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Continued fractions and Fourier transforms

  • R. Kaufman (a1)

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.

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1.Good, I. J.. “The fractional dimensional theory of continued fractions”, Proc. Cambridge Phil. Soc., 37 (1941), 199228.
2.Kahane, J.-P. and Katznelson, Y.. “Sur les ensembles d'unicité U(ε) de Zygmund”, C. R. Acad. Sci. Pahs, 227 (1973), 893895.
3.Rogers, C. A.. “Some sets of continued fractions”, Proc. London Math. Soc., 14 (1964), 2944.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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