1-Saturating sets, caps, and doubling-critical sets in binary spaces


We show that, for a positive integer r, every minimal 1-saturating set in PG(r-1, 2) of size at least 11/36 2r + 3 either is a complete cap or can be obtained from a complete cap S by fixing some s ε S and replacing every point s' ε S \ {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 ⊆ G\{0} satisfies A ∪ 2A = G and is minimal subject to this condition. If |A| ≥ 11/36 |G| + 3, then either A is a maximal sum-free set or there are a maximal sum-free set S ⊆ G and an element s ε S such that A = {s} ∪ (|s + (S \ {s})). 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. © 2010 Society for Industrial and Applied Mathematics.

Publication Title

SIAM Journal on Discrete Mathematics