Modular values of continuants with fixed prefixes and endings

This article is dedicated to the memory of Professor N. M. Korobov. Consider the set of finite words in a finite alphabet A. Add a prefix V and an ending W, which are some fixed finite words in the alphabet N, to each word. We treat 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 integer N∈ N (which is an increasing parameter) by DNA,V,W. We prove that under certain conditions on A, V and W, for each prime number Q proportional to a fixed fractional power of N the set DNA,V,W contains almost all possible residues modulo Q, and the remainder in this asymptotic formula involves a power reduction with respect to Q. Bibliography: 35 titles.