A strengthening of a theorem of Bourgain and Kontorovich. III

We prove that the set of positive integers contains a positive proportion of denominators of the finite continued fractions all of whose partial quotients belong to the alphabet $\{1,2,3,4,10\}$. The corresponding theorem was previousy known only for the alphabet $\{1,2,3,4,5\}$ and for alphabets of larger cardinality.