Some results in support of the Kakeya Conjecture

TL;DR

The paper provides new results supporting the Kakeya conjecture, demonstrating that Besicovitch sets typically have full dimension and establishing lower bounds on their dimensions using simple, intuitive proofs.

Contribution

It offers simple proofs for dimension bounds of Besicovitch sets and shows that typical sets have full upper box-counting dimension, also analyzing a weaker half-infinite line variant.

Findings

Lower bound of (d+1)/2 for packing and lower box-counting dimension

Typical Besicovitch sets have full upper box-counting dimension

Half-extended Besicovitch sets have full Assouad dimension

Abstract

A Besicovitch set is a subset of $\R^d$ 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.