Abstract:

One of the definitions of  a  badly approximable number $\alpha$  is related to its continued fraction expansion
$$
\alpha = \frac{1}{\displaystyle{a_1}+\frac{1}{\displaystyle{a_2}+\frac{1}{\displaystyle{a_3 + \cdots}}}}, \,\,\,\,\, a_j \in \mathbb{Z}_+.
$$
Real irrational $\alpha$ is badly approximable if its partial quotients in continued fraction expansion are bounded, that is
$\sup_{\nu} a_\nu < \infty$. Define $q_\nu$ to be the denominator of $\nu$-th convergent $p_\nu/q_\nu$
 to $\alpha$ and  $\xi_\nu = |q_\nu \alpha - p_\nu|$. From the well known formula
 $q_{\nu} = a_{\nu}q_{\nu-1}+q_{\nu-2}$ we immediately deduce the equivalence
 $$
 \alpha \,\,\, \text{is badly approximable}\,\,\,
 \Longleftrightarrow \,\,\,
 \sup_{\nu} \frac{q_{\nu+1}}{q_\nu} <\infty
 \,\,\,
 \Longleftrightarrow \,\,\,
\inf_{\nu } \frac{\xi_{\nu+1}}{\xi_\nu} >0.
$$
In our lecture we  will discuss how these equivalences look like in the multidimensional settings related to simultaneous approximations and, more generally, systems of linear forms.
Our talk is based on recent  results of the speaker obtained jointly with R. Akhunzhanov, A. Marnat and J. Schleischitz as well as two recent paper by W.M. Schmidt and L. Summerer.