_{l}(z+2lk+i)$ as a function of $z$ along $A_{n}$, the sequence of averages fluctuates across the gap $(\alpha,\beta)$ $N$ times: Averaging along $A_{n}$ of odd $n$ gives

[TABLE]

while for even $n$,

[TABLE]

(for the error summands $\frac{4}{l}$, we used $M_{n-1}\geq M_{0}=l^{2}$ and assumed $l\geq 4$.) Taking $l$ large enough, one get that the claim above takes place with $\alpha=\frac{1}{2}-\frac{\lambda}{5(4+\lambda)}$ and $\beta=\frac{1}{2}+\frac{\lambda}{5(4+\lambda)}$.

Now construct the whole sequence as follows: Take $l_{0}>100$ and also large enough so that property (c) takes place, and then define $(l_{m})_{m=1}^{\infty}$ recursively by the rule

[TABLE]

We define $(F_{n})$ to be the concatenation of the sequences $\{(A_{n}^{l_{m},2l_{m}})_{n=1}^{2l_{m}}\}_{m=0}^{\infty}$. Using properties (a) and (b) above together with the definition of $(l_{m})_{m=1}^{\infty}$, one can observe that this sequence is a $(1+\lambda)$-tempered F lner sequence.

To construct the function which satisfies the conclusion of the theorem, we need some notation which will be of use here and in the rest of the paper: For a given function $f$ on a m.p.s. $(X,\mathscr{B},\mu,\{T_{g}\}_{g\in G})$, a gap $(\alpha,\beta)\subset\mathbb{R}$, a sequence $\left(F_{n}\right)$ of subsets of $G$, and $N,M\in\mathbb{N}$, we shall write

[TABLE]

where the sequence $\left(F_{n}\right)$, the function $f$ and the gap $(\alpha,\beta)$ are understood from the context. At some places we shall write $D_{N}^{f}$ and $D_{N,M}^{f}$ to specify the function $f$ for which the sets refer to.

We will construct the function in question by applying iteratively infinitely many times the following lemma:

Lemma 2.1**.**

Let $f:X\rightarrow[0,1]$. For any $\epsilon>0$, $1>\delta>0$, and $n^{\prime},N^{\prime},N^{\prime\prime}\in\mathbb{N}$, there exists a measurable function $\hat{f}:X\rightarrow[0,1]$ s.t. the following holds:

1. (i)

$\mu\left(D_{N^{\prime\prime}}^{\hat{f}}\right)>\frac{1}{10}\delta$. 2. (ii)

$\mu\left(\left(f(T^{i}x)\right)_{i=0}^{L-1}\neq\left(\hat{f}(T^{i}x)\right)_{i=0}^{L-1}\right)\leq\delta$, where $L:=\max\left(\bigcup_{n=1}^{n^{\prime}}F_{n}\right)$. 3. (iii)

For all $N\leq N^{\prime}$, $\mu\left(D_{N}^{\hat{f}}\right)\geq\min\left\{\mu(D_{N,n^{\prime}}^{f})-\epsilon,\frac{1}{10}\right\}$.

Proof.

We will assume w.l.o.g. that $\epsilon$ is small enough so that $\epsilon<\min\left\{\frac{\delta}{100},1-\delta\right\}$. Take an $m\in\mathbb{N}$ that satisfies $l_{m}\geq N^{\prime\prime}$. Let $B\subset X$ be a base for a Rokhlin tower of height $h$ and total measure $>1-\epsilon/4$, where $h$ is large enough to satisfy

[TABLE]

and also large enough to guarantee that

[TABLE]

in words, for all $N\leq N^{\prime}$, for at least $1-\epsilon/4$ of the $x$’s in $B$, their orbit along the tower spends more than $\mu(D_{N,n^{\prime}}^{f})-\epsilon/4$ of the time in the set $D_{N,n^{\prime}}^{f}$ (For the validity of such a requirement, see for example [4, Theorem 7.13]).

