Continued fractions and Fourier transforms

Let F N 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 F N and certain measures carried by F N . 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 F N , however, metrical properties can be used to construct the measure λ.