Real Analysis Exchange

Some Results in Support of the Kakeya Conjecture

Jonathan M. Fraser, Eric J. Olson, and James C. Robinson

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Abstract

A Besicovitch set is a subset of \(\mathbb{R}^d\) that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least \((d+1)/2\) (better estimates are available in the literature). We also study the “generic validity” of the Kakeya conjecture in the setting of Baire Category and prove that typical Besicovitch sets have full upper box-counting dimension. We also study a weaker version of the Kakeya problem where unit line segments are replaced by half-infinite lines. We prove that such “half-extended Besicovitch sets” have full Assouad dimension. This can be viewed as full resolution of a (much weakened) version of the Kakeya problem.

Article information

Source
Real Anal. Exchange, Volume 42, Number 2 (2017), 253-268.

Dates
First available in Project Euclid: 10 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1525917687

Digital Object Identifier
doi:10.14321/realanalexch.42.2.0253

Mathematical Reviews number (MathSciNet)
MR3721801

Zentralblatt MATH identifier
06870329

Subjects
Primary: 28A80: Fractals [See also 37Fxx] 54E52: Baire category, Baire spaces 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]

Keywords
Kakeya conjecture box-counting dimension Hausdorff dimension Assouad dimension Baire category

Citation

Fraser, Jonathan M.; Olson, Eric J.; Robinson, James C. Some Results in Support of the Kakeya Conjecture. Real Anal. Exchange 42 (2017), no. 2, 253--268. doi:10.14321/realanalexch.42.2.0253. https://projecteuclid.org/euclid.rae/1525917687


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