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RATIONAL DISTANCES WITH RATIONAL ANGLES

Part of: Discrete geometry

Published online by Cambridge University Press:  28 November 2011

Ryan Schwartz ,
József Solymosi  and
Frank de Zeeuw
Ryan Schwartz
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T1Z2, Canada (email: ryano@math.ubc.ca)
József Solymosi
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T1Z2, Canada (email: solymosi@math.ubc.ca)
Frank de Zeeuw
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T1Z2, Canada (email: fdezeeuw@math.ubc.ca)

Abstract

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In 1946 Erdős asked for the maximum number of unit distances, u(n), among n points in the plane. He showed that u(n)>n1+c/log log n and conjectured that this was the true magnitude. The best known upper bound is u(n)<cn4/3, due to Spencer, Szemerédi and Trotter. We show that the upper bound $n^{1+6/\sqrt {\log n}}$ holds if we only consider unit distances with rational angle, by which we mean that the line through the pair of points makes a rational angle in degrees with the x-axis. Using an algebraic theorem of Mann we get a uniform bound on the number of paths between two fixed vertices in the unit distance graph, giving a contradiction if there are too many unit distances with rational angle. This bound holds if we consider rational distances instead of unit distances as long as there are no three points on a line. A superlinear lower bound is given, due to Erdős and Purdy. If we have at most nα points on a line then we get the bound O(n1+α) or $n^{1+\alpha +6/\sqrt {\log n}}$ for the number of rational distances with rational angle depending on whether α≥1/2or α<1/2respectively.

MSC classification

Secondary: 52C10: Erdős problems and related topics of discrete geometry 52C30: Planar arrangements of lines and pseudolines

Information

Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 409 - 418
Copyright
Copyright © University College London 2012

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