Parties d'ensembles B-normaux

Abstract

For a given subset A of the set R of real numbers, we defined M(A) as the infimum of all the lengths of the finite intervals I such that there exists a sequence Λ of real numbers in I such that A is the associated normal set B(Λ). We prove that if A′ is a subset of A, such that all the multiples k.a' belongs to A′ (for each non zero integer k and each element a′ of A′), then M(A′) is less or equal to 2.M(A). Thus the family of the subsets A such that M(A) is finite is closed under intersection and finite union.