Single and multiple recurrence along non-polynomial sequences

Vitaly Bergelson*, Joel Moreira, Florian K. Richter

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We establish new recurrence and multiple recurrence results for a rather large family F of non-polynomial functions which contains tempered functions and (non-polynomial) functions from a Hardy field with polynomial growth. In particular, we show that, somewhat surprisingly (and in the contrast to the multiple recurrence along polynomials), the sets of return times along functions from F are thick, i.e., contain arbitrarily long intervals. A major component of our paper is a new result about equidistribution of sparse sequences on nilmanifolds, whose proof borrows ideas from the work of Green and Tao [26]. Among other things, we show that for any f∈F, any invertible probability measure preserving system (X,B,μ,T), any A∈B with μ(A)>0, and any ε>0, the sets of returns {n∈N:μ(A∩T−⌊f(n)⌋A)>μ2(A)−ε} {n∈N:μ(A∩T−⌊f(n)⌋A∩T−⌊f(n+1)⌋A∩⋯∩T−⌊f(n+k)⌋A)>0} are thick. Our recurrence theorems imply, via Furstenberg's correspondence principle, some new combinatorial results. For example, we show that given a set E⊂N with positive upper density, for every k∈N there are a,n∈N such that {a,a+⌊f(n)⌋,⋯,a+⌊f(n+k)⌋}⊂E. When f(n)=nc, with c>0 non-integer, this result provides a positive answer to a question posed by Frantzikinakis [16, Problem 23].

Original languageEnglish (US)
Article number107146
JournalAdvances in Mathematics
Volume368
DOIs
StatePublished - Jul 15 2020

Keywords

  • Multiple recurrence
  • Nilsystems
  • Non-conventional ergodic averages
  • Thick sets
  • Weighted averages

ASJC Scopus subject areas

  • Mathematics(all)

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