A strengthening of a theorem of Bourgain and Kontorovich. IV

We prove that the denominators of finite continued fractions all of whose partial quotients belong to the alphabet $\{1,2,3,4\}$ form a set of positive density. The analogous theorem was known earlier only for alphabets of larger cardinality. The first result of this kind was obtained in 2011 for the alphabet $\{1,2,\dots,50\}$ by Bourgain and Kontorovich. In 2013, the present author, together with Frolenkov, proved the corresponding theorem for the alphabet $\{1,2,3,4,5\}$. A 2014 result of the present author dealt with the alphabet $\{1,2,3,4,10\}$.