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Some metrical theorems in Diophantine approximation. I

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Trinity CollegeCambridge
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Extract

In this paper we are interested in statements about Diophantine approximation which are ‘almost always’ or ‘almost never’ true. Let

be s non-negative functions of the positive integer n, and let

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 2 , April 1950 , pp. 209 - 218
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

REFERENCES

(1)Bromwich, T. J. I'A.Infinite series (2nd ed.London, 1926), p. 28.Google Scholar
(2)Duffin, R. J. and Schaeffer, A. C.Khintchine's problem in metric Diophantine approximation. Duke Math. J. 8 (1941), 243–55.CrossRefGoogle Scholar
(3)Hardy, G. H. and Littlewood, J. E.Some problems of Diophantine approximation. Acta Math. 37 (1914), 155–91.CrossRefGoogle Scholar
(4)Hardy, G. H., Littlewood, J. E. and Polya, G.Inequalities (Cambridge, 1934), §§ 1·1 and 2·4.Google Scholar
(5)Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers (1st ed.Oxford, 1938), § 18·5.Google Scholar
(6)Khintchine, A.Zur Theorie der Diophantischen Approximationen. Rec. Soc. Math. Moscou, 32 (1925), 277–8.Google Scholar
(7)Khintchine, A.Zur Metrischen Theorie der Diophantischen Approximationen. Math. Z. 24 (1926), 706–14.CrossRefGoogle Scholar
(8)Knopp, K.Mengentheoretische Behandlung einiger Probleme der Diophantischen Approximationen usw. Math. Ann. 95 (1926), 409–26.CrossRefGoogle Scholar
(9)Koksma, J. F.Contribution à la théorie métrique des approximations diophantiques non-lineaires. Proc. K. Akad. Wet. Amsterdam, 45 (1942), 176–83 and 263–8.Google Scholar
(10)Paley, R. E. A. C. and Zygmund, A.On some series of functions. Proc. Cambridge Phil. Soc. 26 (1930), 337–57, 458–74 and 28 (1932), 190–205, See especially Lemma 19.CrossRefGoogle Scholar
(11)Saks, A.Theory of the integral (2nd ed.Warsaw, 1937), chapter iv, § 10.Google Scholar
(12)Weyl, H.Ueber die Gleichverteilung mod. Eins. Math. Ann. 77 (1915–1916), 313–56. For a general discussion of related topics see also:CrossRefGoogle Scholar
(13)Drewes, A. Diophantische Benaderingsproblemen. Dissertation (Amsterdam, 1945).Google Scholar
(14)Koksma, J. F. Diophantische Approximationen. Ergebn. Math, iv, 4 (Berlin, 1937).Google Scholar