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Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces

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Abstract

We prove an analogue of a theorem of Pollington and Velani (Sel Math (N.S.) 11:297–307, 2005), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof incorporates the framework of intrinsic approximation on such hypersurfaces first developed in the authors’ joint work with Kleinbock (Intrinsic Diophantine approximation on manifolds, 2014. arXiv:1405.7650v2) with ideas from work of Kleinbock et al. (Sel Math (N.S.) 10:479–523, 2004).

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References

  1. Beresnevich, V.: Rational points near manifolds and metric Diophantine approximation. Ann. Math. (2) 175(1), 187–235 (2012)

    Article  MathSciNet  Google Scholar 

  2. Beresnevich, V.: Badly approximable points on manifolds, MR 3425389. Invent. Math. 202(3), 1199–1240 (2015)

    Article  MathSciNet  Google Scholar 

  3. Bernik, V., Dodson, M.: Metric Diophantine Approximation on Manifolds, Cambridge Tracts in Mathematics, vol. 137. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  4. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990)

    MATH  Google Scholar 

  5. Fishman, L., Kleinbock, D., Merrill, K., Simmons, D.: Intrinsic Diophantine approximation on manifolds, arXiv:1405.7650v2, preprint (2014)

  6. Fishman, L., Simmons, D., Urbański, M.: Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces. Mem. Am. Math. Soc. 254, 1215 (2018)

    MathSciNet  Google Scholar 

  7. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    Article  MathSciNet  Google Scholar 

  8. Kleinbock, D., de Saxcé, N.: Rational approximation on quadrics: a simplex lemma and its consequences, arXiv:1808.07070, preprint (2018)

  9. Kleinbock, D., Lindenstrauss, E., Weiss, B.: On fractal measures and Diophantine approximation. Sel. Math. (N.S.) 10, 479–523 (2004)

    Article  MathSciNet  Google Scholar 

  10. Kleinbock, D., Margulis, G.: Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. Math. 148(1), 339–360 (1998)

    Article  MathSciNet  Google Scholar 

  11. Kleinbock, D., Weiss, B.: Badly approximable vectors on fractals. Israel J. Math. 149, 137–170 (2005)

    Article  MathSciNet  Google Scholar 

  12. Kristensen, S., Thorn, R., Velani, S.: Diophantine approximation and badly approximable sets. Adv. Math. 203, 132–169 (2006)

    Article  MathSciNet  Google Scholar 

  13. Pollington, A., Velani, S.: Metric diophantine approximation and “absolutely friendly” measures. Sel. Math. (N.S.) 11, 297–307 (2005)

    Article  MathSciNet  Google Scholar 

  14. Stratmann, B., Urbański, M.: Diophantine extremality of the Patterson measure. Math. Proc. Camb. Philos. Soc. 140, 297–304 (2006)

    Article  MathSciNet  Google Scholar 

  15. Urbański, M.: Diophantine approximation and self-conformal measures. J. Numb. Theory 110, 219–235 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first-named author was supported in part by the Simons Foundation Grant #245708.

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Authors and Affiliations

  1. Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX, 76203-5017, USA

    Lior Fishman

  2. Department of Mathematics, Brandeis University, 415 South St, Waltham, MA, 02453, USA

    Keith Merrill

  3. Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, OH, 43210-1174, USA

    David Simmons

Authors
  1. Lior Fishman
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  2. Keith Merrill
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  3. David Simmons
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Correspondence to Keith Merrill.

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Fishman, L., Merrill, K. & Simmons, D. Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces. Sel. Math. New Ser. 24, 3875–3888 (2018). https://doi.org/10.1007/s00029-018-0446-7

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