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The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$

Manik Dhar

TL;DR

The paper resolves the Kakeya set conjecture over the finite ring ${\mathbb{Z}}/N{\mathbb{Z}}$ for general $N$ by uniting prime-power methods with square-free techniques and introducing multiplicity-aware polynomial methods on a complex torus. It develops a robust algebraic framework using crank and rank concepts, a tensor-product Crank bound, and a decoding scheme that analyzes line-averaged evaluations to bound the size of Kakeya sets. Quantitative bounds are established for $(m,ε)$-Kakeya sets over ${\mathbb{Z}}/p^k{\mathbb{Z}}$ and extended to general $N$ via the Chinese Remainder Theorem, yielding near-optimal constructions that demonstrate sharpness in many regimes. The results advance the understanding of Kakeya phenomena in non-field rings, with implications for p-adic and Minkowski-dimension analogues, and provide a foundation for maximal Kakeya bounds in broader settings.

Abstract

We prove the Kakeya set conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. This entails extending and combining the techniques of Arsovski [Ars21a] for $N=p^k$ and the author and Dvir [DD21] for the case of square-free $N$. We also prove stronger lower bounds for the size of $(m,ε)$-Kakeya sets over $\mathbb{Z}/p^k\mathbb{Z}$ by extending the techniques of [Ars21a] using multiplicities as was done in [SS08, DKSS13]. In addition, we show our bounds are almost sharp by providing a new construction for Kakeya sets over $\mathbb{Z}/p^k\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$.

The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$

TL;DR

The paper resolves the Kakeya set conjecture over the finite ring for general by uniting prime-power methods with square-free techniques and introducing multiplicity-aware polynomial methods on a complex torus. It develops a robust algebraic framework using crank and rank concepts, a tensor-product Crank bound, and a decoding scheme that analyzes line-averaged evaluations to bound the size of Kakeya sets. Quantitative bounds are established for -Kakeya sets over and extended to general via the Chinese Remainder Theorem, yielding near-optimal constructions that demonstrate sharpness in many regimes. The results advance the understanding of Kakeya phenomena in non-field rings, with implications for p-adic and Minkowski-dimension analogues, and provide a foundation for maximal Kakeya bounds in broader settings.

Abstract

We prove the Kakeya set conjecture for for general as stated by Hickman and Wright [HW18]. This entails extending and combining the techniques of Arsovski [Ars21a] for and the author and Dvir [DD21] for the case of square-free . We also prove stronger lower bounds for the size of -Kakeya sets over by extending the techniques of [Ars21a] using multiplicities as was done in [SS08, DKSS13]. In addition, we show our bounds are almost sharp by providing a new construction for Kakeya sets over and .

Paper Structure

This paper contains 10 sections, 29 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Proof Overview
  3. Organization of the paper
  4. Acknowledgements
  5. Rank and crank of matrices with polynomial entries
  6. Polynomial Method on the complex torus
  7. Kakeya Set bounds over ${\mathbb{Z}}/p^k{\mathbb{Z}}$
  8. Kakeya Set bounds over ${\mathbb{Z}}/N{\mathbb{Z}}$ for general $N$
  9. Constructing small Kakeya sets
  10. Proof of Lemma \ref{['lem:1rank']}

Key Result

Theorem 1.5

Let $N\in {\mathbb{N}}$ where $N=p_1\hdots p_r$ for distinct primes $p_1,\hdots,p_r$. Any Kakeya set $S$ in $({\mathbb{Z}}/N{\mathbb{Z}})^n$ for $n\in {\mathbb{N}}$ satisfies,

Theorems & Definitions (70)

  • Definition 1.1: Projective space ${\mathbb{P}} ({\mathbb{Z}}/N{\mathbb{Z}})^{n-1}$
  • Definition 1.2: $m$-rich lines
  • Definition 1.3: $(m,\epsilon)$-Kakeya Sets
  • Conjecture 1.4: Kakeya set conjecture over ${\mathbb{Z}}/N{\mathbb{Z}}$
  • Theorem 1.5: Kakeya set bounds for square-free $N$ dhar2021proof
  • Theorem 1.6: Kakeya Set bounds over ${\mathbb{Z}}/p^k{\mathbb{Z}}$,arsovski2021padic
  • Theorem 1.7: $(m,\epsilon)$-Kakeya Set bounds over ${\mathbb{Z}}/p^k{\mathbb{Z}}$,arsovskiNew
  • Theorem 1.8: Stronger $(m,\epsilon)$-Kakeya Set bounds over ${\mathbb{Z}}/p^k{\mathbb{Z}}$
  • Theorem 1.9: Kakeya set bounds for ${\mathbb{Z}}/N{\mathbb{Z}}$
  • Theorem 1.11: Small Kakeya sets over ${\mathbb{Z}}/p^k{\mathbb{Z}}$
  • ...and 60 more