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 AG(n,2). Following the theme of Erd\H{o}s, F\"uredi, Rothschild and T. S\'os, 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 takes 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.