THE DENSITY OF SETS AVOIDING DISTANCE 1 IN EUCLIDEAN SPACE

Christine Bachoc (1) , Alberto Passuello (1) , Alain Thiéry (1)

1 IMB - Institut de Mathématiques de Bordeaux

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

Domaines: Théorie des nombres [math.NT]

Christine Bachoc, Alberto Passuello, Alain Thiéry. THE DENSITY OF SETS AVOIDING DISTANCE 1 IN EUCLIDEAN SPACE. 2014. ⟨hal-01003676⟩