A dimension gap for continued fractions with independent digits

Kinney and Pitcher (1966) determined the dimension of measures on [0,1] which make the digits in the continued fraction expansion i.i.d. variables. Prom 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 absolutely continuous measure μ under which the digits form a stationary process. Denote by E_δ the set of reals where the asymptotic frequency of some digit in the f-expansion differs by at least δ from the frequency prescribed by μ. Then E_δ has Hausdorff dimension less than 1 for any δ > 0.