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Simultaneous Approximation

Chapter
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Part of the Lecture Notes in Mathematics book series (LNM, volume 785)

Abstract

Suppose that α 1,...,α n are n real numbers and that Q > 1 is an integer. Then there exist integers q,p1,...,Pn with
$$ 1 \leqslant q < Q^n \underline {and} \left| {\alpha _i q - p_i } \right| \leqslant \frac{1} {Q} (1 \leqslant i \leqslant n) . $$
(1.1)

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References

  1. L.G.P. Dirichlet (1842). Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. S. B. Preuss. Akad. Wiss. 93–95.Google Scholar
  2. H. Minkowski (1896 & 1910). Geometrie der Zahlen. Teubner: Leipzig u. Berlin (The 1910 ed. prepared posthumously by Hilbert and Speiser).Google Scholar
  3. O. Perron (1921b). Über diophantische Approximationen. Math. Ann. 83, 77–84.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

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