Nonstandard Hausdorff dimension in ${\mathbb{R}}^n$ and an application to the Kakeya conjecture in ${\mathbb{R}}^3$

Abstract

Through the use of a nonstandard version of Frostman's lemma, the notion of Hausdorff dimension is formulated in nonstandard euclidean space of arbitrary dimension. This allows for a nonstandard proof of the Kakeya conjecture in two dimensions, which can then be extended to the three-dimensional case.

This paper has been withdrawn by the author due to a crucial error in section 5