On the arithmetic Kakeya conjecture of Katz and Tao

Ben Green, I. Ruzsa

Research output: Contribution to journalArticle

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.

Original languageEnglish
JournalPeriodica Mathematica Hungarica
DOIs
Publication statusAccepted/In press - Jan 1 2018

Fingerprint

Minkowski Dimension
Galois field
Finite Set
Lower bound
Imply
Form

Keywords

  • Arithmetic progression
  • Kakeya problem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the arithmetic Kakeya conjecture of Katz and Tao. / Green, Ben; Ruzsa, I.

In: Periodica Mathematica Hungarica, 01.01.2018.

Research output: Contribution to journalArticle

@article{756c67222c524b1db81c11280e5374a4,
title = "On the arithmetic Kakeya conjecture of Katz and Tao",
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.",
keywords = "Arithmetic progression, Kakeya problem",
author = "Ben Green and I. Ruzsa",
year = "2018",
month = "1",
day = "1",
doi = "10.1007/s10998-018-0270-z",
language = "English",
journal = "Periodica Mathematica Hungarica",
issn = "0031-5303",
publisher = "Springer Netherlands",

}

TY - JOUR

T1 - On the arithmetic Kakeya conjecture of Katz and Tao

AU - Green, Ben

AU - Ruzsa, I.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Arithmetic progression

KW - Kakeya problem

UR - http://www.scopus.com/inward/record.url?scp=85055964895&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055964895&partnerID=8YFLogxK

U2 - 10.1007/s10998-018-0270-z

DO - 10.1007/s10998-018-0270-z

M3 - Article

AN - SCOPUS:85055964895

JO - Periodica Mathematica Hungarica

JF - Periodica Mathematica Hungarica

SN - 0031-5303

ER -