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

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