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 $${\mathbf {R}}^n$$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.

中文翻译：

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卡茨和道的算术挂屋猜想

Katz 和 Tao 提出的算术 Kakeya 猜想（Math Res Lett 6(5–6):625–630, 1999）是关于有限集加法的陈述。已知隐含了 Kakeya 猜想的一种形式，即 $${\mathbf {R}}^n$$Rn 中 Besicovitch 集的上 Minkowski 维数是 n。在这篇笔记中，我们讨论了这个猜想，并给出了它的许多等价形式。我们表明它的自然有限域变体确实成立。我们还给出了一些下限。