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

Fernando M. de Oliveira Filho(f.m.de.oliveira.filho***at***cwi.nl)
Frank Vallentin(f.vallentin***at***cwi.nl)

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

Keywords: Nelson-Hadwiger problem, measurable chromatic number, linear programming, positive almost periodic functions

Category 1: Linear, Cone and Semidefinite Programming


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Entry Submitted: 08/13/2008
Entry Accepted: 08/13/2008
Entry Last Modified: 08/13/2008

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