Arithmetic progressions in Salem-type subsets of the integers

TL;DR

This paper introduces fractional density as a measure for sparse integer sets, linking it to Hausdorff dimension, and extends results on 3-term arithmetic progressions to these sets under Fourier decay conditions.

Contribution

It defines fractional density for zero-density sets and generalizes a theorem on arithmetic progressions to these sets with Fourier decay constraints.

Findings

Fractional density relates to Hausdorff dimension.

Arithmetic progressions exist in sparse sets with fractional density.

Theorem extension from unit interval to integer subsets.

Abstract

Given a subset of the integers of zero density, we define the weaker notion of fractional density of such a set. It is shown how this notion corresponds to that of the Hausdorff dimension of a compact subset of the reals. We then show that a version of a theorem of {\L}aba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density and satisfying certain Fourier-decay conditions.