A one-dimensional planar Besicovitch-type set

TL;DR

This paper constructs a non-trivial 1-rectifiable set in the plane that contains rotated copies of a specific set in every direction, advancing the understanding of Besicovitch-type sets in geometric measure theory.

Contribution

It introduces a new example of a 1-rectifiable set with a Besicovitch-type property, expanding the class of known Besicovitch sets in the plane.

Findings

Existence of a 1-rectifiable $ ext{Gamma}$-Besicovitch set in the plane.

Construction of a non-trivial $ ext{Gamma}$-Besicovitch set with specific rectifiability.

Advancement in understanding geometric configurations containing rotated copies in all directions.

Abstract

A $\Gamma$-Besicovitch set is a set which contains a rotated copy of $\Gamma$ in every direction. Our main result is the construction of a non-trivial $1$-rectifiable set $\Gamma$ in the plane, for which there exists a 1-dimensional $\Gamma$-Besicovitch set.