2015
DOI: 10.1007/s00493-014-2686-2
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The Szemerédi-Trotter theorem in the complex plane

Abstract: It is shown that n points and e lines in the complex Euclidean plane C 2 determine O(n 2/3 e 2/3 + n + e) point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemerédi and Trotter about point-line incidences in the real Euclidean plane R 2 .

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Cited by 56 publications
(49 citation statements)
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References 21 publications
(29 reference statements)
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“…This is Toth's complex generalization of the Szemerédi-Trotter theorem [Toth03]. If L is a set of L complex lines in C 2 , Toth proved that |S r (L)| L 2 r −3 + Lr −1 -the same estimate as for real lines in R 2 .…”
Section: Some Methods and Problems In Incidence Geometrymentioning
confidence: 62%
“…This is Toth's complex generalization of the Szemerédi-Trotter theorem [Toth03]. If L is a set of L complex lines in C 2 , Toth proved that |S r (L)| L 2 r −3 + Lr −1 -the same estimate as for real lines in R 2 .…”
Section: Some Methods and Problems In Incidence Geometrymentioning
confidence: 62%
“…The known sum-product results over the real or complex field are somewhat stronger than in fields of positive characteristic largely due to order properties of reals, which so far have been indispensable for proofs of the celebrated Szemerédi-Trotter theorem in the plane. See [30], [32] for the original proof for reals and subsequent extension to the complex field.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…This section has two parts. First, we develop an application of Theorem 1 to the problem of counting distinct values of a non-degenerate bilinear form on pairs of points lying in a plane point set, extending to positive characteristic the estimates one easily obtains over R (as well as C, for it applies there as well [48]) via the Szemerédi-Trotter theorem. Then we consider the positive characteristic version of the Erdős distance problem in dimensions three and two.…”
Section: Applications To Erdős-type Geometric Questionsmentioning
confidence: 99%