Fourier analysis, linear programming, and densities of distance avoiding sets in R^n

TL;DR

This paper develops a linear programming approach using Fourier analysis to derive new upper bounds on the density of sets in R^n that avoid specific distances, impacting measurable chromatic numbers and generalizing previous theorems.

Contribution

It introduces a novel linear programming method leveraging Fourier analysis to bound densities of distance-avoiding sets in high-dimensional Euclidean spaces.

Findings

New upper bounds for sets avoiding the unit distance in dimensions 2 to 24

Improved lower bounds for the measurable chromatic number in dimensions 3 to 24

A concise proof of a generalized theorem on sets avoiding multiple distances

Abstract

In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances.