Abstract. A Kakeya set is a subset of � n, where � 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 Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1.

Abstract. We survey recent developments on the Kakeya problem. 1.

We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is bounded from L p to L q when 1 ≤ p ≤ kn+k+1 k(k+1) and 1 ≤ q ≤ (n − k)p ′. 1

A (d, k) set is a subset of Rd containing a translate of every k-dimensional disc of diameter 1. We show that if (1 + √ 2) k−1 + k> d and k ≥ 2, then every (d, k) set has positive Lebesgue measure. This improves a result of Bourgain, who showed that the analogous statement holds when 2k−1 +k ≥ d and k ≥ 2. We obtain this improvement in two parts. First, we replace Bourgain’s main estimate with a simple recursive maximal operator bound involving mixed-norm estimates for the X-ray transform. This method allows us to simplify Bourgain’s proof, allows us to obtain improved bounds for the maximal operator associated with (d, k) sets, and demonstrates that improved estimates for (d, k) sets would follow from new bounds for the X-ray transform. Second, we adapt arithmetic-combinatorial methods of Katz and Tao to obtain improved bounds for the X-ray transform suitable for use with the recursive maximal operator bound. Acknowledgements ii I would first like to thank my advisor, Andreas Seeger. Through my years in Madison, he has continuously sought opportunities for me to grow as a mathematician, and he has always looked out for my best interests. I hope to help my future students as much