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

Zeev Dvir,Manik Dhar

doi:10.5070/c61055361

Abstract

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ 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 $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- 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 $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

Highlights

Given a finite abelian ring R, we consider the space of n-tuples over R, denoted Rn. In this space we may define a line in direction b ∈ Rn \ {0} to be a subset of the form {a + tb|t ∈ R} where a ∈ Rn
The problem of lower bounding the size of Kakeya sets for the rings Z/pkZ and Fq[x]/ xk was first proposed in [EOT10] as a step in the direction of the Euclidean problem as these rings contain ‘scales’ in a way that does not exist over a finite field and is reminiscent of the real numbers
Since Wpk,n has zero-one entries, we can view it as a matrix over any field and, in particular, compute its rank over Fp. We show that this rank lower bounds the size of any Kakeya set

Summary

Introduction

Given a finite abelian ring R, we consider the space of n-tuples over R, denoted Rn. In this space we may define a line in direction b ∈ Rn \ {0} to be a subset of the form {a + tb|t ∈ R} where a ∈ Rn. The problem of lower bounding the size of Kakeya sets for the rings Z/pkZ and Fq[x]/ xk was first proposed in [EOT10] as a step in the direction of the Euclidean problem as these rings contain ‘scales’ in a way that does not exist over a finite field and is reminiscent of the real numbers. Sets modulo each prime factor, a general Kakeya set in (Z/N Z)n might not have this product structure (otherwise the proof of the lower bound would be trivial). Proving a lower bound of, say, p(1− )kn for small > 0 on the rank of the incidence matrix Wpk,n would lead to new bounds for Kakeya sets in (Z/pkZ)n. We note that our reduction is only in one direction – showing that Wpk,n has low rank would not imply the existence of small Kakeya sets using our theorem

Overview of the proof

Organization

Warm-up 1

Warm-up 2

The general square-free N case