Rigid-recurrent sequences for actions of finite exponent groups

The focus of this paper is to better understand the coexistence of rigidity, weak mixing, and recurrence by constructing thin sets in the product of countably many copies of the finite cyclic group of order $q$. A Kronecker-type set is a subset $K$ of this group on which every continuous function into the $q$th roots of unity equals the restriction, to $K$, of a character in the group's Pontryagin dual. Ackelsberg (2022) proves that if, for all $q > 1$, there exists a perfect Kronecker-type set generating a dense subgroup of $D_q$, then there exist large rigidity sequences for weak mixing systems of actions by countable discrete abelian groups of finite exponent. Ackelsberg shows the existence of such sets for prime values of $q$, while we construct them for all $q > 1$.