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Diophantine exponents for mildly restricted approximation. (English) Zbl 1304.11062
The authors study the problem of Diophantine approximation of vectors in \(\mathbb R^n\) by rational numbers with restrictions on the denominators. They introduce new Diophantine exponents as follows. For \(1 \leq l < n\) and \((\alpha_1, \ldots, \alpha_n) \in \mathbb R^n\) let \(\mu_{n,l}(\alpha)\) denote the supremum of real numbers \(\mu\) such that the inequality
\[ 0 < \|x_1\alpha_1 + \dots + x_n\alpha_n \| \leq H(\underline{x})^{-\mu} \]
has infinitely many solutions \(\underline{x} = (x_1,\dots, x_n) \in \mathbb{Z}^n\) satisfying
\[ \max\{|x_{l+1}|, \dots, |x_n|\} < \max\{|x_1|, \dots, |x_l|\}. \]
Here \(H(\underline{x})\) denotes the supremum norm of \(\underline{x}\). Special cases of the above exponents were introduced earlier by W. M. Schmidt [Monatsh. Math. 82, 237–245 (1976; Zbl 0337.10022)] and P. Thurnheer [Acta Arith. 54, No. 3, 241–250 (1990; Zbl 0643.10027)]. The authors prove several results regarding the spectrum of the exponent, i.e. the set of values taken by these exponents on the set of vectors \((\alpha_1, \ldots, \alpha_n)\) which together with \(1\), are linearly independent over \(\mathbb{Q}\).
MSC:
11J13 Simultaneous homogeneous approximation, linear forms
11J83 Metric theory
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