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

The authors show that, for a positive integer r, every minimal 1-saturating set in ... of size at least ... either is a complete cap or can be obtained from a complete cap S by fixing some ... and replacing every point ... by the third point on the line through s and ... 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 ... satisfies ... and is minimal subject to this condition. If ..., then either A is a maximal sum-free set or there are a maximal sum-free set ... and an element ... such that ... Their 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. (ProQuest: ... denotes formulae/symbols omitted.)