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

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.