Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J3 of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that(a)the set {0}∪{±3−(an+1)|n∈N} is quasi-convex in T if and only if a0>0 and gn>1 for every n∈N;(b)the set {0}∪{±an3|n∈N} is quasi-convex in the group J3 of 3-adic integers if and only if gn>1 for every n∈N. Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347–1356] providing a complete characterization of the sequences such that {0}∪{±2−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±2−(an+1)|n∈N} is quasi-convex in R.