The intersection of typical Besicovitch sets with lines

TL;DR

This paper investigates the measure-theoretic properties of typical Besicovitch sets, showing that they intersect lines in measure zero unless the line is contained in the set, revealing intricate geometric structures.

Contribution

It establishes new measure-zero intersection properties of typical Besicovitch sets with lines, advancing understanding of their geometric and measure-theoretic structure.

Findings

Typical Besicovitch sets intersect lines not contained in them in measure zero.

Lines within a Besicovitch set intersect the union of other lines in measure zero.

The results highlight the complex geometric structure of Besicovitch sets.

Abstract

We show that a typical Besicovitch set $B$ has intersections of measure zero with every line not contained in it. Moreover, every line in $B$ intersects the union of all the other lines in $B$ in a set of measure zero.