Geodesic
Elementary methods for incidence problems in finite fields
Javier Cilleruelo1 ; Alex Iosevich2 ; Ben Lund3 ; Oliver Roche-Newton4 ; Misha Rudnev5
1 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain
2 Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A.
3 Department of Computer Science Rutgers, The State University of New Jersey Piscataway, NJ 08854, USA
4 Institute for Financial Mathematics and Applied Number Theory Johannes Kepler Universität 4040 Linz, Austria
5 School of Mathematics University of Bristol University Walk Bristol, UK, BS8 1TW
Acta Arithmetica, Tome 177 (2017) no. 2, pp. 133-142.

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

We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb F_q^n$. As an application, we show that any point set $P\subset \mathbb F_q^2$ with $|P|\geq 5q$ 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.
DOI : 10.4064/aa8225-10-2016
Keywords: elementary methods prove incidence theorem points spheres mathbb application point set subset mathbb geq determines positive proportion circles latter result analogue beck theorem circles which optimal multiplicative constants

Javier Cilleruelo 1 ; Alex Iosevich 2 ; Ben Lund 3 ; Oliver Roche-Newton 4 ; Misha Rudnev 5

1 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain
2 Department of Mathematics University of Rochester Rochester, NY 14627, U.S.A.
3 Department of Computer Science Rutgers, The State University of New Jersey Piscataway, NJ 08854, USA
4 Institute for Financial Mathematics and Applied Number Theory Johannes Kepler Universität 4040 Linz, Austria
5 School of Mathematics University of Bristol University Walk Bristol, UK, BS8 1TW 
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Javier Cilleruelo; Alex Iosevich; Ben Lund; Oliver Roche-Newton; Misha Rudnev. Elementary methods for incidence problems in finite fields. Acta Arithmetica, Tome 177 (2017) no. 2, pp. 133-142. doi : 10.4064/aa8225-10-2016. http://geodesic.mathdoc.fr/articles/10.4064/aa8225-10-2016/

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