, Volume 12, Issue 1, pp 39–44 | Cite as

Bounds for arrays of dots with distinct slopes or lengths

  • Paul Erdős
  • Ron Graham
  • Imre Z. Ruzsa
  • Herbert Taylor


Ann×m sonar sequence is a subset of then×m grid with exactly one point in each column, such that the\(\mathop 2\limits^m \) vectors determined by them are all distinct. We show that for fixedn the maximalm for which a sonar sequence exists satisfiesnCn11/20<m<n+4n2/3 for alln andm>n+c logn log logn for infinitely manyn.

Another problem concerns the maximal numberD of points that can be selected from then×m grid so that all the\(\mathop 2\limits^D \) vectors have slopes. We proven1/2Dn4/5

AMS subject classification code (1991)

05 C 


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  1. [1]
    J. Singer: A Theorem in Finite Projective Geometry and Some Applications to Number Theory,Trans. Amer. Math. Soc.,43 (1938), 377–385.Google Scholar
  2. [2]
    P. Erdős, andP. Turán: On a Problem of Sidon in Additive Number Theory and Some Related Problems,J. London Math. Soc.,16 (1941), 212–215.Google Scholar
  3. [3]
    S. W. Golomb, andH. Taylor: Two-dimensional Synchronization Patterns for Minimum Ambiguity,IEEE Trans. Inform. Theory IT-28 (1982), 600–604.Google Scholar
  4. [4]
    P. Erdős, andR. K. Guy: Distinct Distances Between Lattice Points,Elemente Der Mathematik,25, (1970), 121–123.Google Scholar
  5. [5]
    S. W. Graham, andC. J. Ringrose: Lower Bounds for Least Quadratic Non-residues, to appear inNumber Theory at Allerton Park, Proceedings of a conference in honor of Paul T. Bateman, Allerton Park, 1989, Birkhauser Verlag, Boston.Google Scholar
  6. [6]
    G. H. Hardy, andE. M. Wright:An Introduction to the Theory of Numbers, 3rd Edition, 1954, Oxford.Google Scholar

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • Paul Erdős
    • 1
  • Ron Graham
    • 2
  • Imre Z. Ruzsa
    • 1
  • Herbert Taylor
    • 3
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.AT & T Bell LaboratoriesMurray HillUSA
  3. 3.Communication Sciences Institute EEB 500University of Southern CaliforniaLos AngelesUSA

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