1-Saturating Sets, Caps, and Doubling-Critical Sets in Binary Spaces

Abstract

We show that, for a positive integer r, every minimal 1-saturating set in ${PG}(r-1,2)$ of size at least $\frac{11}{36}\,2^r+3$ either is a complete cap or can be obtained from a complete cap S by fixing some $s\in S$ and replacing every point $s'\in S\setminus\{s\}$ by the third point on the line through s and $s'$. 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 abelian 2-group is known, this provides a complete description of large minimal 1-saturating sets. An algebraic restatement is as follows. Suppose that G is an elementary abelian 2-group and a subset $A\subseteq G\setminus\{0\}$ satisfies $A\cup2A=G$ and is minimal subject to this condition. If $|A|\ge\frac{11}{36}\,|G|+3$, 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 $A=\{s\}\cup\bigl(s+(S\setminus\{s\})\bigr)$. 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 $2A$.