Geodesic
Distances from points to planes
P. Birklbauer1 ; A. Iosevich1 ; T. Pham2
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.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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

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. 
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P. Birklbauer; A. Iosevich; T. Pham. Distances from points to planes. Acta Arithmetica, Tome 186 (2018) no. 3, pp. 219-224. doi : 10.4064/aa171110-23-8. http://geodesic.mathdoc.fr/articles/10.4064/aa171110-23-8/

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