Distances from points to planes

We prove that if $E\subset {\mathbb{F}}_q^d$, $d\ge 2$, $F \subset \operatorname{Graff}(d-1,d)$,thesetofaffine$d-1$-dimensionalplanesin${\Bbb F}_q^d$,then$|\Delta(E,F)| \ge \frac{q}{2}$if$|E||F|>q^{d+1}$,where$\Delta(E,F)$thesetofdistancesfrompointsin$E$tolinesin$F$. In dimension three and higher this significantly improves the exponent obtained by Pham, Phuong, Sang, Vinh and Valculescu.