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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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1435-5337
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Volume 29, Issue 2

Issues

Incidences between points and generalized spheres over finite fields and related problems

Nguyen D. Phuong / Pham Thang / Le A. Vinh
Published Online: 2016-06-02 | DOI: https://doi.org/10.1515/forum-2015-0024

Abstract

Let 𝔽q be a finite field of q elements, where q is a large odd prime power and

Q=a1x1c1++adxdcd𝔽q[x1,,xd],

where 2ciN, gcd(ci,q)=1, and ai𝔽q for all 1id. A Q -sphere is a set of the form

{𝒙𝔽qdQ(𝒙-𝒃)=r},

where 𝒃𝔽qd,r𝔽q. We prove bounds on the number of incidences between a point set 𝒫 and a Q-sphere set 𝒮, denoted by I(𝒫,𝒮), as the following:

|I(𝒫,𝒮)-|𝒫||𝒮|q|qd/2|𝒫||𝒮|.

We also give a version of this estimate over finite cyclic rings /q, where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in 𝔽qd. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

Keywords: Point-circle incidences; sum-product estimate; finite fields

MSC 2010: 52C10

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About the article


Received: 2015-02-05

Revised: 2016-01-26

Published Online: 2016-06-02

Published in Print: 2017-03-01


The first and third author were supported by Vietnam National Foundation for Science and Technology grant 101.99-2013.21. The work of the second author was partially supported by Swiss National Science Foundation grants 200020-144531 and 200021-137574.


Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 449–456, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0024.

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