Density estimates of 1-avoiding sets via higher order correlations

TL;DR

This paper improves the upper bound on the density of planar sets avoiding unit distances by employing novel triple-order correlation constraints through Fourier analysis and linear programming.

Contribution

It introduces the use of triple-order correlations to derive new linear constraints, advancing the analysis of distance-avoiding sets.

Findings

Upper bound on density improved to 0.25442

New linear constraints from triple-order correlations

Combination of Fourier analysis and linear programming techniques

Abstract

We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied.