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On a Hardy-Littlewood problem of diophantine approximation

Published online by Cambridge University Press:  24 October 2008

D. C. Spencer
D. C. Spencer
Affiliation:
Trinity CollegeCambridge
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Extract

1. Let .

When r is a positive integer, various writers have considered sums of the form

where ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 35 , Issue 4 , November 1939 , pp. 527 - 547
Copyright
Copyright © Cambridge Philosophical Society 1939

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References

REFERENCES

(1)Behnke, H. “Zur Theorie der diophantische Approximationen”, Hamburg math. Abhandlungen, 3 (1924), 261–318.CrossRefGoogle Scholar
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