Borel sets without perfectly many overlapping translations, III

TL;DR

This paper extends previous results on Borel sets and their overlapping translations to all perfect Abelian Polish groups, demonstrating the existence of specific sets with controlled translation overlaps using forcing.

Contribution

It generalizes earlier findings to all perfect Abelian Polish groups and constructs sets with a prescribed number of overlapping translations via forcing.

Findings

Existence of a ccc forcing adding a $$ set with specified translation overlaps

Sets with $\u0010$ many pairwise $k$--overlapping translations

No perfect set of such overlapping translations exists

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 $4\leq k<\omega$, then there is a ccc forcing notion adding a $\Sigma^0_2$ set $B\subseteq H$ which has $\aleph_\alpha$ many pairwise $k$--overlapping translations but not a perfect set of such translations.