A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = (un) n∈N 0 in N such that unx → 0 uniformly on A. In particular, A is contained in the subgroup tu(T) := {x ∈ T : unx → 0}, which is the subgroup of T characterized by u. Using strictly increasing sequences u in N such that un divides un+1 for every n ∈ N, we find in T a family of closed perfect D-sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D-sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Moreover, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.

Dirichlet sets vs Characterized subgroups

BARBIERI, Giuseppina Gerarda;DIKRANJAN, Dikran;GIORDANO BRUNO, Anna
;
Weber, Hans
2017-01-01

Abstract

A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = (un) n∈N 0 in N such that unx → 0 uniformly on A. In particular, A is contained in the subgroup tu(T) := {x ∈ T : unx → 0}, which is the subgroup of T characterized by u. Using strictly increasing sequences u in N such that un divides un+1 for every n ∈ N, we find in T a family of closed perfect D-sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D-sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Moreover, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.
File in questo prodotto:
File Dimensione Formato  
1-s2.0-S0166864117303978-main.pdf

accesso aperto

Descrizione: Articolo principale, Accesso Aperto MIUR
Tipologia: Versione Editoriale (PDF)
Licenza: Creative commons
Dimensione 583.39 kB
Formato Adobe PDF
583.39 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/1086220
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 5
social impact