Take $B^{\prime}\subset B$ of measure $\mu(B^{\prime})=0.99\delta/h$ (this can be achieved because $1-\epsilon>\delta$), and define $\hat{f}$ to be:

[TABLE]

The validity of property (c) above for $\left(A_{n}^{l_{m},2l_{m}}\right)_{n=1}^{2l_{m}}$ and thus for $\left(F_{n}\right)$, together with the definition of $\hat{f}$ as $\phi_{l_{m}}$ on the tower above $B^{\prime}$, implies that for any $0\leq i\leq l_{m}/4$ and any $k\geq 0$ s.t. $2kl_{m}+i<h-\max A_{l_{m}}^{l_{m},2l_{m}}$, one has:

[TABLE]

The density of these levels in the tower is at least

[TABLE]

and since $l_{m}\geq l_{0}>100$ and $\frac{l_{m}}{h}\leq\frac{\max A_{l_{m}}^{l_{m},2l_{m}}}{h}\leq\epsilon/4<\frac{1}{100}$, the last expression is at least $\frac{1}{9}$. Thus

[TABLE]

which gives property (i) of the conclusion.

To see why property (ii) of the conclusion holds, notice that

[TABLE]

thus

[TABLE]

Finally, by (4) we have for all $N$

[TABLE]

and by (2) and (1), we have for all $N\leq N^{\prime}$,

[TABLE]

That, together with the first inequality in (3) gives for all $N\leq N^{\prime}$

[TABLE]

which gives property (iii) of the conclusion. ∎

Let $\omega(n)\searrow 0$ be any sequence which decreases to 0. Define $\left(N_{k}\right)_{k=1}^{\infty}$ by

[TABLE]

We will construct a function $f$ which satisfies for all $k$

[TABLE]

and by monotonicity of $\mu\left(D_{N}^{f}\right)$ and $\omega(N)$, for any $N_{k}\leq N<N_{k+1}$, $k\geq 1$,

[TABLE]

and the conclusion of Theorem 1.4 follows.

Take $f_{0}\equiv 0$, and define inductively $\left(f_{k}\right)_{k=0}^{\infty}$ : Given $f_{k-1}$, assume that

[TABLE]

Take $\epsilon>0$ small enough so that for all $i\leq k-1$,

[TABLE]

and apply Lemma 2.1 with $f:=f_{k-1}$,$N^{\prime}=N_{k-1}$, $N^{\prime\prime}=N_{k}$, $n^{\prime}=n_{k-1}$, $\delta=2^{-k}$ while letting $f_{k}$ be the resulting finction $\hat{f}$. This $f_{k}$ satisfies the hypothesis (5) in the inductive step: By property (iii) of the lemma, for all $i\leq k-1$,

[TABLE]

and by property (i) of the lemma,

[TABLE]

Since $\mu(D_{N,n}^{f_{k}})\overset{{\scriptscriptstyle n\rightarrow\infty}}{\longrightarrow}\mu(D_{N}^{f_{k}})$, there exists large enough $n_{k}$ s.t. (6) and (7) will be satisfied with $D_{N,n_{k}}^{f_{k}}$ in place of $D_{N}^{f_{k}}$. Thus the hypothesis (5) of the induction step is indeed satisfied with $k$ in place of $k-1$.

We end up with a sequence $\left(f_{k}\right)_{k=0}^{\infty}$ together with a sequence $\left(n_{k}\right)$ which we can assume to be increasing. By property ((ii)) of the lemma, $\left(f_{k}\right)$ converges a.e. to some limit, call it $f$. For each $k$, let

[TABLE]

then again by property ((ii)) of the lemma, $f$ satisfies

[TABLE]

(in the second inequality we used the assumption that $n_{i}\geq n_{i-1}$ for all $i$). Thus for any $i$,

[TABLE]

taking $k\rightarrow\infty$ gives

[TABLE]

and the proof of Theorem 1.4 is complete.$\hfill\square$

3 Proof of Theorem 1.2

In this section, we will prove Theorem 1.2. Towards this end, we need few definitions and lemmas.

Definition 3.1**.**

Given $0<\lambda<1$, we say that a sequence $\left(F_{n}\right)$* is $\lambda$-good*, if the following two conditions hold:

1. (i)

For any $n$, $\left|\bigcup_{i<n}F_{i}^{-1}F_{n}\,\backslash\,F_{n}\right|\leq\lambda\left|F_{n}\right|$. 2. (ii)

For any $i<n$ and $f\in F_{i}$, $\left|F_{n}\backslash F_{n}f\right|<\lambda\left|F_{n}\right|$.

Proposition 3.2**.**

Let $0<\lambda<\lambda^{\prime}<1$. For any $(1+\lambda)$-bi-tempered two-sided F lner sequence $\left(F_{n}\right)$, there is some $n_{0}$ s.t. $\left(F_{n}\right)_{n\geq n_{0}}$ is $\lambda^{\prime}$-good.

Proof.

