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Kifer, Peres, and Weiss proved that there exists $c_0>0,$ such that $\dim\mu\leq 1-c_0$ for any probability measure $\mu$ which makes the digits of thecontinued fraction expansion i.i.d. random variables. In this paper we provethat amongst this class of measures, there exists one whose dimension ismaximal. Our results also apply in the more general setting of countablebranched systems.

Maximising Bernoulli measures and dimension gaps for countable branched systems