2024/11/20 by Cosmin Pohoata, Dmitrii Zakharov, Pohoata, Cosmin +1
Computer Science · Decision Sciences · Economics, Econometrics and Finance · #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Housing Market and Economics #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2411.13395
openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as the `Arithmetic Kakeya Conjecture'. We consider a higher dimensional variant of this problem and prove an upper bound using a certain iterative argument. The main new ingredient in our proof is a general way to strengthen the sum-difference inequalities of Katz and Tao which might be of independent interest. As a corollary, we obtain a new lower bound for the Minkowski dimension of (n, d)-Besicovitch sets.