## 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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