The $p$-adic Kakeya conjecture

TL;DR

This paper proves that in the p-adic setting, sets containing line segments in all directions have full Hausdorff and Minkowski dimension, confirming an analogue of the Kakeya conjecture over p-adic fields.

Contribution

It establishes the p-adic Kakeya conjecture, showing that such sets necessarily have full dimension, a significant extension of classical Euclidean results to p-adic spaces.

Findings

Sets with line segments in all directions in b_p^n have Hausdorff dimension n

Sets with line segments in all directions in b_p^n have Minkowski dimension n

Confirms the p-adic Kakeya conjecture for bounded sets

Abstract

We prove that all bounded subsets of $\mathbb{Q}_p^n$ containing a line segment of unit length in every direction have Hausdorff and Minkowski dimension $n$. This is the analogue of the classical Kakeya conjecture with $\mathbb{R}$ replaced by $\mathbb{Q}_p$.