JournalsjemsVol. 12 , No. 6DOI 10.4171/jems/236

Fourier analysis, linear programming, and densities of distance avoiding sets in ℝ<sup><em>n</em></sup>

  • Fernando Mário de Oliveira Filho

    Centrum voor Wiskunde en Informatica, Amsterdam, Netherlands
  • Frank Vallentin

    Centrum voor Wiskunde en Informatica, Amsterdam, Netherlands
Fourier analysis, linear programming, and densities of distance avoiding sets in ℝ<sup><em>n</em></sup> cover

Abstract

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 short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss, Bourgain and Falconer about sets avoiding many distances.