Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group GG, the following statements are equivalent: (i)GG contains no sequence {xn}n=0∞ such that {0}∪{±xn∣n∈N}{0}∪{±xn∣n∈N} is infinite and quasi-convex in GG, and xn⟶0xn⟶0;(ii)one of the subgroups {g∈G∣2g=0}{g∈G∣2g=0} or {g∈G∣3g=0}{g∈G∣3g=0} is open in GG;(iii)GG contains an open compact subgroup of the form Z2κ or Z3κ for some cardinal κκ.