Ideals with Smital properties

A $$\sigma $$ σ -ideal $$\mathcal {I}$$ I on a Polish group $$(X,+)$$ ( X , + ) has the Smital Property if for every dense set D and a Borel $$\mathcal {I}$$ I -positive set B the algebraic sum $$D+B$$ D + B is a complement of a set from $$\mathcal {I}$$ 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 $$\mathfrak {c}$$ c many maximal invariant $$\sigma $$ σ -ideals with Borel bases on the Cantor space $$2^\omega $$ 2 ω .