Countably infinite quasi-convex sets in some locally compact abelian groups

We produce a class of countably infinite quasi-convex sets (sequences converging to zero) in the circle group T and in the group J2 of 2-adic integers determined by sequences of integers satisfying a mild lacunarity condition. We also extend our results to the group R of real numbers. All these quasi-convex sets have a stronger property: Every infinite (necessarily) symmetric subset containing 0 is still quasi-convex.