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 J 3 of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence a ̲ = { a n } n = 0 ∞ ⊆ Z , put g n = a n + 1 − a n . We prove that(a) t { 0 } ∪ { ± 3 − ( a n + 1 ) | n ∈ N } is quasi-convex in T if and only if a 0 > 0 and g n > 1 for every n ∈ N ; t { 0 } ∪ { ± 3 a n | n ∈ N } is quasi-convex in the group J 3 of 3-adic integers if and only if g n > 1 for every n ∈ N . ver, 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 a ̲ such that { 0 } ∪ { ± 2 − ( a n + 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 − ( a n + 1 ) | n ∈ N } is quasi-convex in R .