Topological torsion related to some recursive sequences of integers

Abstract

Abstract For a recursively defined sequence u : = ( u n ) of integers, we describe the subgroup t u (𝕋) of the elements x of the circle group 𝕋 satisfying lim n u n x = 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 t u (𝕋) is determined by the asymptotic behaviour of the ratios q n = ( u n / u n –1 ) and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)