A note on three problems in metric Diophantine approximation

TY - CHAP

T1 - A note on three problems in metric Diophantine approximation

AU - Beresnevich, Victor

AU - Velani, Sanju

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - metric Diophantine approximation

KW - Inhomogenous appromultiplicative and inhomogeneous simultaneous approximationximations

KW - Hausdorff measure and dimension

U2 - 10.1090/conm/631/12605

DO - 10.1090/conm/631/12605

M3 - Chapter (peer-reviewed)

SN - 978-1-4704-0931-9

VL - 631

T3 - Contemporary Mathematics

SP - 211

EP - 229

BT - Recent Trends in Ergodic Theory and Dynamical Systems

A2 - Bhattacharya, Siddhartha

A2 - Das, Tarun

A2 - Ghosh, Anish

A2 - Shah, Riddhi

PB - American Mathematical Society

ER -