Borel sets without perfectly many overlapping translations, III

Abstract

We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $\alpha <{\omega }_1$ and $4\le k<\omega$, then there is a ccc forcing notion adding a ${\Sigma }_2^0$ set $B\subseteq H$ which has ${\aleph }_{\alpha }$ many pairwise $k$--overlapping translations but not a perfect set of such translations.