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Simultaneous Diophantine approximation. (English) Zbl 0048.03204

Proof of the theorem: “Let \(c > 46^{-1/4}\). Then, for every pair of real irrational numbers \(\alpha, \beta\), there exist infinitely many solutions \(p, q, r > 0\) of \(r(p-\alpha r)^2 < c\), \(r(q- \beta r)^2 < c\) in integers.”
This result slightly improves one by P. Mullender [Ann. Math. (2) 52, 417-426 (1950; Zbl 0037.17102)]. The theorem is proved by showing that the lattice determinant of the star domain \((\xi s-\eta c)^2\vert \eta\vert\le 1\), \((\xi c-\eta s)^2\vert \eta\vert\le 1\) in the \((\xi, \eta)\)-plane is not less than \(\sqrt 2\) if \(c = \cos \theta\) and \(s = \sin \theta\), and \(\theta\) is an arbitrary angle.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms

Citations:

Zbl 0037.17102
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