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

5 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

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics

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10.5070/C61055361

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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",

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AU - Dhar, Manik

AU - Dvir, Zeev

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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.

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JO - Combinatorial Theory

JF - Combinatorial Theory

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ER -

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

Abstract

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