Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T15:30:25.490Z Has data issue: false hasContentIssue false

Continued fractions and Fourier transforms

Published online by Cambridge University Press:  26 February 2010

R. Kaufman
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.
Get access

Extract

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
Copyright
Copyright © University College London 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Good, I. J.. “The fractional dimensional theory of continued fractions”, Proc. Cambridge Phil. Soc., 37 (1941), 199228.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), 893895.Google Scholar
3.Rogers, C. A.. “Some sets of continued fractions”, Proc. London Math. Soc., 14 (1964), 2944.Google Scholar