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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
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On some Diophantine spectra N. G. Moshchevitin M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics |
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Abstract: 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 $$ \mathbb{L}=\Bigl\{\lambda\in\mathbb{R}:\exists\,\alpha\in\mathbb{R}\setminus\mathbb{Q}\ \liminf_{t\to\infty}t\psi_\alpha(t)=\lambda\Bigr\}, $$ 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 $$ \mathbb{M}=\Bigl\{m\in\mathbb{R}:\exists\,\alpha\in\mathbb{R}\setminus\mathbb{Q}\ \limsup_{t\to\infty}t\mu_\alpha(t)=m\Bigr\}, $$ connected with the function |
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