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We study the set of intersection sizes of a k-dimensional affine subspace anda point set of size m \in [0, 2^n] of the n-dimensional binary affine spaceAG(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 subspacesare unavoidable in all $m$-element point sets in the n-dimensional affinespace. We also show constructions of point sets for which the intersectionsizes with $k$-dimensional affine subspaces takes values from a set of a smallsize compared to 2^k. These are built up from affine subspaces and so-calledsubspace evasive sets. Meanwhile, we improve the best known upper bounds onsubspace evasive sets and apply results concerning the canonical signed-digit(CSD) representation of numbers.

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