On polynomial $d$-chaos via $d$-dissociated character subsystems on compact abelian groups

In this paper, we study polynomial chaoses of degree $d$ constructed from sequences of functions; that is, sets of all possible $d$-fold products of sequence elements, allowing repeated factors. The tetrahedral chaos of degree $d$ is defined as the subset consisting of products with pairwise distinct factors. We prove that polynomial $d$-chaoses (and, consequently, the tetrahedral chaoses) with respect to $d$-dissociated subsystems of characters on compact abelian groups are $q$-lacunary and $2d\mathrm{/}(d+1)$-Sidon systems.