Density estimates of 1-avoiding sets via higher order correlations

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.

11 pages, 2 figures

References in corpus (1)

• Complete positivity and distance-avoiding sets

Cited by in corpus (3)

• Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs
• The density of planar sets avoiding unit distances
• On lower bounds of the density of planar periodic sets without unit distances