eBook ISBN:  9781470447465 
Product Code:  MEMO/254/1215.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470447465 
Product Code:  MEMO/254/1215.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 254; 2018; 137 ppMSC: Primary 11; 20; Secondary 28; 37
In this paper, the authors 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 1976 paper to more recent results of Hersonsky and Paulin (2002, 2004, 2007). The authors consider concrete examples of situations which have not been considered before. These 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 infinitedimensional hyperbolic space.
Moreover, in addition to providing much greater generality than any prior work of which the authors are aware, the 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 (1997) 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 Patterson–Sullivan measure unless the group is quasiconvexcocompact. The latter is an application of a Diophantine theorem.

Table of Contents

Chapters

1. Introduction

2. Gromov hyperbolic metric spaces

3. Basic facts about Diophantine approximation

4. Schmidt’s game and McMullen’s absolute game

5. Partition structures

6. Proof of Theorem (Absolute winning of $\mathrm {BA}_\xi $)

7. Proof of Theorem (Generalization of the Jarník–Besicovitch Theorem)

8. Proof of Theorem (Generalization of Khinchin’s Theorem)

9. Proof of Theorem ($\mathrm {BA}_d$ has full dimension in $\Lambda _{\mathrm {r}}(G)$)

A. Any function is an orbital counting function for some parabolic group

B. Real, complex, and quaternionic hyperbolic spaces

C. The potential function game

D. Proof of Theorem using the $\mathcal {H}$potential game, where $\mathcal {H}$ = points

E. Winning sets and partition structures


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In this paper, the authors 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 1976 paper to more recent results of Hersonsky and Paulin (2002, 2004, 2007). The authors consider concrete examples of situations which have not been considered before. These 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 infinitedimensional hyperbolic space.
Moreover, in addition to providing much greater generality than any prior work of which the authors are aware, the 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 (1997) 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 Patterson–Sullivan measure unless the group is quasiconvexcocompact. The latter is an application of a Diophantine theorem.

Chapters

1. Introduction

2. Gromov hyperbolic metric spaces

3. Basic facts about Diophantine approximation

4. Schmidt’s game and McMullen’s absolute game

5. Partition structures

6. Proof of Theorem (Absolute winning of $\mathrm {BA}_\xi $)

7. Proof of Theorem (Generalization of the Jarník–Besicovitch Theorem)

8. Proof of Theorem (Generalization of Khinchin’s Theorem)

9. Proof of Theorem ($\mathrm {BA}_d$ has full dimension in $\Lambda _{\mathrm {r}}(G)$)

A. Any function is an orbital counting function for some parabolic group

B. Real, complex, and quaternionic hyperbolic spaces

C. The potential function game

D. Proof of Theorem using the $\mathcal {H}$potential game, where $\mathcal {H}$ = points

E. Winning sets and partition structures