Topological torsion related to some recursive sequences of integers

Abstract

For a recursively defined sequence u := (un) of integers, we describe the subgroup tu(T) of the elements x of the circle group T satisfying limn unx = 0. More attention is dedicated to the sequences satisfying a secondorder recurrence relation. In this case, we show that the size and the free-rank of tu(T) is determined by the asymptotic behaviour of the ratios qn = un u n−1 and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion.

Introduction to topological groups

“…Definition 6.10. [7,9] A sequence A = {a n } n in an abelian group G is called a TB-sequence is there exists a precompct group topology on G such that a n → 0.…”

“…The advantage of TB-sequences over the T-sequences is in the easier way of determining sufficient condition for a sequence to be a TB-sequence [7,9]. For example, a a sequence (a n ) in Z is a TB-sequence iff the subgroup t a (T) of T is infinite.…”

“…[9] There exists a T B-sequence (a n ) in Z with lim n an+1 an = 1 . Here is an example of a T-sequence in Z that is not a TB-sequence.…”

“…Unaware of his result, Larcher [74], and later Kraaikamp and Liardet [71], proved that some cyclic subgroups of T are characterizable (see also [16,15,12,14,13] for related results). The paper [9] describes the algebraic structure of the subgroup t u (T) when the sequence u := (u n ) verifies a linear recurrence relation of order ≤ k,…”

Characterizing subgroups of compact abelian groups

“…The subgroups s u (X ) have been studied by many authors, especially in the case X = T, where X is identified with Z (see [1,6,7,10,11,15,18,24,25]). When x ∈ T is a non-torsion element, the behavior of the sequences of the form u n (x) : n ∈ ω is related to Diophantine approximation and dynamical systems (more specifically, the Sturmian flow [26]), as well as to the study of precompact group topologies with converging sequences [4,5,16].…”