Pick some $g_{0}\in F_{1}$. Since the sequence is (left) F lner, there is some $n_{1}$ s.t. for all $n\geq n_{1}$,

[TABLE]

By the (left) temperedness property of $\left(F_{n}\right)$, we have

[TABLE]

and (i) of Definition 3.1 takes place. The same proof applies from the right, thus we get some $n_{2}$ s.t. for any $n\geq n_{2}$

[TABLE]

but now, for any $i<n$ and $f\in F_{i}$,

[TABLE]

which is (ii) of Definition 3.1. Now take $n_{0}=max\left\{n_{1},n_{2}\right\}$. ∎

The following theorem is a version of Theorem 1.2 for $\lambda$-good F lner sequences, from which we will deduce Theorem 1.2:

Theorem 3.3**.**

For any $\alpha<\beta$ and $S>0$, there exist $\lambda>0$, $0<c_{0}<1$ and $c_{1}>0$, s.t. for any $\lambda$-good (left) F lner sequence $\left(F_{n}\right)$, any m.p.s. $(X,\mu,\mathscr{B},(T_{g})_{g\in G})$ and any $f\in L_{\mu}^{\infty}(X)$ with $||f||_{\infty}\leq S$, one has

[TABLE]

We remark that as opposed to Theorem 1.2, here the constant $c_{0}$ doesn’t depend on $\left(F_{n}\right)$.

Once Theorem 3.3 is valid, the proof of Theorem 1.2 is immidiate:

Proof of Theorem 1.2: For $[\alpha,\beta]$ and $S>0$, let $\lambda^{\prime}$ be the value for which any $\lambda^{\prime}$-good F lner sequence satisfies the conclusion of Theorem 3.3 with $c_{0}^{\prime}$ and $c_{1}^{\prime}$. Take $0<\lambda<\lambda^{\prime}$, then by Proposition 3.2, for any $(1+\lambda)$-bi-tempered two-sided F lner sequence $\left(F_{n}\right)$, there is some $n_{0}$, s.t. $\left(F_{n}\right)_{n\geq n_{0}}$ is $\lambda^{\prime}$-good, and thus for any m.p.s. $(X,\mu,\mathscr{B},(T_{g})_{g\in G})$, any $f\in L_{\mu}^{\infty}(X)$ with $||f||_{\infty}\leq S$ and any $N$,

[TABLE]

thus for $c_{0}=c_{0}^{\prime}$ , $c_{1}=c^{\prime}c_{0}^{\prime-n_{0}}$ the conclusion follows.$\hfill\square$

Thus it remains to prove Theorem 3.3, which will be our task for the rest of the paper.

Definition 3.4**.**

Given $\epsilon>0$, a collection $\left(F_{j}\right)_{j=1}^{L}$ of finite subsets of $G$ is said to be $\epsilon$-disjoint if there are pairwise disjoint sets $E_{j}\subset F_{j}$ s.t. $\left|E_{j}\right|\geq(1-\epsilon)\left|F_{j}\right|$ for all $1\leq j\leq L$.

We record here a version of the $\epsilon$-disjointification lemma [5, Lemma 9.2], which will be uses again and again:

Lemma 3.5**.**

($\epsilon$-disjointification lemma) Let $F_{1},...,F_{L}$ be a sequence of finite subsets of a group $G$ which is $2$-tempered, let $C\subset G$ be finite, and suppose that $C_{1},...,C_{L}$ are disjoint subsets of $C$. For any $0<\epsilon\leq\frac{1}{2}$, there are subsets $D_{j}\subset C_{j}$, s.t. :

1. (i)

The collection $\left\{F_{j}d:\,d\in D_{j},\,1\leq j\leq L\right\}$ is $\epsilon$-disjoint, 2. (ii)

$\left|\bigcup_{j=1}^{L}F_{j}D_{j}\right|\geq\frac{\epsilon}{5}\left|C\right|$.

The following proposition, which is analogous to the effective Vitali covering argument of Kalikow and Weiss [1], will be used as a key step through.

Proposition 3.6**.**

For any $\epsilon>0$, once $\lambda>0$ is small enough and $q\in\mathbb{N}$ is large enough, the following holds for any $\lambda$-good F lner sequence $(F_{n})$:

Let $C\subset G$ be a finite subset, and suppose that for each $c\in C$ there is associated a subsequence of $(F_{n}c)$ of length $q$:

[TABLE]

Then there exists an $\epsilon$-disjoint collection $\{F_{n(d)}d\}_{d\in D}$ where $D\subset C$ and $n(d)\in\left\{n_{1}(d),...,n_{q}(d)\right\}$ , which satisfies at least one of the following properties:

1. Either $\left|\bigcup_{d\in D}F_{n(d)}d\right|\geq 2|C|$,

2. or $\left|\bigcup_{d\in D}F_{n(d)}d\cap C\right|\geq(1-\epsilon)|C|$.

