Almost all non-archimedean Kakeya sets have measure zero

TL;DR

This paper investigates Kakeya sets over non-archimedean fields, showing that almost all such sets have measure zero under a probabilistic measure, and discusses implications for the Kakeya conjecture.

Contribution

It introduces a probabilistic framework for non-archimedean Kakeya sets and proves that almost all have measure zero, advancing understanding of their measure-theoretic properties.

Findings

Almost all non-archimedean Kakeya sets have measure zero

Probabilistic measure is used to analyze Kakeya sets

Discussion on relations to the non-archimedean Kakeya conjecture

Abstract

We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.