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On the arithmetic Kakeya conjecture of Katz and Tao
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abstract
The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in $\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.
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Improved bounds for the Kakeya maximal conjecture in higher dimensions
New multiscale polynomial Wolff axioms lead to Kakeya maximal estimates for p ≥ 1 + O(1/n), improving prior bounds in dimensions n=5 and n≥7.
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