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Continued fractions and Fourier transforms

Published online by Cambridge University Press:  26 February 2010

R. Kaufman
R. Kaufman
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.
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Let FN be the set of real numbers x whose continued fraction expansion x = [a0; a1, a2,…, an,…] contains only elements ai = 1,2,…, N. Here N ≥ 2. Considerable effort, [1,3], has centred on metrical properties of FN and certain measures carried by FN. A classification of sets of measure zero, as venerable as the dimensional theory, depends on Fourier-Stieltjes transforms: a closed set E of real numbers is called an M0-set, if E carries a probability measure λ whose transform vanishes at infinity. Aside from the Riemann-Lebesgue Lemma, no purely metrical property of E can ensure that E is an M0-set. For the sets FN, however, metrical properties can be used to construct the measure λ.

Type
Research Article
Information
Mathematika , Volume 27 , Issue 2 , December 1980 , pp. 262 - 267
Copyright
Copyright © University College London 1980

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References

1.Good, I. J.. “The fractional dimensional theory of continued fractions”, Proc. Cambridge Phil. Soc., 37 (1941), 199–228.Google Scholar
2.Kahane, J.-P. and Katznelson, Y.. “Sur les ensembles d'unicité U(ε) de Zygmund”, C. R. Acad. Sci. Pahs, 227 (1973), 893–895.Google Scholar
3.Rogers, C. A.. “Some sets of continued fractions”, Proc. London Math. Soc., 14 (1964), 29–44.Google Scholar