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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10990/657

Autori: Impieri, Daniele
Supervisore afferente all'Università: DIKRANJAN, DIKRAN
Titolo: Characterized Subgroups
Abstract (in inglese): Let T = R / Z be the written additively circle group and u = (un) be a sequence of integers. Many authors in various areas of Mathematics gave their attention to the following subgroups of T and their subsets t u( T ) = { x ∈ T | unx → 0 } . These subgroups are known with various names, here I refer to these subgroups as topologically u-torsion subgroups, because of their strong connection with torsion subgroups. Here, be- sides these subgroups in the circle group, I consider their nat- ural generalization for an arbitrary topological abelian group, with particular attention to the compact case: for a topologi- cal abelian group X and a sequence of characters v = (vn) the following subgroup s v(X) = { x ∈ X | vn(x) → 0 } is called characterized subgroup. Here I present some of my research results. In particular, I give a complete description of the subgroups t u( T ) where u is an arithmetic sequence, that is a strictly increasing sequence where un | un+1 for every n ∈ N. I give also some new results on the study of the Borel complexity of these subgroups, both in the compact case and in the circle group. Moreover, I present a structure theorem for the subgroups that admit a finer locally compact Polish group topology. The latter is a sufficient condi- tion for a subgroup to be characterized. Furthermore, I give a complete description of closed characterized subgroups in arbi- trary topological abelian groups and various useful reductions to the metrizable case. Presenting these results, I take the op- portunity to give an exhaustive description of the state of the art in this topic and to show some applications to other areas of Mathematics, with the aim of providing a useful handbook to an expert audience and a starting point for potential research purposes to non-expert users.
Parole chiave: Characterized subgroup, T-characterized, K-characterized, N-characterized, Circle group, Compact groups, Locally compact groups, Precompact group, MAP, MinAp, AMAP, Converging sequence, Arithmetic sequence, Topologically torsion, Topologically u-torsion, Sequence of integers, Characters, Pontryagin duality, Polishability, Locally quasi-convex, Borel complexity, von Neumann radical, Eggleston, Number Theory, Harmonic Analysis, Dynamical Systems, Topology, Uniform distribution, Continued fractions, Thin set, Trigonometric series, Arbault, A-set, D-set, Armacost, Prüfer, Furstenberg, Marcinkiewiz, Diophantine approximation, T-sequence, TB-sequence, Autocharacterized, Ergodic Theory, Fibonacci, Torsion, Topologcal group, Haar measure, Countable modulo compact, p-adic
MIUR : Settore MAT/02 - Algebra
Lingua: eng
Data: 8-lug-2015
Corso di dottorato: Dottorato di ricerca in Matematica e fisica
Ciclo di dottorato: 27
Università di conseguimento titolo: Università degli Studi di Udine
Luogo di discussione: Udine
Citazione: Impieri, D. Characterized Subgroups. (Doctoral Thesis, Università degli Studi di Udine, 2015).
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