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

TL;DR

This paper characterizes quasi-convex sequences converging to zero in the circle group and 3-adic integers, providing conditions on integer sequences and solving an open problem about 2-adic quasi-convexity.

Contribution

It offers a complete characterization of quasi-convex sequences in the circle group and 3-adic integers, and solves an open problem regarding 2-adic sequences.

Findings

Quasi-convexity in T depends on sequence growth conditions.

Complete characterization of sequences in J_3 for quasi-convexity.

Resolution of an open problem on 2-adic quasi-convex sequences.

Abstract

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J_3 of 3-adic integers. These sequences are determined by an increasing sequences of integers. For an increasing sequence \underline{a}=\{a_n\} of integers, put g_n=a_{n+1}-a_n. We prove that: (a) the set \{0\}\cup\{\pm 3^{-(a_n+1)} : n\in N\} is quasi-convex in T if and only if a_0>0 and g_n>1 for every n\in N; (b) the set \{0\}\cup\{\pm 3^{a_n} : n\in N\} is quasi-convex in the group J_3 of 3-adic integers if and only if g_n>1 for every n\in N. Moreover, we solve an open problem of Dikranjan and de Leo by providing a complete characterization of the sequences \underline{a} such that \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences \underline{a} such that the set \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in R.