JournalsjemsVol. 12, No. 6pp. 1417–1428

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
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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.

Cite this article

Fernando Mário de Oliveira Filho, Frank Vallentin, Fourier analysis, linear programming, and densities of distance avoiding sets in ℝ<sup><em>n</em></sup>. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1417–1428

DOI 10.4171/JEMS/236