Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications

Abstract

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and nonpolynomial) extensions of Szemerédi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume• for any measure preserving system (X, B, µ, T ) and h 1 , . . . , h k ∈ L ∞ (X), the limitWe also show that if f 1 , . . . , f k belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system (X, B, µ, T ) and any A ∈ B, lim sup

Joint ergodicity of Hardy field sequences

“…A similar result was established in [4] with lim sup in place of the limit, but under more general conditions on the functions a 1 , ..., a k . Utilizing Furstenberg's correspondence principle, we can deduce a combinatorial result about large sets of integers.…”

“…Theorem 1.2 was also proven in [9] when all functions a 1 , ..., a k have different growth rates and satisfy t N i +ε ≪ a i (t) ≺ t N i +1 for non-negative integers N i and some ε > 0. In addition, Theorem 1.2 was established in [4] under a variant of our condition. More precisely, an independence condition on the functions a 1 , ..., a k and on all of their derivatives was imposed.…”

“…General overview of the proof and organization of the paper. Similarly to the work in [10,9,4], our approach is to show that the Host-Kra factor introduced in [18] (see also [20] for a presentation of the general theory) is characteristic for convergence of our averages. This is a technique used extensively in the literature to reduce the problem of convergence in general measure preserving systems to the case where the system is a rotation in a nilpotent homogeneous space.…”

“…Furthermore, there are also several differences between our methods and the methods used in [4] and [10] to establish seminorm estimates, where a standard PET induction argument was utilized to reduce to the case of functions with sub-linear growth rate. This technique restricts the cases that can be handled, because the van der Corput operation may eventually yield functions that do not satisfy the condition (A) (a typical example in this case is the pair of functions (t log t, t log log t), which "drop" to functions that have growth rate smaller than log t after applications of the van der Corput inequality).…”

Joint ergodicity of fractional powers of primes

“…We also prove more general statements of this sort involving two or more linearly independent polynomials with fractional exponents evaluated at primes (the corresponding result for the integers was established in [4,25]).…”