Elementary methods for incidence problems in finite fields

Cilleruelo Javier; Alex Iosevich; Ben Lund; Oliver Roche-Newton; Misha Rudnev

doi:10.4064/aa8225-10-2016

Elementary methods for incidence problems in finite fields

Cilleruelo Javier, Alex Iosevich, Ben Lund, Oliver Roche-Newton, Misha Rudnev

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We use elementary methods to prove an incidence theorem for points and spheres in Fnq. As an application, we show that any point set P⊂F2q with |P|≥5q determines a positive proportion of all circles. The latter result is an analogue of Beck’s Theorem for circles which is optimal up to multiplicative constants.

Original language	English
Pages (from-to)	133-142
Number of pages	10
Journal	Acta Arithmetica
Volume	177
Early online date	22 Dec 2016
DOIs	https://doi.org/10.4064/aa8225-10-2016
Publication status	Published - 2017

Keywords

incidences
circles
finite fields

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10.4064/aa8225-10-2016

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This is the author accepted manuscript (AAM). The final published version (version of record) is available online via PANIM at https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/177/2/91950/elementary-methods-for-incidence-problems-in-finite-fields. Please refer to any applicable terms of use of the publisher.
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@article{c45eaa9018a1418da9b67cd0d0d396b1,

title = "Elementary methods for incidence problems in finite fields",

abstract = "We use elementary methods to prove an incidence theorem for points and spheres in Fnq. As an application, we show that any point set P⊂F2q with |P|≥5q determines a positive proportion of all circles. The latter result is an analogue of Beck{\textquoteright}s Theorem for circles which is optimal up to multiplicative constants.",

keywords = "incidences, circles, finite fields",

author = "Cilleruelo Javier and Alex Iosevich and Ben Lund and Oliver Roche-Newton and Misha Rudnev",

year = "2017",

doi = "10.4064/aa8225-10-2016",

language = "English",

volume = "177",

pages = "133--142",

journal = "Acta Arithmetica",

issn = "0065-1036",

publisher = "Instytut Matematyczny",

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TY - JOUR

T1 - Elementary methods for incidence problems in finite fields

AU - Javier, Cilleruelo

AU - Iosevich, Alex

AU - Lund, Ben

AU - Roche-Newton, Oliver

AU - Rudnev, Misha

PY - 2017

Y1 - 2017

N2 - We use elementary methods to prove an incidence theorem for points and spheres in Fnq. As an application, we show that any point set P⊂F2q with |P|≥5q determines a positive proportion of all circles. The latter result is an analogue of Beck’s Theorem for circles which is optimal up to multiplicative constants.

AB - We use elementary methods to prove an incidence theorem for points and spheres in Fnq. As an application, we show that any point set P⊂F2q with |P|≥5q determines a positive proportion of all circles. The latter result is an analogue of Beck’s Theorem for circles which is optimal up to multiplicative constants.

KW - incidences

KW - circles

KW - finite fields

U2 - 10.4064/aa8225-10-2016

DO - 10.4064/aa8225-10-2016

M3 - Article (Academic Journal)

SN - 0065-1036

VL - 177

SP - 133

EP - 142

JO - Acta Arithmetica

JF - Acta Arithmetica

ER -

Elementary methods for incidence problems in finite fields

Abstract

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