On the size of Kakeya sets in finite fields

TL;DR

This paper proves that Kakeya sets in finite fields must be proportionally large, with size at least a constant times q^n, improving previous lower bounds significantly.

Contribution

It establishes a new lower bound on the size of Kakeya sets in finite fields, advancing understanding of their minimal possible size.

Findings

Kakeya sets have size at least C_n * q^n

Improves previous lower bound of approximately q^{4n/7}

Provides a bound depending only on the dimension n

Abstract

A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the previously best lower bound for general n of ~q^{4n/7}.