Luzin and Sierpi\'nski sets, some nonmeasurable subsets of the plane and additive properties on the line

TL;DR

This paper constructs nonmeasurable subsets of the plane with special properties, explores additive features of Luzin and Sierpiński sets, and introduces generalized I-Luzin sets, advancing understanding of nonmeasurable sets and their algebraic structures.

Contribution

It introduces new nonmeasurable plane subsets with specific properties and investigates additive properties of Luzin, Sierpiński, and generalized I-Luzin sets.

Findings

Construction of nonmeasurable plane sets with strong Luzin/Sierpiński intersection properties

Analysis of additive properties of Luzin and Sierpiński sets on the real line

Introduction of generalized I-Luzin sets

Abstract

In this paper we shall introduce some nonmeasurable and completely nonmeasurable subsets of the plane with various additional properties, e.g. being Hamel basis, intersecting each line in a strong Luzin / Sierpi\'nski set. Also some additive properties of Luzin and Sierpi\'nski sets and their generalization I-Luzin sets, on the line are investigated.