  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 K is a subset of this group on which every continuous function into the
complex unit circle equals the restriction, to K, of a character in the group's
Pontryagin dual. Ackelsberg proves that if, for all q > 1, there exists a
perfect Kronecker-type set generating a dense subgroup, then there exist large
rigidity sequences for weak mixing systems of actions by countable discrete
abelian groups. Ackelsberg shows the existence of such sets for prime values of
q, while we construct them for all q > 1.
