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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 30, 2015 15:00–15:25, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences

On some Diophantine spectra

N. G. Moshchevitin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
 Video records: Flash Video 138.5 Mb Flash Video 829.7 Mb MP4 138.5 Mb

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Abstract: Let $\alpha$ be an irrational number, and let
$$\psi_\alpha(t)=\min_{\mathbb{Z}_+\ni q\le t}\|q\alpha\|$$
be the function of measure of its irrationality. In the talk, we discuss some old and new results concerning Lagrange spectrum
$$\mathbb{L}=\{\lambda\in\mathbb{R}:\exists \alpha\in\mathbb{R}\setminus\mathbb{Q} \liminf_{t\to\infty}t\psi_\alpha(t)=\lambda\},$$
Dirchlet spectrum
$$\mathbb{D} = \{ d\in \mathbb{R}: \exists \alpha \in \mathbb{R}\setminus\mathbb{Q} \limsup_{t\to \infty} t\psi_\alpha (t) = d\},$$
and the spectrum
$$\mathbb{M}=\{m\in\mathbb{R}:\exists \alpha\in\mathbb{R}\setminus\mathbb{Q} \limsup_{t\to\infty}t\mu_\alpha(t)=m\},$$
connected with the function $\mu_\alpha(t)$, arising in the analysis of Minkowski diagonal fraction.

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