Evasive Sets, Covering by Subspaces, and Point-Hyperplane Incidences

Abstract

AbstractGiven positive integers $$k\le d$$ k ≤ d and a finite field $$\mathbb {F}$$ F , a set $$S\subset \mathbb {F}^{d}$$ S ⊂ F d is (k, c)-subspace evasive if every k-dimensional affine subspace contains at most c elements of S. By a simple averaging argument, the maximum size of a (k, c)-subspace evasive set is at most $$c |\mathbb {F}|^{d-k}$$ c | F | d - k . When k and d are fixed, and c is sufficiently large, the matching lower bound $$\Omega (|\mathbb {F}|^{d-k})$$ Ω ( | F | d - k ) is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of (k, c)-evasive sets in case d is large, extending results of Ben-Aroya and Shinkar. The existence of optimal evasive sets has several interesting consequences in combinatorial geometry. We show that the minimum number of k-dimensional linear hyperplanes needed to cover the grid $$[n]^{d}\subset \mathbb {R}^{d}$$ [ n ] d ⊂ R d is $$\Omega _{d}\big (n^{\frac{d(d-k)}{d-1}}\big )$$ Ω d ( n d ( d - k ) d - 1 ) , which matches the upper bound proved by Balko et al., and settles a problem proposed by Brass et al. Furthermore, we improve the best known lower bound on the maximum number of incidences between points and hyperplanes in $$\mathbb {R}^{d}$$ R d assuming their incidence graph avoids the complete bipartite graph $$K_{c,c}$$ K c , c for some large constant $$c=c(d)$$ c = c ( d ) .