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The fractional dimensional theory of continued fractions

Published online by Cambridge University Press:  24 October 2008

I. J. Good
I. J. Good
Affiliation:
Jesus CollegeCambridge
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Extract

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 3 , July 1941 , pp. 199 - 228
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

REFERENCES

(1)Jarnik, . Zur metrischen Theorie der diophantischen Approximationen. Prace Mat.-Fiz. 36 (1928), 91–106.Google Scholar
(2)Borel, . Rendiconti del Circolo Mat. di Palermo, 27 (1909), 246–71.Google Scholar
Bernstein, . Math. Ann. 92 (1912), 417–39.Google Scholar
(3)Besicovitch, . On rational approximation to real numbers. J. London Math. Soc. 9 (1934), 126–31.CrossRefGoogle Scholar
(4)Perron, . Die Lehre von den Kettenbruchen (Teubner, 1929), 9 and 15.Google Scholar
(5)Sylvester, . Math. Papers, 3 (Cambridge, 1909), 249.Google Scholar