A note on three problems in metric Diophantine approximation

Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)

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Publication details

Title of host publicationRecent Trends in Ergodic Theory and Dynamical Systems
DatePublished - 2015
Pages211-229
PublisherAmerican Mathematical Society
EditorsSiddhartha Bhattacharya, Tarun Das, Anish Ghosh, Riddhi Shah
Volume631
Original languageEnglish
ISBN (Print)978-1-4704-0931-9

Publication series

NameContemporary Mathematics
PublisherAmerican Mathematical Society
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

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 well-approximable and badly-approximable 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 non-degenerate 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.

    Research areas

  • metric Diophantine approximation, Inhomogenous appromultiplicative and inhomogeneous simultaneous approximationximations, Hausdorff measure and dimension

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