Under recurrence in the Khintchine recurrence theorem

The Khintchine recurrence theorem asserts that on a measure preserving system, for every set $A$ and $\varepsilon>0$, we have $\mu(A\cap T^{-n}A)\geq \mu(A)^2-\varepsilon$ for infinitely many $n\in \mathbb{N}$. We show that there are systems having under-recurrent sets $A$, in the sense that the inequality $\mu(A\cap T^{-n}A)< \mu(A)^2$ holds for every $n\in \mathbb{N}$. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V.~Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.Comment: 18 pages. Referee's comments incorporated. To appear in the Israel Journal of Mathematic

Rates of divergence of non-conventional ergodic averages

We first study the rate of growth of ergodic sums along a sequence (an) of times: SNf(x)= μn≤Nf(Tanx). We characterize the maximal rate of growth and identify a number of sequences such as an=2n, along which the maximal rate of growth is achieved. To point out though the general character of our techniques, we then turn to Khintchines strong uniform distribution conjecture that the averages (1/N) ∑n≤Nf(nx mod 1) converge pointwise to f for integrable functions f. We give a simple, intuitive counterexample and prove that, in fact, divergence occurs at the maximal rate. © 2009 Cambridge University Press

A Hardy field extension of Szemerédi\u27s theorem

In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p (n) where p (n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi\u27s theorem can be of the form [nδ] where δ is any positive real number and [x] denotes the integer part of x. More generally, the common difference can be of the form [a (n)] where a (x) is any function that is a member of a Hardy field and satisfies a (x) / xk → ∞ and a (x) / xk + 1 → 0 for some non-negative integer k. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds

On convergence of oscillatory ergodic Hilbert transforms

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let p(t) be a Hardy field function which grows “super-linearly” and stays “sufficiently far” from polynomials. We show that for each measure-preserving system, (X, Σ, µ, τ), with τ a measure-preserving Z-action, the modulated one-sided ergodic Hilbert transform ∞ X e2πip(n) τnf (x) n n=1 converges µ-almost everywhere for each f ∈ Lr (X), 1 ≤ r \u3c ∞. This affirmatively answers a question of J. Rosenblatt [22]. In the second part of the paper, we establish almost-sure sparse bounds for a random one-sided ergodic Hilbert ∞ nX=1Xnn τnf (x), where {Xn} are uniformly bounded, independent, and mean-zero random variables

Sets of k-recurrence but not (k + 1)-recurrence

For every k ∈ ℕ, we produce a set of integers which is k-recurrent but not (k + 1)-recurrent. This extends a result of Furstenberg who produced a 1-recurrent set which is not 2-recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi\u27s theorem

Random differences in Szemerédi’s theorem and related results

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u1 ≥ u2 ≥ · · · } be a decreasing sequence of probabilities satisfying un · n1/(l+1)→∞. Let R = Rω be the random sequence obtained by selecting the natural number n with probability un. Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,.., a+lr of length l + 1 with difference r ∈ Rω. The best previous result (by M. Christ and us) was the condition un·n2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r1 \u3c r2 \u3c · · ·} of positive integers such that rn/n2−ϵ → ∞ for all ϵ \u3e 0 and, for any measure preserving transformation T of a probability space, the averages (Formula Presented) converge for almost every x. Our best previous result was the growth rate rn/n(1+1/14)−ϵ → ∞ of the sequence {rn}

SETS OF K-RECURRENCE BUT NOT (K+1)-RECURRENCE

We dedicate this paper to Y. Katznelson. Our work began at the conference organized for his 70th birthday, and we wish to honor him for his fundamental contribution to ergodic theory. Abstract. For every k ∈ N, we produce a set of integers which is k-recurrent but not (k+1)-recurrent. This extends a result of Furstenberg who produced a 1-recurrent set which is not 2-recurrent. We discuss a similar result for convergence of multiple ergodic averages. Finally, we also point out a combinatorial consequence related to Szemerédi’s theorem. Content