As it can be seen from the proof below, for $\left(F_{n}\right)$ to satisfy the conclusion, one can assume that $\left(F_{n}\right)$ is a F lner sequence that merely admits property (i) of being $\lambda$-good (Definition 3.1).

Proof.

Define

[TABLE]

let $m=\max\{n:\,\exists c\in G,\,(c,n)\in\mathscr{C}\}$, and consider the $m$-section of $\mathscr{C}$:

[TABLE]

Assuming $\lambda\leq 1$, the $\epsilon$-disjointification lemma guarantees there is a subset $D_{m}\subset C_{m}$, s.t.

(a) The collection $\{F_{m}d\}_{d\in D_{m}}$ is $\epsilon$-disjoint, and

(b) $|F_{m}D_{m}|\geq\frac{\epsilon}{5}\left|C_{m}\right|$.

Let $1\leq k\leq m-1$, and suppose we have already defined subsets $\left(D_{m-i}\right)_{i=0}^{k-1}$ of $C$. Denote:

[TABLE]

and use again the $\epsilon$-disjointification lemma to take some $D_{m-k}\subset C_{m-k}$ so that:

(a)’ The collection $\{F_{m-k}d\}_{d\in D_{m-k}}$ is $\epsilon$-disjoint, and

(b)’ $|F_{m-k}D_{m-k}|\geq\frac{\epsilon}{5}\left|C_{m-k}\right|$.

The restriction $C_{m-k}\subset W_{m-k+1}$ (9) together with (8) guarantees that

[TABLE]

We end up (after $m$ steps) with a pairwise disjoint subsets $D_{1}\subset C_{1},...,D_{m}\subset C_{m}$ where $\bigsqcup_{n=1}^{m}C_{n}\times\{n\}\subset\mathscr{C}$, and s.t. each $\{F_{n}d\}_{d\in D_{n}}$ is $\epsilon-$disjoint, the unions $\bigcup_{d\in D_{n}}F_{n}d=F_{n}D_{n}$ are disjoint to each other and are of size $\left|F_{n}D_{n}\right|\geq\frac{\epsilon}{5}\left|C_{n}\right|$. Let $\mathscr{D}=\bigsqcup_{n=1}^{m}D_{n}\times\{n\}$. We claim that the collection $\{F_{n}d\}_{(d,n)\in\mathscr{D}}$ satisfies the conclusion of the Lemma: We just pointed out that it is indeed an $\epsilon$-disjoint collection. Suppose it doesn’t satisfy property 2 of the conclusion, that is,

[TABLE]

We distinguish between two cases:

I. One has:

[TABLE]

then, together with (10) one get:

[TABLE]

and for small enough $\lambda$ ($\lambda\leq\frac{\epsilon}{8}$), the last inequality gives property 1 in the conclusion, so we’re done.

II. For the other case,

[TABLE]

we bound from below the size of $W_{2}=C\backslash\,\bigcup_{n=2}^{m}\,\bigcup_{i<n}F_{i}^{-1}F_{n}D_{n}$ :

[TABLE]

(the second inequality follows from property (i) of Definition 3.1, the third by the $\epsilon$-disjointness of the collection, and the last one by (11), together with the assumption $\epsilon\leq\frac{1}{2}$ ). Any element in $W_{2}$ appears as the left coordinate of $q$ different elements in $\bigcup_{k=1}^{m}C_{k}\times\{k\}$, thus,

[TABLE]

assuming $q\geq\frac{20}{\epsilon^{2}}$, the Lemma is proved. ∎

Proof of Theorem 3.3: For any $x\in X$, the number of fluctuations of $\mathsf{A}_{n}f(x)$ across $(\alpha,\beta)$ is equal to the number of fluctuations of $\mathsf{A}_{n}[f+||f||_{\infty}](x)$ across $(\alpha+||f||_{\infty},\beta+||f||_{\infty})$. Consequently, for any $N$,

[TABLE]

Notice that $||f+||f||_{\infty}||_{\infty}\leq 2||f||_{\infty}$, and besides trivial cases, one has $0<\alpha+||f||_{\infty}$. Hence, for any $S>0$ and $\alpha<\beta$, any estimate of $\mu\left(D_{N}\right)$, where $D_{N}$ is defined w.r.t. any non negative function $0\leq f\leq 2S$ and the gap $[\alpha+S,\beta+S]\subset(0,\infty)$, is an estimate of $\mu\left(D_{N}\right)$, where $D_{N}$ is defined w.r.t. any function $||f||_{\infty}\leq S$ and the gap $[\alpha,\beta]\subset\mathbb{R}$. Thus from now on, we shall assume $0\leq f\leq S$ and $0<\alpha<\beta$.

