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Inequalities for Diophantine exponents

Nikolay Moshchevitin
(Moscow State University)
Analysis Seminar
Thursday, 18 April, 2013, at 16:15-17:15, in Aud. D3 (1531-215)

We consider linear form \[ L({\bf x}) = x_0+x_1\theta_1+\dots+x_n\theta_n \] in integer variables $x_j$ with real coefficients $\theta_j$. For this linear form we consider the ordinary Diophantine exponent \[ {\omega} = \sup \left\{\gamma:\,\,\liminf_{t\to\infty}\,\,\left(t^\gamma \min_{0<|{\bf x}|\le t}{|L({\bf x})|}\right) <\infty \right\} \] and the uniform Diophantine exponent \[\hat{\omega} = \sup \left\{\gamma:\,\,\limsup_{t\to\infty}\,\,\left( t^\gamma \min_{0<|{\bf x}|\le t}{|L({\bf x})|}\right) <\infty \right\} .\] We discuss various inequalities involving $\omega $ and $\hat{\omega}$ as well as dual values for simultaneous approximations. We also consider Diophantine exponent $\omega_+$ related to approximation with positive integers and exponents related to approximation to a real number by algebraic numbers of bounded degree.

We suppose to give a brief survey on different results by Khintchine, Jarnik, Davenport, Schmidt, Laurent, Bugeaud, and introduce some new inequalities.

Contact person: Simon Kristensen