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The k-dimensional Duffin and Schaeffer conjecture

Published online by Cambridge University Press:  26 February 2010

A. D. Pollington  and
R. C. Vaughan
A. D. Pollington
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah, 84602, U.S.A.
R. C. Vaughan
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2BZ
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Extract

In 1941 Duffin and Schaeffer [2] considered the case k = 1 of the assertion that when the statement;

(Ak) “For almost all there are infinitely many natural numbers q for which there exist integers a1,…, ak such that (a1 … ak, q) = 1 and

where

holds, if, and only if,

diverges.

Type
Research Article
Information
Mathematika , Volume 37 , Issue 2 , December 1990 , pp. 190 - 200
Copyright
Copyright © University College London 1990

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References

1. Cassels, J. W. S.. Some metrical theorems in diphantine approximation. Proc. Cam. Phil. Soc., 46 (1950), 209–218.CrossRefGoogle Scholar
2. Duffin, R. J. and Schaeffer, A. C.. Khintchine's problem in metric Diophantine approximation. Duke Math. J., 8 (1941), 243–255.CrossRefGoogle Scholar
3. Erdős, P.. On the distribution of convergents of almost all real numbers. J. Number Theory, 2 (1970), 425–441.CrossRefGoogle Scholar
4. Gallagher, P. X.. Approximation by reduced fractions. J. Math. Soc. of Japan, 13 (1961), 342–345.CrossRefGoogle Scholar
5. Halberstam, H. and Richert, H.-E.. Sieve Methods (Academic Press, London, 1974).Google Scholar
6. Khinchin, A.. Einige Satze über Kettenbruche, mit Anwendungen auf die Theorie der Diophantischen Approximation. Math. Ann., 92 (1923), 115–125.CrossRefGoogle Scholar
7. Montgomery, H. L. and Vaughan, R. C.. On the distribution of reduced residues. Annals of Math., 123 (1986), 311–333.CrossRefGoogle Scholar
8. Sprindzuk, V. G., Metric theory of Diophantine Approximation (V. H. Winston and Sons, Washington, DC, 1979).Google Scholar
9. Strauch, O.. Some new criterions for sequences which satisfy Duffin-Schaefler conjecture, I. Acta Math. Univ. Com., 42–43. (1983), 87–95.Google Scholar
10. Vaaler, J. D.. On the metric theory of Diophantine approximation. Pacific J. Math., 76 (1978), 527–539.CrossRefGoogle Scholar
11. Vilchinski, V. T.. On simultaneous approximations. Vest. Akad. Nauk BSSR Ser. Fiz-Mat., (1981), 41–47.Google Scholar
12. Vilchinski, V. T.. The Duffin and Schaeffer conjecture and simultaneous approximations. Dokl. Akad. Nauk BSSR, 25 (1981), 780–783.Google Scholar