Distances from points to planes

Abstract

We prove that if $E \subset {\Bbb F}_q^d$, $d \ge 2$, $F \subset \operatorname{Graff}(d-1,d)$, the set of affine $d-1$-dimensional planes in ${\Bbb F}_q^d$, then $|\Delta(E,F)| \ge \frac{q}{2}$ if $|E||F|>q^{d+1}$, where $\Delta(E,F)$ the set of distances from points in $E$ to lines in $F$. In dimension three and higher this significantly improves the exponent obtained by Pham, Phuong, Sang, Vinh and Valculescu.