The Kakeya Set conjecture over Z/NZ for general N
2024; Linguagem: Inglês
10.19086/aic.2024.2
ISSN2517-5599
Autores Tópico(s)Advanced Topology and Set Theory
ResumoA [Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set) is a set of points of an Euclidean space that contains a unit line segment in every direction. In 1919, [Besicovitch](https://en.wikipedia.org/wiki/Abram_Samoilovitch_Besicovitch) constructed a Kakeya set of measure zero for every dimension, and in addition, he constructed sets in the plane with arbitrarily small measure such that a unit segment can rotate full 360 degrees within the set. While Kakeya sets can have measure zero, the famous [Kakeya Conjecture](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture) asserts that every Kakeya set in ${\mathbb R}^n$ has both [Hausdorff dimension](https://en.wikipedia.org/wiki/Hausdorff_dimension) and [Minkowski dimension](https://en.wikipedia.org/wiki/Minkowski_dimension) equal to $n$. The conjecture is open for $n\ge 3$. This paper concerns Kakeya sets in the finite setting. In 1999, Wolff conjectured that every Kakeya set in ${\mathbb F}^n$, i.e., a set containing a line in every direction, has size at least $c_n\cdot\lvert {\mathbb F}\rvert^n$. The conjecture was proven by Dvir with $c_n=1/n!$ in 2008 (an exposition of the proof can be found in [this blog post on Terence Tao's blog](https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/)). Subsequently, Ellenberg, Oberlin and Tao proposed studying Kakeya sets over the rings ${\mathbb Z}/p^k{\mathbb Z}$, and Hickman and Wright over ${\mathbb Z}/N{\mathbb Z}$ for an arbitrary $N$. The paper resolves a conjecture of Hickman and Wright by giving an exponential lower bound on the size of a Kakeya set in this most general setting.
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