Kakeya sets over non-archimedean local rings

TL;DR

This paper constructs measure-zero Kakeya sets in non-archimedean local rings and proves their maximal Minkowski dimension in certain cases, confirming the existence of Besicovitch phenomena in these settings.

Contribution

It explicitly constructs measure-zero Kakeya sets in F_q[[t]]^n and establishes their maximal Minkowski dimension in two-dimensional cases, answering open questions.

Findings

Existence of measure-zero Kakeya sets in F_q[[t]]^n

Kakeya sets in F_q[[t]]^2 have Minkowski dimension 2

Confirmed Besicovitch phenomena in non-archimedean local rings

Abstract

In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set in F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2.