On the arithmetic Kakeya conjecture of Katz and Tao

The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5-6):625-630, 1999), 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 Rn 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.