Diophantine Extremality of the Patterson Measure

Finer Diophantine and Regularity Properties of 1-dimensional Parabolic Ifs

Recall that a Borel probability measure μ on IR is called extremal if μ-almost every number in IR is not very well approximable. In this paper, we prove extremality (and implying it the exponentially fast decay property (efd)) of conformal measures induced by 1-dimensional finite parabolic iterated function systems. We also investigate the doubling property of these measures and we estimate fro...

Metric Diophantine Approximation for Systems of Linear Forms via Dynamics

The goal of this paper is to generalize the main results of [KM1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish ‘joint strong extremality’ of arbitrary finite collection of smooth nondegenerate submanifolds of R. The proofs are based on quantitative nondivergence estimates for quasi-polyn...

Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces

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 what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Dio...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

A Generalized Fibonacci Sequence and the Diophantine Equations $x^2pm kxy-y^2pm x=0$

In this paper some properties of a generalization of Fibonacci sequence are investigated. Then we solve the Diophantine equations $x^2pmkxy-y^2pm x=0$, where $k$ is positive integer, and describe the structure of solutions.