On the arithmetic Kakeya conjecture of Katz and Tao

Ben Green, Imre Z. Ruzsa

Type: Article

Publication Date: 2018-11-02

Citations: 6

DOI: https://doi.org/10.1007/s10998-018-0270-z

Abstract

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.

Locations

Periodica Mathematica Hungarica - View - PDF
PubMed Central - View
PubMed - View

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Works Cited by This (18)

Action	Title	Year	Authors
+	Remarks on Montgomery's conjectures on dirichlet sums	1991	Jean Bourgain
+	From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE	2000	Terence Tao
+	On the size of Kakeya sets in finite fields	2008	Zeev Dvir
+ PDF Chat	New bounds for Kakeya problems	2002	N. Katz Terence Tao
+ PDF Chat	New counterexamples for sums-differences	2015	Marius Lemm
+ PDF Chat	Restriction and Kakeya phenomena for finite fields	2004	Gerd Mockenhaupt Terence Tao
+	On the Distribution of Dirichlet Sums	1993	Jean Bourgain
+	On the Dimension of Kakeya Sets and Related Maximal Inequalities	1999	Jean Bourgain
+ PDF Chat	The primes contain arbitrarily long arithmetic progressions	2008	Benjamin Green Terence Tao
+	None	2009	Dan Hefetz Michael Krivelevich Miloš Stojaković Tibor Szabó