Distances from points to planes

Abstract

We prove that if $E\subset {\mathbb{F}}_q^d$, $d\ge 2$, $F\subset \mathrm{Graff}(d-1,d)$, the set of affine $d-1$-dimensional planes in ${\mathbb{F}}_q^d$, 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.