
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.
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