THE (d, k) KAKEYA PROBLEM AND ESTIMATES FOR THE X-RAY TRANSFORM

A (d, k) set is a subset of R 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.