Ensembles de multiples de suites finies Original Research Article

A Behrend sequence is a (necessarily infinite) integer sequence A with elements exceeding 1 and whose set of multiples M A) has logarithmic density μ(A) = 1. By a famous theorem of Davenport and Erdös, this implies that (M A) also has natural density equal to 1. An ϵ-pseudo-Behrend sequence is a finite sequence of integers exceeding 1 with μ(A) > 1 − ϵ. We show that, for any given ϵϵ]0,1[ and any function ξN → ∞, the maximal number of disjoint ϵ-pseudo-Behrend sequences included in [1,N] is (log N)log2eO(ξN√log2N). We also prove that, for any given positive real number α, there is a positive constant c = c(α) such that c < μ(AN) < 1 − c where AN = AN(α) is the set of all products ab with N < a ⩽ N1 + α, a < b ⩽ a(1 + ϵN), (a,b) = 1 and ϵN: = (log N) 1 − log 3eξN√log2N. This provides, in a strong quantitative form, a finite analogue of the Maier-Tenenbaum theorem confirming Erdösʹ conjecture on the propinquity of divisors. A similar result holds for the natural density of the set of all integers n such that F(n) has a divisor in the interval [N,N1 + α], where F is any polynomial with integer coefficients, and we establish in full generality that this quantity tends to a limit as N approaches infinity.