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The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is astatement about addition of finite sets. It is known to imply a form of theKakeya conjecture, namely that the upper Minkowski dimension of a Besicovitchset in $\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving anumber of equivalent forms of it. We show that a natural finite field variantof it does hold. We also give some lower bounds.

On the arithmetic Kakeya conjecture of Katz and Tao