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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 the group, and groups acting on infinitedimensional 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 PattersonSullivan measure unless the group is quasiconvexcocompact. The latter is an application of a Diophantine theorem.
Original language  English 

Pages (fromto)  1150 
Number of pages  150 
Journal  Memoirs of the American Mathematical Society 
Volume  254 
Issue number  1215 
DOIs  
Publication status  Published  11 Jun 2018 
Keywords
 math.DS
 Hyperbolic geometry
 Schmidt's game
 Diophantine approximation
 Gromov hyperbolic metric spaces
Profiles
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research