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

الأصل · EN

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.