Solutions to congruences using sets with the property of Baire (Q2772965)

It was shown in [\textit{R. Dougherty} and \textit{M. Foreman}, ``Banach-Tarski decompositions using sets with the property of Baire'', J.~Am.\ Math.\ Soc.~7, No.~1, 75-124 (1994; Zbl 0795.28003)] that the Banach-Tarski paradox can be carried out using pieces with the property of Baire. The author of the paper under review uses the same methods to characterize those systems of congruences which have solutions in the sphere under rotations using sets with the property of Baire. More generally, for suitable systems of congruences the author finds solutions using sets with the property of Baire in Polish spaces~\(\mathcal X\) with a~non Abelian free group of homeomorphisms of~\(\mathcal X\) which acts locally commutatively on~\(\mathcal X\) and freely on a~comeager subset of~\(\mathcal X\).