Research Article

Continued fractions and Fourier transforms

R. Kaufmana1

a1 Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.

Let FN be the set of real numbers x whose continued fraction expansion x = [a 0; a 1, a 2,…, a n,…] contains only elements a i = 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 M 0-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 M 0-set. For the sets FN , however, metrical properties can be used to construct the measure λ.

Key Words:

  • 10K15: NUMBER THEORY; Probabilistic Theory, measure, dimension, etc.; Diophantine approximation