Points, Lines, and Polynomials: The Kakeya Problem and the Polynomial Method

Abstract

The Kakeya Conjecture is a long-standing problem in analysis about the size and dimension of sets in \( \mathbb{R}^n \) containing a unit line segment in every direction, known as Kakeya or Besicovitch sets. This thesis studies the analogous discrete problem of the size of such sets in \( \mathbb{F}_q^n \), where a Kakeya set is similarly defined as containing an entire line in every direction. This problem is solved using the polynomial method, which seeks to obtain results about combinatorial objects by describing their structure through the vanishing sets of polynomials. We discuss various aspects of the polynomial method, its application to the discrete Kakeya problem, and applications of Kakeya sets in Information Theory.