Abstract
We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m∈[0,2n] of the n-dimensional binary affine space 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 2k. 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.
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