Ideals with Smital properties

Abstract

A sigma -ideal mathcal {I} on a Polish group (X,+) has the Smital Property if for every dense set D and a Borel mathcal {I}-positive set B the algebraic sum D+B is a complement of a set from mathcal {I}. We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are mathfrak {c} many maximal invariant sigma -ideals with Borel bases on the Cantor space 2^omega .