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

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

Töredékek a térben tér a töredékekben

Tizennyolc éve foglalkozom az esztergomi királyi palota festett töredékeivel, a királyi kápolnával és hozzá tartozó helyiségeivel, valamint a Studioloval. A kápolna különösen a szívemhez nőtt, gyönyörű darabjaival, összetettségével és az ezzel járó végeláthatatlan feladatokkal. Nem hinném, hogy létezik ma élő ember, aki jobban ismerné minden zegét-zúgát, az összes kicsi és nagyobb bajával, négyzetcentiméterek megható fordulataival, a “talpától” egészen a “feje búbjáig”. Ez a tér, elfogadja hiányosságait, nem törekszik a “mindenáron egészre”, vagy kiegészítettre.Ebben az egységben a kiegészített rekonstrukciós felületek is új funkciót kapnak, a fragmentált jelenetekkel valami egésszé formálódva, egy új műalkotássá állnak össze: a Magyar királyi kápolna- töredékké

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