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Simultaneous diophantine approximations. (Problèmes diophantiens simultanés.) (French) Zbl 1162.11361

Summary: We study a mixed simultaneous diophantine problem, with an approximation condition and a divisibility condition. We solve this problem for quadratic numbers.
Theorem: Suppose that the sequence \(\mathcal B\) is bounded. Let \(\alpha\) be a quadratic real number. Then there exists an infinite set of integers \(q > 1\) with \(\| q\alpha\|\ll 1/q\) and \(| q|_{\mathcal B}\ll 1/\ln q\). In particular we have \(\liminf_{q\to +\infty}q\ln q\| q\alpha\|| q|_{\mathcal B}<\infty\).

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations
11J68 Approximation to algebraic numbers
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