Answer to Raczkowski's questions on convergent sequences of integers

Abstract

1) Every infinite, Abelian compact (Hausdorff) group K admits 2 |K|many dense, non-Haar-measurable subgroups of cardinality |K|. When

Dirichlet sets vs characterized subgroups

“…We recall Problem 1.11 below, which was proposed in [3]. To better express it, we introduce the following notion.…”

“…The behavior of the sequence of multiples (u n α) n∈N 0 , where (u n ) n∈N 0 is a sequence of integers and α ∈ [0, 1], considered modulo 1 has deep roots in Topology (precompact topologies on Z with or without converging sequences [4,15]), Harmonic Analysis (sets of convergence of trigonometric series, thin sets) and Topological Algebra (topo-logically torsion elements and characterized subgroups), as well as in Number Theory (uniform distribution of sequences) and Dynamical Systems.…”

Introduction to topological groups

“…Let us see that G is a Markov group (i.e., M G is discrete) 7 . Assume T be a Hausdorff group topology on G. There…”

“…(1−kε) 2 = 3 2r k 2 and pick sufficiently large n to have (8). Now apply the Bogoliouboff Lemma 5.4 to find s = 3 2r k 2 characters ξ 1n , .…”

“…Our next aim is to eliminate the dependence of the number m of characters on the free rank of the group A in Bogoliouboff -Følner's lemma. The price to pay for this is taking V (8) instead of V (4) .…”