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A note on a problem of littlewood about diophantine approximation

Published online by Cambridge University Press:  26 February 2010

Walter Philipp
Affiliation:
Universität Wien, Montana State University, and Missoula, Montana.
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A famous problem of Littlewood is whether or not inf u¬¬ux ¬¬¬¬=0, (1) for all real numbers α, β, where the infimum is taken over all positive integers u, and ¬¬ε¬¬, as usual, denotes the distance from ε to the nearest integer. By a well-known transference principle (see [2, p. 78], with an obvious modification), problem (1) is equivalent to whether or not inf ¬xy¬ ¬¬xx+¬¬=0 (2) for all real numbers α, β, with 1, α, β linearly independent over the rationals, where the infimum is taken over all non-zero integers x, y.

Type
Research Article
Copyright
Copyright © University College London 1964

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References

1. Cassels, J. W. S., “On a result of Marshall Hall”, Mathematika, 3 (1956), 109110.CrossRefGoogle Scholar
2. Cassels, J. W. S., An introduction to Diophantine approximation, (Cambridge, 1957).Google Scholar
3. Davenport, H., “Simultaneous Diophantine approximation”, Mathematika, 1 (1954), 5172.Google Scholar
4. Davenport, H., “A note on Diophantine approximation”, Studies in mathematical analysis and related topics (Stanford, California, 1962).Google Scholar
5. Hardy, G. H., Wright, E. M., An introduction to the theory of numbers, 2nd ed. (Oxford 1945).Google Scholar