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

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 $0(\mathrm{mod}p)$ 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 $\mathbb{Z}$-linear combinations of a plane and weighted differences of parallel lines.