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Distance sets of well-distributed planar sets for polygonal norms

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Abstract

LetX be a two-dimensional normed space, and letBX be the unit ball inX. We discuss the question of how large the set of extremal points ofBX may be ifX contains a well-distributed set whose distance set Δ satisfies the estimate |Δ∩[0,N]|≤CN 3/2-ε. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩[0,N]|≤CN.

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References

  1. J. Bourgain,On the dimension of Kakeya sets and related maximal inequalities, Geometric and Functional Analysis9 (1999), 256–282.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Elekes and P. Erdős,Similar configurations and pseudo grids, inIntuitive Geometry (Szeged, 1991), Colloquia Mathematica Societatis János Bolyai, 63, North-Holland, Amsterdam, 1994, pp. 85–104.

    Google Scholar 

  3. P. Erdős,On sets of distances of n points, The American Mathematical Monthly53 (1946), 248–250.

    Article  MathSciNet  Google Scholar 

  4. K. J. Falconer,Dimensions of intersections and distance sets for polyhedral norms, preprint, 2004.

  5. G. Freiman,Foundations of a Structural Theory of Set Addition (translation from Russian), Translations of Mathematical Monographs, Vol. 37, American Mathematical Society, Providence, RI, 1973.

    MATH  Google Scholar 

  6. J. Garibaldi,Erdős Distance Problem for Convex Metrics, Ph.D. thesis, UCLA, 2004.

  7. A. Iosevich,Curvature, combinatorics and the Fourier transform, Notices of the American Mathematical Society46 (2001), 577–583.

    MathSciNet  Google Scholar 

  8. A. Iosevich and I. Łaba,Distance sets of well-distributed planar point sets, Discrete and Computational Geometry31 (2004), 243–250.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Iosevich and I. Łaba,K-distance sets, Falconer conjecture and discrete analogs, Integers: Electronic Journal of Combinatorial Number Theory, to appear.

  10. N. H. Katz and G. Tardos,A new entropy inequality for the Erdős distance problem, inTowards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, American Mathematical Society, Providence, RI, 2004.

    Google Scholar 

  11. M. Laczkovich and I. Z. Ruzsa,The number of homothetic subsets, inThe Mathematics of Paul Erdős, II, Algorithms and Combinatorics, 14, Springer, Berlin, 1997, pp. 294–302.

    Google Scholar 

  12. M. Nathanson,Additive Number Theory, II: Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996.

    Google Scholar 

  13. I. Ruzsa,Sum of sets in several dimensions, Combinatorica14 (1994), 485–490.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Solymosi and Cs. Tóth,Distinct distances in the plane, Discrete and Computational Geometry25 (2001), 629–634.

    MATH  MathSciNet  Google Scholar 

  15. J. Solymosi and V. Vu,Distinct distances in high dimensional homogeneous sets, inTowards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics, Vol. 342, American Mathematical Society, Providence, RI, 2004.

    Google Scholar 

  16. T. Wolff,Decay of circular means of Fourier transforms of measures, International Mathematics Research Notices10 (1999), 547–567.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia

    Sergei Konyagin

  2. Department of Mathematics, University of British Columbia, V6T 1Z2, Vancouver, B.C., Canada

    Izabella Łaba

Authors
  1. Sergei Konyagin
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  2. Izabella Łaba
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Correspondence to Sergei Konyagin.

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Konyagin, S., Łaba, I. Distance sets of well-distributed planar sets for polygonal norms. Isr. J. Math. 152, 157–179 (2006). https://doi.org/10.1007/BF02771981

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  • DOI: https://doi.org/10.1007/BF02771981

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