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Continued fractions and Fourier transforms

  • R. Kaufman (a1)
Abstract

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. I. J. Good . “The fractional dimensional theory of continued fractions”, Proc. Cambridge Phil. Soc., 37 (1941), 199228.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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