Fix $x\in X$, $M\in\mathbb{N}$, and let $\Omega\subset G$ be a set which is sufficiently invariant w.r.t. $\bigcup_{n=1}^{M}F_{n}$, so that the set

[TABLE]

has size close to $|\Omega|$. We will give an upper bound to the relative density $\frac{|C|}{|\Omega|}$, where

[TABLE]

This upper bound won’t depend on $x$ or $M$, and thus by the transference principle, it will give an upper bound for $\mu\left(D_{N}\right)$, as it is shown at the end of the proof.

Take

[TABLE]

and choose $\frac{1}{4}>\epsilon>0$ small enough so that the following three inequalities hold:

[TABLE]

Take $q\in\mathbb{N}$ and $0<\lambda\leq\epsilon/2$ so that the conclusion of Lemma 3.6 will take place with $\epsilon/2$.

The first step is to replace $C$ with a union of $\epsilon$-disjoint collections of size not much less than $|C|$, where for each set in the collection, the average of $f$ at $x$ on it is above $\beta$. For that, use the first group of $q$ fluctuations to find for each $c\in C$ an increasing sequence $n_{1}(c)<,...,<n_{q}(c)$ s.t. $\mathsf{A}_{n_{i}(c)}f(cx)\geq\beta$ for each $1\leq i\leq q$. Then, by applying Proposition 3.6, one take an $\epsilon$-disjoint collection $\left(F_{n}c\right)_{(c,n)\in\mathscr{B}_{1}}$, where its union $C_{1}=\bigcup_{(c,n)\in\mathscr{B}_{1}}F_{n}c$ is in $\Omega$ and of size $\left|C_{1}\right|\geq(1-\epsilon)\left|C\right|$. The next step will be done recursively $(\lfloor\frac{N}{2q}\rfloor-1)$ times, thus we introduce it in a more general form:

Lemma 3.7**.**

Let $f,\left(F_{n}\right),C,\alpha,\beta,\delta,\epsilon,N$ and $q$ be as above. Let $N_{k}\leq N-2q$, and suppose that $\mathscr{B}_{k}\subset C\times\mathbb{N}$ is a collection of tuples s.t. :

(i) For each $(c,n)\in\mathscr{B}_{k}$ the average $\mathsf{A}_{n}f(cx)$ is one of $cx$’s first $N_{k}$ upcrossings to above $\beta$.

(ii) the collection $\left(F_{n}c\right)_{(c,n)\in\mathscr{B}_{k}}$ is $\epsilon$-disjoint.

Then there exists a collection $\mathscr{B}_{k+1}\subset C\times\mathbb{N}$ of tuples s.t. :

(i) For each $(c,n)\in\mathscr{B}_{k+1}$, the average $\mathsf{A}_{n}f(cx)$ is one of $cx$’s first $N_{k}+2q$ upcrossings to above $\beta$.

(ii) The collection $\left(F_{n}c\right)_{(c,n)\in\mathscr{B}_{k+1}}$ is $\epsilon$-disjoint.

(iii) $\left|\bigcup_{(c,n)\in\mathscr{B}_{k+1}}F_{n}c\right|\geq(1+\delta/2)\left|\bigcup_{(c,n)\in\mathscr{B}_{k}}F_{n}c\right|$.

Proof of Lemma 3.7. Denote $C_{k}=\bigcup_{(c,n)\in\mathscr{B}_{k}}F_{n}c$. To each $g\in C_{k}$ we will associate a subsequence of $\left(F_{n}g\right)$ of length $q$, in order to apply Lemma 3.6 to the set $C_{k}$: For any $g\in C_{k}$, choose some $c=c(g)$ so that $(c,n)\in\mathscr{B}_{k}$ for some $n$ and $g\in F_{n}c$. Associate to $g$ the indices of the next $q$ downcrossings to below $\alpha$ of $c$, $n<n_{1}(c)<...<n_{q}(c)$. By Proposition 3.6, there is an $\nicefrac{{\epsilon}}{{2}}$-disjoint collection $\left(F_{n}g\right)_{(g,n)\in\mathscr{B}_{k}^{\prime}}$ , with union $C_{k}^{\prime}=\bigcup_{(g,n)\in\mathscr{B}_{k}^{\prime}}F_{n}g\subset\Omega$ that satisfies one of the two options in the conclusion of Proposition 3.6. Next, we define another index set $\mathscr{B}_{k}^{\prime\prime}$ to be

