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Abstract
The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding wellapproximable and badlyapproximable points. In this paper we consider three inhomogeneous problems that further develop these classical results. Firstly, we obtain a Jarnik type theorem for the set of multiplicatively approximable points in the plane. This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting. Next, we establish a Jarnik type theorem for that set restricted to a nondegenerate planar curve. This completes the Hausdorff theory for planar curves. Finally, we show that the set of (i,j)inhomogeneously badly approximable points in the plane is of full dimension. The underlying philosophy behind the proof has other applications; e.g. towards establishing the inhomogeneous version of Schmidt's Conjecture. The higher dimensional analogues of the planar results are also discussed.
Original language  English 

Title of host publication  Recent Trends in Ergodic Theory and Dynamical Systems 
Subtitle of host publication  Conference in Honor of S.G. Dani's 65th Birthday Recent Trends in Ergodic Theory and Dynamical Systems, December 2629, 2012, Vadodara, India 
Editors  Siddhartha Bhattacharya, Tarun Das, Anish Ghosh, Riddhi Shah 
Publisher  American Mathematical Society 
Pages  211229 
Volume  631 
ISBN (Print)  9781470409319 
DOIs  
Publication status  Published  2015 
Publication series
Name  Contemporary Mathematics 

Publisher  American Mathematical Society 
ISSN (Print)  02714132 
ISSN (Electronic)  10983627 
Keywords
 metric Diophantine approximation
 Inhomogenous appromultiplicative and inhomogeneous simultaneous approximationximations
 Hausdorff measure and dimension
Profiles
Projects
 1 Finished

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