Some results in support of the Kakeya Conjecture

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.

17 pages. Incorporated referee suggestions, improved results in section 4, to appear in Real Analysis Exchange