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