Some properties of I-Luzin sets

In this paper we consider a notion of I-Luzin set which generalizes the classical notion of Luzin set and Sierpiński set on Euclidean spaces. We show that there is a translation invariant σ-ideal I with Borel base for which I-Luzin set can be I-measurable. If we additionally assume that I has the Smital property (or its weaker version) then I-Luzin sets are I-nonmeasurable. We give some constructions of I-Luzin sets involving additive structure of Rn. Moreover, we show that if c is regular, L is a generalized Luzin set and S is a generalized Sierpiński set then the complex sum L+S belongs to Marczewski ideal s0.