Powers of sequences and recurrence

We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mend\`es-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.Comment: 30 pages. Numerous small changes made. To appear in the Proceedings of the London Mathematical Societ

Random differences in Szemerédi's theorem and related results

À paraître dans le Journal d'Analyse MathématiqueWe 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 one concerns existence of arithmetic progresseions of given length in any set of integers of positive density, with differences in a random subset of the integers. The second one concerns almost everywhere convergence of double ergodic averages along partially random sequences

Random Sequences and Pointwise Convergence of Multiple Ergodic Averages

International audienceWe prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preserving transformations, and $a_n$ is a random version of the sequence $[n^c]$ for some appropriate $c>1$. We also prove similar mean convergence results for averages of the form $\frac{1}{N}\sum_{n=1}^N f(T^{a_n}x)\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$ are powers of the same transformation. The deterministic versions of these results, where one replaces $a_n$ with $[n^c]$, remain open, and we hope that our method will indicate a fruitful way to approach these problems as well

Additive bases arising from functions in a Hardy field

A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show that they are asymptotic bases.Comment: 11 pages, reference problem fixe

Powers of sequences and convergence of ergodic averages

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $\frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages.Comment: After a few minor corrections, to appear in Ergodic Theory and Dynamical System

Random ergodic theorems with universally representative sequences

When elements of a measure-preserving action of Rd or Zd are selected in a random way, according to a stationary stochastic process, a.e. convergence of the averages of an LP function along the resulting orbits may almost surely hold, in every system; in such a case we call the sampling scheme universally representative. We show that i.i.d. integervalued sampling schemes are universally representative (with p > 1) if and only if they have nonzero mean, and we discuss a variety of other sampling schemes which have or lack this property

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions in L p, p\u3e1. This extends a result of Bourgain where he proved a similar theorem for p\u3e(1+√3)/2. © 1988 The Weizmann Science Press of Israel