Arithmetic properties of sparse subsets of $\mathbb{Z}^n$

TL;DR

This paper explores the relationship between the arithmetic structure of sparse subsets of integer lattices and their geometric properties, demonstrating the existence of specific configurations under certain Fourier decay and density conditions.

Contribution

It establishes a precise correspondence between discrete Fourier decay and geometric properties, extending Salem's construction to higher dimensions for analyzing arithmetic progressions.

Findings

Sparse subsets with Fourier decay contain arithmetic progressions

Higher-dimensional configurations like parallelograms exist in certain sparse sets

Mapping between integer sets and continuum preserves key dimensional and arithmetic properties

Abstract

Arithmetic progressions of length $3$ may be found in compact subsets of the reals that satisfy certain Fourier -- as well as Hausdorff -- dimensional requirements. It has been shown that a very similar result holds in the integers under analogous conditions, with Fourier dimension being replaced by the decay of a discrete Fourier transform. In this paper we make this correspondence more precise, using a well-known construction by Salem. Specifically, we show that a subset of the integers can be mapped to a compact subset of the continuum in a way which preserves certain dimensional properties as well as arithmetic progressions of arbitrary length. The higher-dimensional version of this construction is then used to show that certain parallelogram configurations must exist in sparse subsets of $\mathbb{Z}^n$ satisfying appropriate density and Fourier-decay conditions.