Reflexive group topologies on the integers generated by sequences

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 is an element of 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+1 bn are integers and diverge to oo (whereas the same conclusion was obtained in [21] under the more stringent condition �n >= 1 qn � oo). The character group of (Z, Tb) is the subgroup ttb(T ) := 1 {x + Z E T : bnx + Z - 0} of the torus T. If the ratios qn are integers and for ) some .e E N the sequence of quotients (bn+` diverges to oo, then ttb(T) with the bn compact-open topology is reflexive. (c) 2023 Elsevier B.V. All rights reserved.