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Dirichlet's diophantine approximation theorem

Published online by Cambridge University Press:  17 April 2009

T.W. Cusick
T.W. Cusick
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Amherst, New York, USA.
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Abstract

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One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if α1, α2, …, αn are any real numbers and m ≥ 2 is an integer, then there exist integers q, p1, p2, …, pn such that 1 ≤ q < m and |qαi.-pi| ≤ m–1/n holds for 1 < i < n. The paper considers the problem of the extent to which this theorem can be improved by replacing m–1/n by a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 16 , Issue 2 , April 1977 , pp. 219 - 224
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Davenport, H. and Schmidt, Wolfgang M., “Dirichlet's theorem on diophantine approximation”, Symposia Mathematica, Volume IV: Teoria dei nimeri, 113–132 (INDAM, Roma, 1968/1969. Academic Press, London and New York, 1970).Google Scholar
[2]Davenport, H. and Schmidt, W.M., “Dirichlet's theorem on diophantine approximation. II”, Acta Arith. 16 (1909/1970), 413–424.CrossRefGoogle Scholar
[3]Mahler, Kurt, “A theorem on diophantine approximations”, Bull. Austral. Math. Soc. 14 (1976), 463–465.CrossRefGoogle Scholar