Sur un ensemble de Besicovitch

TL;DR

This paper presents a corrected proof and an in-depth analysis of a plane set with zero area, constructed from line segments with directions covering an angle, building on Besicovitch's projection theorem and related to Kenyon's work.

Contribution

It provides a new, rigorous proof of a previously misproved example of a zero-area set with directional segments, and explores its properties in various dimensions.

Findings

The set has vanishing area despite covering a range of directions.

The proof utilizes Besicovitch's projection theorem and detailed analysis of horizontal sections.

Variations of the construction are applicable in higher dimensions.

Abstract

Here is an example of a plane set of vanishing area and consisting of line-segments whose directions cover an angle : let E be a Cantor set of dissection ratio 1/4 (therefore dimension 1/2) carried by the horizontal axis and E' the image of E by an homothetic transformation of ratio 2 whose center is not on the horizontal axis ; the union of the line segments joining E and E' is the set in question. The example was given in 1969 by the author with a wrong proof. The present article contains a short proof based on the projection theorem of Besicovitch, and a long proof resulting from the investigation of the arithmetic, geometric and analytic properties of the horizontal sections of the set. This investigation copies the study by Richard Kenyon of a similar problem. Variations of the construction are provided in the plane and in multidimensional spaces.