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arXiv · 2601.09910

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

Gergely Kiss·Ádám Markó·Zoltán Lóránt Nagy·Gábor Somlai
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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.

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Gergely Kiss, Ádám Markó, Zoltán Lóránt Nagy, Gábor Somlai. 2026-01-14. Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$. https://arxiv.org/abs/2601.09910

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