Local existence of $\mathcal{K}$-sets, projective tensor products, and Arens regularity for $A(E_{1}+\dots +E_{n})$

Theorem. If $X_{1},\dots ,X_{n}$ are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets $Y_{1}\subseteq X_{1},\dots ,Y_{n}\subseteq X_{n}$such that (i) $Y_{j}$ is either a Kronecker set or (ii) for some $p_{j}\ge 2$, $Y_{j}$ is a translate of a $K_{p_{j}}$-set all of whose elements have order $p_{j}$, and (iii) $A(Y_{1}+\dots +Y_{n})$ is isomorphic to the projective tensor product $C(Y_{1}) \hat \otimes \cdots \hat \otimes C(Y_{n})$.

This extends what was previously known for groups such as $\mathbb{T}$ or for the case $n=2$to the general locally compact abelian group. Old results concerning the local existence of Kronecker and $K_{p}$-sets are improved.