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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)
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Abstract

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 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 for the number of rational distances with rational angle depending on whether α≥1/2 or α<1/2 respectively.

MSC classification

Secondary: 52C10: Erdős problems and related topics of discrete geometry 52C30: Planar arrangements of lines and pseudolines
Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 409 - 418
Copyright
Copyright © University College London 2012

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References

[1]Apostol, T. M., Introduction to Analytic Number Theory, Springer (New York, NY, 1976), Ch. 4, 82–85.Google Scholar
[2]Brass, P., Moser, W. and Pach, J., Research Problems in Discrete Geometry, Springer (2006), Ch. 5, 183–257.Google Scholar
[3]Erdős, P., On sets of distances of n points. Amer. Math. Monthly 53(5) (1946), 248–250.CrossRefGoogle Scholar
[4]Erdős, P. and Purdy, G. B., Some extremal problems in geometry, IV. In Proc. Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State University, Baton Rouge, LA, 1976) (Congressus Numerantium XVII), Utilitas Math. (Winnipeg, Manitoba, 1976).Google Scholar
[5]Lang, S., Algebra, 3rd edn., Addison-Wesley (Reading, MA, 1994).Google Scholar
[6]Mann, H. B., On linear relations between roots of unity. Mathematika 12 (1965), 107–117.CrossRefGoogle Scholar
[7]Matoušek, J., The number of unit distances is almost linear for most norms. Adv. Math. 226 (2011), 2618–2628.CrossRefGoogle Scholar
[8]Spencer, J., Szemerédi, E. and Trotter, W., Unit distances in the Euclidean plane. In Graph Theory and Combinatorics: Proceedings of the Cambridge Combinatorial Conference, in Honour of Paul Erdős (ed. Bollobás, B.), Academic Press (New York, 1984), 293–303.Google Scholar
[9]Székely, L., Crossing numbers and hard Erdős problems in discrete geometry. Combin. Probab. Comput. 6(3) (1997), 353–358.CrossRefGoogle Scholar