Geodesic
Distances from points to planes
P. Birklbauer1 ; A. Iosevich1 ; T. Pham2
1Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A.
2Department of Mathematics University of California, San Diego La Jolla, CA 92093, U.S.A.
Acta Arithmetica, Tome 186 (2018) no. 3, pp. 219-224
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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

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