[TABLE]

and the union of its associated collection

[TABLE]

For any $(c,n)\in\mathscr{B}_{k}^{\prime\prime}$, let $(g,n)\in\mathscr{B}_{k}^{\prime}$ be such that $c(g)=c$ . Then, $\left(F_{n}\right)$ being $\lambda$-good, by (ii) of Definition 3.1,

[TABLE]

That, together with $\left(F_{n}g\right)_{(g,n)\in\mathscr{B}_{k}^{\prime}}$ being $\epsilon/2$-disjoint, implies that

[TABLE]

and that

[TABLE]

This relation together with $C_{k}^{\prime}$ being as in the conclusion of Proposition 3.6, gives one of the following two options:

1. Either $\left|C_{k}^{\prime}\right|\geq 2|C_{k}|$, in which case (16) implies that

[TABLE]

,

2. or $\left|C_{k}^{\prime}\right|<2|C_{k}|$, but$\left|C_{k}^{\prime}\bigcap C_{k}\right|\geq(1-\epsilon/2)|C_{k}|$, which implies

[TABLE]

In both cases one can conclude that $\left|C_{k}^{\prime\prime}\right|\geq(1+\delta)|C_{k}|$: for the first case (17), $\epsilon<\frac{1}{4}$ and $\delta\leq\frac{1}{2}$ gives

[TABLE]

For the second case (18), this can be observed by the next calculation:

By (15), there are pairwise disjoint sets $E^{\prime\prime}_{(n,c)}\subset F_{n}c$ (for each $(n,c)\in\mathscr{B}_{k}^{\prime\prime}$), with $\left|E^{\prime\prime}_{(n,c)}\right|\geq(1-\epsilon)\left|F_{n}c\right|$. Thus

[TABLE]

On the other hand, the collection $\left(F_{n}c\right)_{(c,n)\in\mathscr{B}_{k}}$ is $\nicefrac{{\epsilon}}{{2}}$-disjoint, and so, there are pairwise disjoint sets $E_{(n,c)}\subset F_{n}c$ (for each $(n,c)\in\mathscr{B}_{k}$), with $\left|E_{(n,c)}\right|\geq(1-\nicefrac{{\epsilon}}{{2}})\left|F_{n}c\right|$. Thus

[TABLE]

if $\left|C_{k}^{\prime\prime}\bigcap C_{k}\right|\geq(1-3\epsilon)|C_{k}|$ as in (18), then:

[TABLE]

Thus, with our choice of $\epsilon$ w.r.t. $\delta$ (12), we get that:

[TABLE]

In the same manner we constructed $\mathscr{B}_{k}^{\prime\prime}$, we use the next $q$ upcrossings to above $\beta$ to construct a collection $\mathscr{B}_{k+1}$ s.t. $\left(F_{n}c\right)_{\mathscr{B}_{k+1}}$ is an $\epsilon$-disjoint collection of upcrossings, with union $C_{k+1}=\bigcup_{\mathscr{B}_{k+1}}F_{n}g$ in $\Omega$ that satisfies one of the two options in the conclusion of Proposition 3.6. In particular, we have:

[TABLE]

(the last inequality follows from the assumption $(1-\epsilon)(1+\delta)\geq(1+\delta/2)$), and Lemma 3.7 is proved.$\hfill\square$

Back to the proof of Theorem 3.3, from Lamma 3.7 it follows that there exist finite subsets of $\Omega,$ $C_{1},...,C_{\lfloor\frac{N}{2q}\rfloor}$ s.t.

[TABLE]

(the last inequality follows partialy from the assumption $(1-\epsilon)\geq(1+\delta/2)^{-1}$). Since

[TABLE]

where $(1-\frac{\left|B\right|}{\left|\Omega\right|})$ can be made arbitrarily small (by taking $\Omega$ to be arbitrarily invariant), one have

[TABLE]

Thus the claim of the theorem takes place with $c_{0}=(1+\delta/2)^{-\frac{1}{2q}},c_{1}=(1+\delta/2)^{3}$.$\hfill\square$