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 recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances.
Additional Metadata
Keywords Nelson-Hadwiger problem, measurable chromatic number, linear programming, positive almost periodic functions
MSC Fourier series and coefficients (msc 42B05)
THEME Logistics (theme 3)
Publisher European Mathematical Society
Journal Journal of the European Mathematical Society
Project Spinoza prijs Lex Schrijver
de Oliveira Filho, F.M, & Vallentin, F. (2010). Fourier analysis, linear programming, and densities of distance avoiding sets in {$R^n$}. Journal of the European Mathematical Society, 12, 1417–1428.