Quasi-convex sequences in the circle and the 3-adic integers
Abstract
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 an increasing sequences of integers. For an increasing sequence a=\an\ of integers, 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\\ 3an : 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 of Dikranjan and de Leo by providing a complete characterization of the sequences a such that \0\\ 2-(an+1) : n∈ N\ is quasi-convex in T. Using this result, we also obtain a characterization of the sequences a such that the set \0\\ 2-(an+1) : n∈ N\ is quasi-convex in R.