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arXiv · 2411.13395

Generalized Arithmetic Kakeya

Cosmin Pohoata·Dmitrii Zakharov
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Abstract

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.

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Cosmin Pohoata, Dmitrii Zakharov. 2024-11-20. Generalized Arithmetic Kakeya. https://arxiv.org/abs/2411.13395

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