Une Propriété Topologique de Certains Ensembles de Mills

In this article , we show that the set of Mills constants (real numbers M such that [M^3ⁿ] 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 M_ϕ(A) = {α ∈ ℝ/ ∀n ∈ ℕ, [ϕ_n(α)] ∈ A} (where ϕ_n = ϕ∘...∘ϕ n times). We show that, under certain assumptions over ϕ and A, for all real w > infM_ϕ(A) the set M_ϕ(A) ∩ [2, w] is homeomorphic to the triadic Cantor’s set.