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Metric simultaneous diophantine approximation (II)

Published online by Cambridge University Press:  26 February 2010

P. X. Gallagher
P. X. Gallagher
Affiliation:
Institute for Advanced Study, Princeton, New Jersey, U.S.A.
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Extract

If an is a sequence of numbers between 0 and 1, then

has infinitely many integral solutions n, l either for almost all real x or for almost no real x[1,4]. Duffin and Schaeffer [2], improving on an earlier theorem of Khintchine [7], proved that for decreasing sequences an, (1) has infinitely many solutions a.e. if and only if Σan diverges. They also gave an example of a sequence an for which Σan diverges, but for which (1) has only finitely many solutions a.e. No general necessary and sufficient condition for (1) to have infinitely many solutions a.e. is known.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 2 , December 1965 , pp. 123 - 127
Copyright
Copyright © University College London 1965

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References

1.Cassels, J. W. S., “Some metrical theorems in diophantine approximation. I”, Proc. Cambridge 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.Erdös, P., “Some results on diophantine approximation”, Acta Arith., 5 (1959), 359–369.CrossRefGoogle Scholar
4.Gallagher, P. X., “Approximation by reduced fractions”, Journal Math. Soc. Japan, 13 (1961), 342–345.Google Scholar
5.Gallagher, P. X., “Metric simultaneous diophantine approximation”, Journal London Math. Soc., 37 (1962), 387–390.CrossRefGoogle Scholar
6.Halmos, P. R., “Lectures on ergodic theory”, Pub. Math. Soc. Japan, 3 (1956).Google Scholar
7.Khintchine, A., “Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen”, Math. Annalen, 92 (1924), 115–125.CrossRefGoogle Scholar
8.Khintchine, A., “Zur metrischen Theorie der diophantischen Approximationen”, Math. Zeitschrift, 24 (1926), 706–714.CrossRefGoogle Scholar
9.LeVeque, W. J., “On the frequency of small fractional parts in certain real sequences, II”, Trans. American Math. Soc., 94 (1959), 130–149.CrossRefGoogle Scholar
10.Schmidt, W. M., “A metrical theorem in diophantine approximation”, Canadian J. of Math., 12 (1960), 619–631.CrossRefGoogle Scholar
11.Schmidt, W. M., “Metrical theorems on fractional parts of sequences”, Trans. American Math. Soc., 110 (1964), 493–518.CrossRefGoogle Scholar