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

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.