Pointwise Ergodic Theorem Research Articles

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits Following our proof of the first Goldbach result in number field setting after 60 years (Mitsui 1960s), We give new information on density ternary Goldbach problem after 10 years (Shao 2013), and also design a new strategy to prove density version of representing a number as sum of “certain” primes. We also apply Rosser Sieve arguments to prove represtation of integers into sum of primes coming from a zero density subset. From Ergodic theory side, we give the first pointwise bilinear theorem along the primes, which is the natural pointwise convergence along prime result after Wierdl (1990) and a natural follow up after recent pointwise bilinear break-through by Krause Mirek-Tao (2021). In another project, the sharpest pointwise convergence average along primes for “near” integrable function is given, which is sharp up to Generalized Riemann Hypothesis.

Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets

We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.

Ergodic averages with the Hecke eigenvalue square weights and the Piltz divisor function weights

In this paper, we prove that the Hecke eigenvalue square for a holomorphic cusp form and the Piltz divisor functions are good weighting functions for the pointwise ergodic theorem. Additionally, we prove similar results for various other arithmetical functions in the last section.

Grid method for divergence of averages

AbstractIn this paper, we will introduce the ‘grid method’ to prove that the extreme case of oscillation occurs for the averages obtained by sampling a flow along the sequence of times of the form $\{n^\alpha : n\in {\mathbb {N}}\}$ , where $\alpha $ is a positive non-integer rational number. Such behavior of a sequence is known as the strong sweeping-out property. By using the same method, we will give an example of a general class of sequences which satisfy the strong sweeping-out property. This class of sequences may be useful to solve a long-standing open problem: for a given irrational $\alpha $ , whether the sequence $(n^\alpha )$ is bad for pointwise ergodic theorem in $L^2$ or not. In the process of proving this result, we will also prove a continuous version of the Conze principle.

A dynamical approach to the asymptotic behavior of the sequence

AbstractWe study the asymptotic behavior of the sequence $ \{\Omega (n) \}_{ n \in \mathbb {N} } $ from a dynamical point of view, where $ \Omega (n) $ denotes the number of prime factors of $ n $ counted with multiplicity. First, we show that for any non-atomic ergodic system $(X, \mathcal {B}, \mu , T)$ , the operators $T^{\Omega (n)}: \mathcal {B} \to L^1(\mu )$ have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along $\Omega (n)$ . Second, we show that the behaviors of $\Omega (n)$ captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.

Some remarks on oscillation inequalities

AbstractIn this paper, we establish uniform oscillation estimates on $L^p(X)$ with $p\in (1,\infty )$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990s [Pointwise ergodic theorems via harmonic analysis. Proceedings of Conference on Ergodic Theory (Alexandria, Egypt, 1993) (London Mathematical Society Lecture Notes, 205). Eds. K. Petersen and I. Salama. Cambridge University Press, Cambridge, 1995, pp. 3–151]. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt, and Wierdl for bounded martingales [Oscillation in ergodic theory. Ergod. Th. & Dynam. Sys.18(4) (1998), 889–935]. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for r-variation inequalities.

Polynomial averages and pointwise ergodic theorems on nilpotent groups

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on \(\sigma \)-finite measure spaces. We also establish corresponding maximal inequalities on \(L^p\) for \(1<p\le \infty \) and \(\rho \)-variational inequalities on \(L^2\) for \(2<\rho <\infty \). This gives an affirmative answer to the Furstenberg–Bergelson–Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

Computability of convergence rates in the ergodic theorem for Martin-Löf random points

In this paper we look at the convergence rates for the ergodic averages in the pointwise ergodic theorem for computable ergodic transformations on the unit interval. While these rates are layerwise computable for Martin-Löf random points and effectively open sets with Lebesgue measure a computable real, they are also layerwise computable for an arbitrary interval. There are however, effectively open sets for which there are \emph{no} effective rates, in particular, not layerwise computable ones. We also show that, when the measure of the effectively open set is any real $\alpha$, the convergence rates are computable in $\alpha$ and the layers relative to $\alpha$.

Pointwise ergodic theorems for non-conventional bilinear polynomial averages

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages $$ A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x) $$ as $N \to \infty$, where $T\colon X\to X$ is a measure-preserving transformation of a $\sigma$-finite measure space $(X,\mu), P(\mathrm{n})\in \mathbb{Z}[\mathrm{n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$ for some $p_1, p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to ``break duality" by handling some ranges of exponents $p_1$, $p_2$ with $\frac{1}{p_1} + \frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly. This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(\mathrm{n})=\mathrm{n}^2)$, which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. This also gives a contribution to the Furstenberg--Bergelson--Leibman conjecture. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $\mathbb{A}_{\mathbb{Z}}$ also plays a role.

