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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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##### References:
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