Metric Diophantine Approximation for Systems of Linear Forms via Dynamics

Self-similar fractals and arithmetic dynamics

‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

Article outline This article gives a brief overview of recent developments in metric number theory, in particular, Diophantine approximation on manifolds, obtained by applying ideas and methods coming from dynamics on homogeneous spaces. Glossary 1. Definition: Metric Diophantine approximation 2. Basic facts 3. Introduction 4. Connection with dynamics on the space of lattices 5. Diophantine app...

Definition of General Operator Space and The s-gap Metric for Measuring Robust Stability of Control Systems with Nonlinear Dynamics

In the recent decades, metrics have been introduced as mathematical tools to determine the robust stability of the closed loop control systems. However, the metrics drawback is their limited applications in the closed loop control systems with nonlinear dynamics. As a solution in the literature, applying the metric theories to the linearized models is suggested. In this paper, we show that usin...

Bounded Orbits Conjecture and Diophantine Approximation

We describe and generalize S.G. Dani’s correspondence between bounded orbits in the space of lattices and systems of linear forms with certain Diophantine properties. The solution to Margulis’ Bounded Orbit Conjecture is used to generalize W. Schmidt’s theorem on abundance of badly approximable systems of linear forms.

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm