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Simultaneous asymptotic Diophantine approximations

Published online by Cambridge University Press:  26 February 2010

William W. Adams
Affiliation:
Institute for Advanced Studies, Princeton. University of California, Berkeley
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Extract

Let θ1, …, θk be k real numbers. Suppose ψ(t) is a positive decreasing function of the positive variable t. Define λ(N), for all positive integers N, to be the number of solutions in integers p1 …, pk, q of the inequalities

and

Type
Research Article
Copyright
Copyright © University College London 1967

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References

1.Cassels, J. W. S., An introduction to diophantine approximations (Cambridge, Cambridge University Press, 1957).Google Scholar
2.Cassels, J. W. S., An introduction to the geometry of numbers (Berlin, Springer-Verlag, 1959).CrossRefGoogle Scholar
3.Cassels, J. W. S., “Simultaneous diophantine approximations II”, Proc. Lond. Math. Soc, 5 (1955), 435448.CrossRefGoogle Scholar
4.Lang, S., “Asymptotic diophantine approximations“, Proc. Nat. Acad. Sci. USA, 55 (1966), 3134.CrossRefGoogle ScholarPubMed
5.Lang, S., Introduction to diophantine approximations (Reading, Mass., Addison-Wesley, 1966).Google Scholar
6.Popken, J., “Zur Transzendenz von e”, Math. Zeit., 29 (1929), 525541.CrossRefGoogle Scholar
7.Schmidt, W. M., “A metrical theorem in diophantine approximations”. Canadian J. Math., 11 (1960), 619631.CrossRefGoogle Scholar
8.Schmidt, W. M., “Simultaneous approximations to a basis of a real algebraic number field”, American J. Math., 88 (1966), 517527.CrossRefGoogle Scholar