<abstract xmlns="http://www.w3.org/1999/xhtml"> 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. </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.

Une Propriété Topologique de Certains Ensembles de Mills

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Une Propriété Topologique de Certains Ensembles de Mills"}

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...

Une Propriété Topologique de Certains Ensembles de Mills

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 139 - 153

Received: Mar 23, 2016

Accepted: Aug 02, 2016

DOI: https://doi.org/10.1515/udt-2017-0009

Keywords
Nombres premiers, théorème de Mills, ensemble triadique de Cantor

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Une Propriété Topologique de Certains Ensembles de Mills

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 139 - 153

Received: Mar 23, 2016

Accepted: Aug 02, 2016

DOI: https://doi.org/10.1515/udt-2017-0009

KeywordsNombres premiers, théorème de Mills, ensemble triadique de Cantor

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Nombres premiers, théorème de Mills, ensemble triadique de Cantor