Proof of the Kakeya set conjecture over rings of integers modulo square-free N

Manik Dhar; Zeev Dvir

doi:10.5070/C61055361

Proof of the Kakeya set conjecture over rings of integers modulo square-free N

Manik Dhar, Zeev Dvir

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

A Kakeya set S ⊂ (Z/NZ)ⁿ is a set containing a line in each direction. We show that, when N is any square-free integer, the size of the smallest Kakeya set in (Z/NZ)ⁿ is at least C_n,ɛ N^n−ɛ for any ɛ – resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the F_p rank of the incidence matrix of points and hyperplanes over (Z/p^k Z)ⁿ.

Original language	English (US)
Article number	#4
Journal	Combinatorial Theory
Volume	1
DOIs	https://doi.org/10.5070/C61055361
State	Published - 2021
Externally published	Yes

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics

Access to Document

10.5070/C61055361

Cite this

@article{a350c6950823421b96e388332507e8f2,

title = "Proof of the Kakeya set conjecture over rings of integers modulo square-free N",

abstract = "A Kakeya set S ⊂ (Z/NZ)n is a set containing a line in each direction. We show that, when N is any square-free integer, the size of the smallest Kakeya set in (Z/NZ)n is at least Cn,ɛ Nn−ɛ for any ɛ – resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the Fp rank of the incidence matrix of points and hyperplanes over (Z/pk Z)n.",

author = "Manik Dhar and Zeev Dvir",

note = "Publisher Copyright: {\textcopyright} The authors. Released under the CC BY license (International 4.0).",

year = "2021",

doi = "10.5070/C61055361",

language = "English (US)",

volume = "1",

journal = "Combinatorial Theory",

issn = "2766-1334",

publisher = "eScholarship University of California",

}

TY - JOUR

T1 - Proof of the Kakeya set conjecture over rings of integers modulo square-free N

AU - Dhar, Manik

AU - Dvir, Zeev

N1 - Publisher Copyright: © The authors. Released under the CC BY license (International 4.0).

PY - 2021

Y1 - 2021

N2 - A Kakeya set S ⊂ (Z/NZ)n is a set containing a line in each direction. We show that, when N is any square-free integer, the size of the smallest Kakeya set in (Z/NZ)n is at least Cn,ɛ Nn−ɛ for any ɛ – resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the Fp rank of the incidence matrix of points and hyperplanes over (Z/pk Z)n.

AB - A Kakeya set S ⊂ (Z/NZ)n is a set containing a line in each direction. We show that, when N is any square-free integer, the size of the smallest Kakeya set in (Z/NZ)n is at least Cn,ɛ Nn−ɛ for any ɛ – resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the Fp rank of the incidence matrix of points and hyperplanes over (Z/pk Z)n.

UR - http://www.scopus.com/inward/record.url?scp=85118723494&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85118723494&partnerID=8YFLogxK

U2 - 10.5070/C61055361

DO - 10.5070/C61055361

M3 - Article

AN - SCOPUS:85118723494

SN - 2766-1334

VL - 1

JO - Combinatorial Theory

JF - Combinatorial Theory

M1 - #4

ER -

Proof of the Kakeya set conjecture over rings of integers modulo square-free N

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this