Claremont Center for the Mathematical Sciences

In this talk I will consider the problem of determining the maximum density of sets in Euclidean space which avoid some given distances, in the sense that there are no two points in the set at the given distances. To find upper bounds for the maximum density we use the Fourier coefficients of the autocorrelation function of the characteristic function of a distance avoiding set together with linear programming. This method is a continuous analog of the sphere packing bound due to Cohn and Elkies.

I will show two applications of the bound: Computing new upper bounds for the density of sets avoiding the unit distance in dimension 2,...,24. Giving an alternative, quantitative, and extremely simple proof of a result of Furstenberg, Katznelson, Weiss, proved by ergodic theoretic methods: Every planar set of positive density does not avoid arbitrary large distances.

This is joint work with Fernando M. de Oliveira Filho
(http://arxiv.org/abs/0808.1822).