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

TL;DR

This paper introduces a nonstandard approach to Hausdorff dimension in Euclidean spaces, providing a new proof of the Kakeya conjecture in two and three dimensions using nonstandard analysis techniques.

Contribution

It develops a nonstandard formulation of Hausdorff dimension and applies it to prove the Kakeya conjecture in 2D and extend the proof to 3D.

Findings

Nonstandard Hausdorff dimension formulation

New proof of Kakeya conjecture in 2D

Extension of proof to 3D

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.