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Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces


About this Title

Lior Fishman, David Simmons and Mariusz Urbański

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 254, Number 1215
ISBNs: 978-1-4704-2886-0 (print); 978-1-4704-4746-5 (online)
DOI: https://doi.org/10.1090/memo/1215
Published electronically: June 11, 2018
Keywords:Diophantine approximation, Schmidt’s game, hyperbolic geometry, Gromov hyperbolic metric spaces

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Gromov hyperbolic metric spaces
  • Chapter 3. Basic facts about Diophantine approximation
  • Chapter 4. Schmidt’s game and McMullen’s absolute game
  • Chapter 5. Partition structures
  • Chapter 6. Proof of Theorem 6.1 (Absolute winning of $\BA _\xi $)
  • Chapter 7. Proof of Theorem 7.1 (Generalization of the Jarník–Besicovitch Theorem)
  • Chapter 8. Proof of Theorem 8.1 (Generalization of Khinchin’s Theorem)
  • Chapter 9. Proof of Theorem 9.3 ($\BA _d$ has full dimension in $\Lr (G)$)
  • Appendix A. Any function is an orbital counting function for some parabolic group
  • Appendix B. Real, complex, and quaternionic hyperbolic spaces
  • Appendix C. The potential function game
  • Appendix D. Proof of Theorem 6.1 using the $\HH $-potential game, where $\HH $ = points
  • Appendix E. Winning sets and partition structures

Abstract


In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero PattersonâĂŞSullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

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