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Dirichlet's diophantine approximation theorem

1977
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Bulletin of the Australian Mathematical Society
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One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if a" a_, ..., a are any real numbers and m 2 2 is an integer, then there exist integers q, p , p , ..., p such that 1 5 q < m and |qa.-p. | £ mh olds for 1 £ i £ n . The paper considers the problem of the extent to which this theorem can be improved by replacing mb y 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

doi:10.1017/s0004972700023224
fatcat:g2j5pd4ckrf57lj4vrj55xtnru