1-Saturating Sets, Caps and Round Sets in Binary Spaces

Abstract

We show that, for a positive integer $r$, every minimal 1-saturating set in $\mathrm{PG}(r-1,2)$ of size at least $11/362^r+3$ is either a complete cap or can be obtained from a complete cap $S$ by fixing some $s\in S$ and replacing every point $s^{\prime }\in S\setminus \{s\}$ by the third point on the line through $s$ and $s^{\prime }$. Stated algebraically: if $G$ is an elementary abelian 2-group and a set $A\subseteq G\setminus \{0\}$ with $|A|>11/36|G|+3$ satisfies $A\cup 2A=G$ and is minimal subject to this condition, then either $A$ is a maximal sum-free set, or there are a maximal sum-free set $S\subseteq G$ and an element $s\in S$ such that $A=\{s\}\cup (s+(S\setminus \{s\}))$. Since, conversely, every set obtained in this way is a minimal 1-saturating set, and the structure of large sum-free sets in an elementary 2-group is known, this provides a complete description of large minimal 1-saturating sets. Our approach is based on characterizing those large sets $A$ in elementary abelian 2-groups such that, for every proper subset $B$ of $A$, the sumset 2B is a proper subset of 2A.

A section presenting the results for the the projective geometry viewpoint added