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Mathematics > Dynamical Systems

Title: Joint ergodicity of fractional powers of primes

Abstract: We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form $\{m,m+[p_n^a], m+[p_n^b]\}$, where $a,b$ are positive non-integers and $p_n$ denotes the $n$-th prime, a property that fails if $a$ or $b$ is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences and the bulk of the proof is devoted to obtaining good seminorm estimates for the related multiple ergodic averages. The input needed from number theory are upper bounds for the number of prime $k$-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.
Comments: 27 pages. Referee's comments incorporated. To appear in Forum of Mathematics, Sigma
Subjects: Dynamical Systems (math.DS); Combinatorics (math.CO); Number Theory (math.NT)
MSC classes: 37A45, 28D05, 05D10, 11B30
Cite as: arXiv:2107.02669 [math.DS]
  (or arXiv:2107.02669v2 [math.DS] for this version)

Submission history

From: Nikos Frantzikinakis [view email]
[v1] Tue, 6 Jul 2021 15:21:34 GMT (36kb)
[v2] Wed, 18 May 2022 15:32:12 GMT (37kb)

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