Some properties of $\mathcal{I}$-Luzin sets

In this paper we consider a notion of $\mathcal{I}$-Luzin set which generalizes the classical notion of Luzin set and Sierpi{\'n}ski set on Euclidean spaces. We show that there is a translation invariant $\sigma$-ideal $\mathcal{I}$ with Borel base for which $\mathcal{I}$-Luzin set can be $\mathcal{I}$-measurable. If we additionally assume that $\mathcal{I}$ has Smital property (or its weaker version) then $\mathcal{I}$-Luzin sets are $\mathcal{I}$-nonmeasurable. We give some constructions of $\mathcal{I}$-Luzin sets involving additive structure of $\mathbb{R}^n$. Moreover, we show that if $L$ is a Luzin set and $S$ is a Sierpi{\'n}ski set then the complex sum $L+S$ cannot be a Bernstein set.