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VIDEO LIBRARY |
À.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
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On irrationality measure functions N. G. Moshchevitin Lomonosov Moscow State University, Faculty of Mechanics and Mathematics |
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Abstract: For a real $$ \psi_{\alpha}(t)\,=\,\min_{1\le q\le t, \, q\in \mathbb{Z}}||q\alpha ||,\quad t\,\ge\,1 $$ (here $$ \liminf_{t\to\infty} t\psi_\alpha (t)\,\,\,\, \text{and}\,\,\,\, \limsup_{t\to\infty} t\psi_\alpha (t). $$ Another interesting result is related to oscillation property of the difference In our lecture, we will discuss certain properties of the functions $$ \psi_\alpha^{[2]}(t)\,=\,\min_{ \begin{array}{c} (q,p): \, q,p\in \mathbb{Z}, 1\le q\le t, \cr (p,q) \neq (p_n, q_n) \,\forall\, n =0,1,2,3,... \end{array} } |q\alpha -p| $$ and $$ \psi_\alpha^{[2]*} (t)\,=\,\min_{ \begin{array}{c} (q,p): \, q,p\in \mathbb{Z}, 1\le q\le t, \cr p/q \neq p_n/q_n \,\forall\, n =0,1,2,3,... \end{array} } |q\alpha -p| $$ related to the “second best” approximations and certain properties of the function Language: English |