Riesz sequences and generalized arithmetic progressions

TL;DR

This paper extends the understanding of Riesz sequences and generalized arithmetic progressions to multidimensional settings, specifically analyzing their growth rates on the two-dimensional torus.

Contribution

It provides the first sharp growth rate estimates for generalized arithmetic progressions in multidimensional exponential systems with Riesz sequence properties.

Findings

Established the sharp growth rate of steps in 2D generalized arithmetic progressions

Extended previous 1D results to multidimensional settings

Analyzed exponential systems on the 2D torus with Riesz sequence properties

Abstract

The purpose of this note is to verify that the results attained in [6] admit an extension to the multidimensional setting. Namely, for subsets of the two dimensional torus we find the sharp growth rate of the step(s) of a generalized arithmetic progression in terms of its size which may be found in an exponential systems satisfying the Riesz sequence property.