Pointwise ergodic theorem for locally countable quasi-pmp graphs

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $ L^1 $ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor.The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon–Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

Compressibility analysis of asymptotically mean stationary processes

This work provides new results for the analysis of random sequences in terms of ℓ p -compressibility. The results characterize the degree in which a random sequence can be approximated by its best k -sparse version under different rates of significant coefficients (compressibility analysis). In particular, the notion of strong ℓ p -characterization is introduced to denote a random sequence that has a well-defined asymptotic limit (sample-wise) of its best k -term approximation error when a fixed rate of significant coefficients is considered (fixed-rate analysis). The main theorem of this work shows that the rich family of asymptotically mean stationary (AMS) processes has a strong ℓ p -characterization. Furthermore, we present results that characterize and analyze the ℓ p -approximation error function for this family of processes. Adding ergodicity in the analysis of AMS processes, we introduce a theorem demonstrating that the approximation error function is constant and determined in closed-form by the stationary mean of the process. Our results and analyses contribute to the theory and understanding of discrete-time sparse processes and, on the technical side, confirm how instrumental the point-wise ergodic theorem is to determine the compressibility expression of discrete-time processes even when stationarity and ergodicity assumptions are relaxed.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

AbstractWe prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Quantitative multiple pointwise convergence and effective multiple correlations

We show that effective 2ℓ-multiple correlations imply quantitative ℓ-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.

Pointwise ergodic theorems for higher levels of mixing

We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly mixing measure preserving systems. We show that our pointwise theorem for weakly mixing systems is strictly stronger than the Wiener-Wintner Theorem. We also show that our pointwise Theorems for weakly mixing and strongly mixing systems characterize weakly mixing systems and strongly mixing systems respectively. The methods of this paper also allow one to prove an enhanced pointwise ergodic theorem for other levels of the ergodic hierarchy such as ergodicity and mild mixing but not K-mixing. The author plans to include these additional pointwise ergodic theorems in his thesis.

Ergodicity Conditions for Controlled Stochastic Nonlinear Systems under Information Constraints

Consider a stochastic nonlinear system controlled over a possibly noisy communication channel. An important problem is to characterize the largest class of channels which admit coding and control policies so that the closed-loop system is stochastically stable. In this paper, we consider the stability notion of (asymptotic) ergodicity. We prove lower bounds on the channel capacity necessary to achieve the stability criterion. Under mild technical assumptions, we obtain that the necessary channel capacity is lower bounded by the log-determinant of the linearization, double-averaged over the state and noise space. Our proof uses a modified version of invariance entropy, and utilizes the almost sure convergence of sample paths guaranteed by the pointwise ergodic theorem. Our results generalize those for linear systems, and are in some cases more refined than those obtained for nonlinear systems via information-theoretic methods.

Unimodular Hausdorff and Minkowski dimensions

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley’s lemma and Frostman’s lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.

Failure of the $$L^1$$ pointwise ergodic theorem for $$\mathrm {PSL}_2(\mathbb {R})$$

Amos Nevo established the pointwise ergodic theorem in $$L^p$$ for measure-preserving actions of $$\mathrm {PSL}_2(\mathbb {R})$$ on probability spaces with respect to ball averages and every $$p>1$$ . This paper shows by explicit example that Nevo’s Theorem cannot be extended to $$p=1$$ .

Dimension-Free L p-Maximal Inequalities for Spherical Means in ℤm+1N

Abstract We extend the main result of Krause—the existence of dimension-free $L^p$-bounds, $p &amp;gt; 1$, for the spherical maximal function in the hypercube, $\{0,1\}^N$—to all cyclic groups, $\mathbb{Z}_{m+1}^N, \ m \geq 1$. Our approach follows that of Krause, which grew out of the arguments of Harrow, Kolla, and Schulman, which were in turn motivated by the spectral technique developed by Nevo and Stein, and by Stein, in the context of pointwise ergodic theorems on general groups.

Dominated and pointwise ergodic theorems with “weighted” averages for bounded Lamperti representations of amenable groups

Group representations by bounded Lamperti operators in the spaces Lα(1≤α<∞) form a wide class of representations, including representations by bounded positive operators and (when α≠2) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Cesàro averages for the bounded Lamperti representations of amenable σ-compact locally compact groups in Lα(1<α<∞) were proved by A. Tempelman in Proc. Amer. Math. Soc. 143 (2015) 4989–5004. By using a completely different, functional-analytical method, developed by A. Shulman in his PhD thesis in 1988, we generalize this result to “weighted” averages of such representations and discuss various conditions on the “weights” under which the DET and the PET hold. We conclude with applications of the general results to the bounded Lamperti representations of groups of polynomial growth and of the groups Rm and Zm.

A double return times theorem

We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$ f_1(T^{a_1 n}x) f_2(T^{a_2 n}x) $$ is a good weight for the pointwise ergodic theorem.