Reflexive group topologies on the integers generated by sequences

Abstract

We establish reflexivity of a family of group topologies on Z generated by sequences, extending results of Gabriyelyan [21]. More precisely, for a T-sequence b=(bn)n∈N of integers and the associated topology Tb on Z (in the sense of [28]), we prove that (Z,Tb) is reflexive whenever the ratios qn=bn+1bn are integers and diverge to ∞ (whereas the same conclusion was obtained in [21] under the more stringent condition ∑n≥11qn<∞). The character group of (Z,Tb) is the subgroup ttb(T):={x+Z∈T:bnx+Z→0} of the torus T. If the ratios qn are integers and for some ℓ∈N the sequence of quotients (bn+ℓbn) diverges to ∞, then ttb(T) with the compact-open topology is reflexive.