TY - JOUR

T1 - Single and multiple recurrence along non-polynomial sequences

AU - Bergelson, Vitaly

AU - Moreira, Joel

AU - Richter, Florian K.

N1 - Funding Information:
The first author gratefully acknowledges the support of the National Science Foundation under grant DMS-1500575 . The second author gratefully acknowledges the support of the National Science Foundation under grant DMS-1700147 . The third author gratefully acknowledges the support of the National Science Foundation under grant DMS-1901453 .
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/7/15

Y1 - 2020/7/15

N2 - 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].

AB - 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].

KW - Multiple recurrence

KW - Nilsystems

KW - Non-conventional ergodic averages

KW - Thick sets

KW - Weighted averages

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U2 - 10.1016/j.aim.2020.107146

DO - 10.1016/j.aim.2020.107146

M3 - Article

AN - SCOPUS:85083109913

VL - 368

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107146

ER -