Une Propriété Topologique de Certains Ensembles de Mills

Abstract

Abstract In this article , we show that the set of Mills constants (real numbers M such that [M3ⁿ] is prime for all n ≥ 0) is the increasing limit of sets homeomorphic to the triadic Cantor’s set. More generally, for a given function ϕ and a set A of integers, we studying the Mills set Mϕ(A) = {α ∈ ℝ/ ∀n ∈ ℕ, [ϕn(α)] ∈ A} (where ϕn = ϕ∘...∘ϕ n times). We show that, under certain assumptions over ϕ and A, for all real w > infMϕ(A) the set Mϕ(A) ∩ [2, w] is homeomorphic to the triadic Cantor’s set.