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AN EXPLICIT INCIDENCE THEOREM IN 𝔽p

Part of: Sequences and sets Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  13 December 2010

Harald Andrés Helfgott  and
Misha Rudnev
Harald Andrés Helfgott
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. (email: h.andres.helfgott@bristol.ac.uk)
Misha Rudnev
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. (email: m.rudnev@bristol.ac.uk)

Abstract

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Let P=A×A⊂𝔽p×𝔽p, p a prime. Assume that P=A×A has n elements, n<p. See P as a set of points in the plane over 𝔽p. We show that the pairs of points in P determine ${\geq } c n^{1 + \frac {1}{267}}$ lines, where c is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of n points and a set of n lines in the projective plane over 𝔽p (n<p)is bounded by $C n^{\frac {3}{2}-\frac {1}{10678}}$, where C is an absolute constant.

MSC classification

Secondary: 68R05: Combinatorics 11B75: Other combinatorial number theory

Information

Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 135 - 145
Copyright
Copyright © University College London 2011

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