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Metric simultaneous diophantine approximation (II)

Published online by Cambridge University Press:  26 February 2010

P. X. Gallagher
P. X. Gallagher
Affiliation:
Institute for Advanced Study, Princeton, New Jersey, U.S.A.
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Extract

If an is a sequence of numbers between 0 and 1, then

has infinitely many integral solutions n, l either for almost all real x or for almost no real x[1,4]. Duffin and Schaeffer [2], improving on an earlier theorem of Khintchine [7], proved that for decreasing sequences an, (1) has infinitely many solutions a.e. if and only if Σan diverges. They also gave an example of a sequence an for which Σan diverges, but for which (1) has only finitely many solutions a.e. No general necessary and sufficient condition for (1) to have infinitely many solutions a.e. is known.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 2 , December 1965 , pp. 123 - 127
Copyright
Copyright © University College London 1965

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References

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