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We show there exists a constant 0 < c0 < 1 such that the dimension of every measure on [0, 1], which makes the digits in the continued fraction expansion independent, is at most 1 − c0. This extends a result of Kifer, Peres and Weiss from 2001, which established this under the additional assumption of stationarity. For k ≥ 1 we prove an analogous statement for measures under which the digits form a *-mixing k-step Markov chain. This is also generalized to the case of f-expansions. In addition, we construct for each k a measure, which makes the continued fraction digits a stationary and *-mixing k-step Markov chain, with dimension at least 1 − 23−k.

A dimension gap for continued fractions with independent digits—the non stationary case