Email this article Modular values of continuants with fixed prefixes and endings
- Authors: Kan I.D.1
-
Affiliations:
- Moscow Aviation Institute (National Research University), Moscow, Russia
- Issue: Vol 217, No 1 (2026)
- Pages: 54-88
- Section: Articles
- URL: https://ogarev-online.ru/0368-8666/article/view/378949
- DOI: https://doi.org/10.4213/sm10170
- ID: 378949
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Abstract
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 $Q$ .
About the authors
Igor' Davidovich Kan
Moscow Aviation Institute (National Research University), Moscow, Russia
Email: igor.kan@list.ru
Candidate of physico-mathematical sciences, Associate professor
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