The exponent $1/4$ in three complexity bounds: integer bases, continued fractions, and the $(3/2)^n$ steering word

Let $ξ$ be a real algebraic irrational number and $p(n,ξ,b)$ the number of distinct blocks of length $n$ in its base-$b$ expansion. Bugeaud and Evertse proved in 2008 that $p(n,ξ,b)/(n(\log n)^η)$ is unbounded for every $η<1/11$. We prove this for every $η<1/4$. We also prove the analogous bound for