Modular values of continuants with fixed prefixes and endings

Consider the set of finite words in a finite alphabet $\mathbf{A}\subseteq\mathbb{N}$. Add a prefix $V$ and an ending $W$, which are some fixed finite words in the alphabet $\mathbb{N}$, to each word. We interpret the resulting words as the expansions in finite continued fractions of some rational numbers in the interval $(0,1)$. Next consider the irreducible denominators of these rational numbers; we denote the set of those denominators that do not exceed some quantity $N\in \mathbb{N}$ (which is an increasing parameter) by $\mathfrak{D}^{N}_{\mathbf{A},V,W}$. We prove that under certain conditions on $\mathbf{A}$, $V$ and $W$, for each prime number $Q$ proportional to a fixed fractional power of $N$ the set $\mathfrak{D}^{N}_{\mathbf{A},V,W}$ contains almost all possible residues modulo $Q$, and the remainder in this asymptotic formula involves a power reduction with respect to $Q$.

trigonometric sum, Zaremba's conjecture, Hausdorff dimension, continued fraction, continuant, Keywo ensemble