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 2 r + 3 either is a complete cap or can be obtained from a complete cap S by fixing some s ∈ S and replacing every point s′ ∈ 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 ∈ 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. 1. Saturating Sets and Caps: The Main Result. Let r ≥ 1 be an integer, q a prime power, and A ⊆ PG(r− 1, q) a set in the (r− 1)dimensional projective space over the q-element field. Given an integer ρ ≥ 1, one says that A is ρ-saturating if every point of PG(r−1, q) is contained in a subspace generated by ρ + 1 points from A. Furthermore, A is said to be a cap if no three points of A are collinear; a cap is complete if it is not properly contained in another cap. Since the property of being ρ-saturating is inherited by supersets and that of being a cap is inherited by subsets, of particular interest are minimal ρ-saturating sets and complete caps. In this paper, we are concerned with the case ρ = 1 and the space PG(r−1, 2) whose points are, essentially, non-zero elements of the elementary abelian 2-group of rank r, and whose lines are triples of points adding up to 0. A large random set in PG(r−1, 2) is 1-saturating with very high probability, but the probability that it is a minimal 1saturating set is extremely low; thus, one can expect that large minimal 1-saturating 2000 Mathematics Subject Classification. 51E20, 11B75, 11P70.