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Simultaneous diophantine approximation with square-free numbers. (English) Zbl 0771.11028

A set \(\alpha_ 1,\dots,\alpha_ s\) of real numbers is said to be weakly compatible if \(\sum_{j=1}^ s \ell_ j \alpha_ j=u/v\) with \((u,v)=1\) implies that \(v\) is square-free. This condition is necessary and sufficient for \[ \liminf_{\mu^ 2(n)=1} \max_ j\|\alpha_ j n\|=0, \] where \(\|\cdot\|\) denotes the fractional part, as usual. It is shown that in general no quantitative form of the above result is possible. However if the \(\alpha_ j\) are algebraic, then \(\max_ j\|\alpha_ j n\|<n^{-A}\) for infinitely many square- free \(n\), for any \(A<1/(d^ 2-d)\), where \(d\) is the dimension of \(\langle 1,\alpha_ 1,\dots,\alpha_ s\rangle\) as a rational vector space. If \(d=2\) then any \(A<2/3\) is in fact admissible. These latter results are proved using exponential sums.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
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