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

2008/11/09 by David J. Grynkiewicz, Grynkiewicz, David J., Vsevolod F. Lev +2
Mathematics · #11B75 #11P70 #51E20 #FOS: Mathematics #Group Theory (math.GR) #Mathematical Approximation and Integration #Number Theory (math.NT) #math.GR #math.NT #msc:11B75 #msc:11P70 #msc:51E20

paper · pdf · doi:10.48550/arxiv.0811.1322

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

openalex publication_date 2008/11/09 · arxiv created 2009/01/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that, for a positive integer r, every minimal 1-saturating set in \rm PG(r-1,2) of size at least 11/36 2^r+3 is either 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'. Stated algebraically: if G is an elementary abelian 2-group and a set A⊆ G∖\0\ with |A|>11/36 |G|+3 satisfies A∪ 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⊆ G and an element s∈ S such that A=\s\∪(s+(S∖\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.

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