Distances from points to planes

P. Birklbauer ¹   ; A. Iosevich ¹   ; T. Pham ²

¹ Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A.
² Department of Mathematics University of California, San Diego La Jolla, CA 92093, U.S.A.

Acta Arithmetica, Tome 186 (2018) no. 3, pp. 219-224

Résumé

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 $|E|\,|F| \ge Cq^{{4d}/{3}}$.

Détail

DOI : 10.4064/aa171110-23-8

Keywords: prove subset bbb subset operatorname graff d set affine d dimensional planes bbb delta where delta set distances points planes dimension three higher significantly improves exponent obtained pham phuong sang valculescu vinh who obtain conclusion under assumption