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Bounds for arrays of dots with distinct slopes or lengths

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Abstract

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 satisfiesn−Cn 11/20<m<n+4n 2/3 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 proven 1/2≪D≪n 4/5

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References

  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. 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. S. W. Golomb, andH. Taylor: Two-dimensional Synchronization Patterns for Minimum Ambiguity,IEEE Trans. Inform. Theory IT-28 (1982), 600–604.

    Google Scholar 

  4. P. Erdős, andR. K. Guy: Distinct Distances Between Lattice Points,Elemente Der Mathematik,25, (1970), 121–123.

    Google Scholar 

  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.

  6. G. H. Hardy, andE. M. Wright:An Introduction to the Theory of Numbers, 3rd Edition, 1954, Oxford.

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    Paul Erdős & Imre Z. Ruzsa

  2. AT & T Bell Laboratories, 600 Mountain Ave., 07974, Murray Hill, NJ, USA

    Ron Graham

  3. Communication Sciences Institute EEB 500, University of Southern California, 90089-2565, Los Angeles, CA, USA

    Herbert Taylor

Authors
  1. Paul Erdős
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  2. Ron Graham
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  3. Imre Z. Ruzsa
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  4. Herbert Taylor
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Additional information

Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901

Research conducted by Herbert Taylor was sponsored in part by the Office of Naval Research under ONR Contract No. N00014-90-J-1341.

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Erdős, P., Graham, R., Ruzsa, I.Z. et al. Bounds for arrays of dots with distinct slopes or lengths. Combinatorica 12, 39–44 (1992). https://doi.org/10.1007/BF01191203

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  • DOI: https://doi.org/10.1007/BF01191203

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