A note on the parameterisation of Besicovitch sets

TL;DR

This paper constructs a set of measure zero in the plane that allows a line segment to be rotated continuously via a Baire-1 map, addressing a question about geometric transformations within measure-zero sets.

Contribution

It demonstrates the existence of a measure-zero set permitting continuous rotation of a line segment, answering a question posed by Terry Tao.

Findings

Existence of a measure-zero set with continuous segment rotation

Construction of a Baire-1 map for segment rotation

Addresses a previously open question in geometric measure theory

Abstract

In response to a question raised (and answered in the negative) by Terry Tao on his blog as to whether it is possible to rotate a line segment continuously within a set of area zero, we show that there is a set of area zero in the plane within which a line segment can be rotated by a Baire-1 map.