Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T04:03:36.075Z Has data issue: false hasContentIssue false

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.
Get access

Extract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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