Bounded ratios and badly approximability

Abstract: We will discuss relatively new criteria of badly approximability in terms of ratios of best approximations. Let qν be convergents of continued fractions to real irrational α. It is well known that

α is badly approximable   iff   supν qν+1/qν is finite   iff   infν||qν+1α||/||qνα||>0.

We will discuss how this property may be generalised to Diophantine Approximation in higher dimensions. The answer seems to be rather non-trivial. Some of the related properties may be expressed in terms of Parametric Geometry of Numbers recently developed by Schmidt, Summerer, Roy and the others. Also we discuss some properties of ratios under the consideration in accordance with the study of multidimensional Dirichlet spectra.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar