Kakeya-Type Sets in Local Fields with Finite Residue Field

TL;DR

This paper constructs measure-zero Kakeya-type sets in local fields with finite residue fields, extending previous ideas and providing alternative constructions over discrete valuation rings, contributing to the understanding of geometric measure theory in non-Archimedean settings.

Contribution

It introduces a new construction of measure-zero Kakeya-type sets in local fields and discrete valuation rings, expanding the scope of geometric measure theory in non-Archimedean spaces.

Findings

Constructed measure-zero Kakeya-type sets in local fields.

Provided an alternative construction over discrete valuation rings.

Extended Kakeya set theory to non-Archimedean local fields.

Abstract

We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^d$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in [12] and [13]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one presented in [4] over $\mathbb{F}_q[[t]]$.