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Distances from points to planes

Volume 186 / 2018

P. Birklbauer, A. Iosevich, T. Pham 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 $|E|\,|F| \ge Cq^{{4d}/{3}}$.

Authors

  • P. BirklbauerDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627, U.S.A.
    e-mail
  • A. IosevichDepartment of Mathematics
    University of Rochester
    Rochester, NY 14627, U.S.A.
    e-mail
  • T. PhamDepartment of Mathematics
    University of California, San Diego
    La Jolla, CA 92093, U.S.A.
    e-mail

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