Avoiding intersections of given size in finite affine spaces AG(n,2)

Abstract

We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size $m\in [0,2^n]$ of the n-dimensional binary affine space $\mathrm{AG}(n,2)$. Following the theme of Erdős, Füredi, Rothschild and T. Sós, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all m-element point sets in the n-dimensional affine space.

We also show constructions of point sets for which the intersection sizes with k-dimensional affine subspaces take values from a set of a small size compared to $2^k$. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers.

Keywords: unavoidable, affine subspaces, evasive sets, random methods, canonical signed-digit number system.