A multidimensional Szemeredi theorem for Hardy sequences of different growth

TL;DR

This paper extends the multidimensional polynomial Szemerédi theorem to Hardy sequences with different growth rates, revealing new combinatorial and dynamical properties of sparse sequences.

Contribution

It introduces a novel approach to analyze Hardy sequences in the context of Szemerédi-type theorems, providing new limit formulas and dynamical consequences.

Findings

Established a limit formula for multiple ergodic averages along Hardy sequences.

Proved denseness of certain orbits in topological dynamics for sparse subsequences.

Showed that syndetic sets and colorings contain specific non-shift invariant and polychromatic patterns.

Abstract

We prove a variant of the multidimensional polynomial Szemer\'edi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some Hardy field and satisfy certain growth conditions. We do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula. A consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences. Another consequence is that every syndetic set of integers contains certain non-shift invariant patterns, and every finite coloring of $\N$, with each color class a syndetic set, contains certain polychromatic patterns, results very particular to our non-polynomial setup.