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

TL;DR

This paper explores generalized Luzin sets relative to a $\sigma$-ideal, examining their measurability properties, constructions involving additive structures, and their sum with Sierpiński sets in Euclidean spaces.

Contribution

It introduces the concept of $\\mathcal{I}$-Luzin sets, analyzes their measurability under various conditions, and investigates their additive properties and interactions with Sierpiński sets.

Findings

Existence of translation invariant $\sigma$-ideals with Borel base where $\mathcal{I}$-Luzin sets can be $\mathcal{I}$-measurable.

Under Smital property, $\mathcal{I}$-Luzin sets are $\mathcal{I}$-nonmeasurable.

The sum of a Luzin set and a Sierpiński set cannot be a Bernstein set.

Abstract

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.