  We improve by an exponential factor the best known asymptotic upper bound for
the density of sets avoiding 1 in Euclidean space. This result is obtained by a
combination of an analytic bound that is an analogue of Lovasz theta number and
of a combinatorial argument involving finite subgraphs of the unit distance
graph. In turn, we straightforwardly obtain an asymptotic improvement for the
measurable chromatic number of Euclidean space. We also tighten previous
results for the dimensions between 4 and 24.
