In this talk I will consider the problem of determining the maximal
density of sets in Euclidean space which avoid some given distances, in
the sense that there should be no two points in the set at the given
distances. To find upper bounds for the maximal density we use the Fourier
coefficients of the auto correlation function of the characteristic
function of a distance avoiding set together with linear programming. This
method is related to the linear programming bound for sphere packings due
to Henry Cohn and Noam Elkies. I give two applications of our bound: In dimensions 2, ..., 24 we compute
new upper bounds for the density of sets avoiding the unit distance, which
results into new lower bounds for the measurable chromatic number of
Euclidean space. Then, we have a new, simple proof of a recent result by
Boris Bukh concerning sets avoiding many distances. His proof resembles
the famous regularity lemma of Szemeredi. Furthermore, it implies a result
by Furstenberg, Katznelson, Weiss, proved by ergodic theoretic methods,
that every planar set of positive density realizes all distances which are
large enough. This is joint work with Fernando M. de Oliveira Filho.