A dimension gap for continued fractions with independent digits
- Cite this article as:
- Kifer, Y., Peres, Y. & Weiss, B. Isr. J. Math. (2001) 124: 61. doi:10.1007/BF02772607
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. From 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 considerf-expansions with a corresponding absolutely continuous measureμ under which the digits form a stationary process. Denote byEδ the set of reals where the asymptotic frequency of some digit in thef-expansion differs by at leastδ from the frequency prescribed byμ. ThenEδ has Hausdorff dimension less than 1 for anyδ>0.