Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact) if there is a sequence u=(un) in G such that τ is the finest precompact group topology on G making u=(un) converge to zero. It is proved that a metrizable precompact abelian group (G,τ) is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups (G,τ) and (G,η) have the same Pontryagin dual groups (in other words, (G,τ) is not a Mackey group in the class of maximally almost periodic groups).We give a complete description of all ss-precompact abelian groups modulo countable ss-precompact groups from which we derive:(1)No infinite pseudocompact abelian group is ss-precompact.(2)An ss-precompact group G is a k-space if and only if G is countable and sequential.(3)An ss-precompact group is hereditarily disconnected.(4)An ss-precompact group has countable tightness.We provide also a description of the sequentially complete ss-precompact abelian groups.