A dimension gap for continued fractions with independent digits

Kinney and Pitcher (1966) determined the dimension of measures on [0,1] whi ch make the digits in the continued fraction expansion i.i.d. variables. Fr om their formula it is not clear that these dimensions are less than 1, but this follows from the thermodynamic formalism for the Gauss map developed by Walters (1978). We prove that, in fact, these dimensions are bounded by 1 - 10(-7). More generally, we consider f-expansions with a corresponding a bsolutely continuous measure mu under which the digits form a stationary pr ocess. Denote by E-delta the set of reals where the asymptotic frequency of some digit in the f-expansion differs by at least delta from the frequency prescribed by mu. Then E-delta has Hausdorff dimension less than 1 for any delta > 0.