Distances from points to planes
Volume 186 / 2018
Acta Arithmetica 186 (2018), 219-224
MSC: Primary 52C10.
DOI: 10.4064/aa171110-23-8
Published online: 5 November 2018
Abstract
We prove that if $E \subset {\Bbb F}_q^d$, $d \ge 2$, and $F \subset \operatorname{Graff}(d-1,d)$, the set of affine $d-1$-dimensional planes in ${\Bbb F}_q^d$, then $|\Delta(E,F)| \ge {q}/{2}$ if $|E|\,|F| \gt q^{d+1}$, where $\Delta(E,F)$ is the set of distances from points in $E$ to planes in $F$. In dimension three and higher this significantly improves the exponent obtained by Pham, Phuong, Sang, Valculescu and Vinh (2018), who obtain the same conclusion under the assumption $|E|\,|F| \ge Cq^{{4d}/{3}}$.
