Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$

TL;DR

The paper studies $p$-divisible sets in $\mathbb{F}_p^3$ and advances the Strong Cylinder Conjecture by providing a precise algebraic structure: any $p$-divisible multiset lies in the $\mathbb{F}_p$-span of cylinders (and cylinder-type multisets), and when the total size is $p^2$, it is a $\mathbb{Z}$-linear combination of cylinders, equivalently a plane plus a $\mathbb{Z}$-linear combination of differences of parallel lines. The authors develop a robust framework that translates $p$-divisibility into orthogonality to affine planes, identifies the orthogonal complement $\mathcal{S}_1^{\perp}$ as polynomials with degree at most $2p-3$, and shows cylinder-type configurations generate this space. They further lift mod $p$ results to integer decompositions using two auxiliary lemmas, establishing the plane-plus-cylinder structure for $p^2$-sized multisets, and connect these geometric findings to vanishing sums of roots of unity and Rédei-type conjectures. Overall, the work provides a concrete decomposition mechanism for $p$-divisible configurations and links finite geometry with additive combinatorics, offering progress toward the strong cylinder conjecture and related additive-tile questions.

Abstract

A set of points $S \subseteq \mathbb{F}_p^n$ is called \emph{$p$-divisible} if every affine hyperplane in $\mathbb{F}_p^n$ intersects $S$ in $0 \pmod p$ points. The Strong Cylinder Conjecture of Ball asserts that if $S$ is a $p$-divisible set of $p^2$ points in $\mathbb{F}_p^3$, then $S$ is a cylinder. In this paper, we show that every $p$-divisible multiset $S$ is both a $\mathbb{F}_p$-linear and $\mathbb{Z}$-linear combination of characteristic functions of cylinders. In addition, the multisets of size $p^2$ are $\Z$-linear combinations of a plane and weighted differences of parallel lines.

Theorems & Definitions (7)

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