Remainders of Continuants with Large Fixed Suffixes
Abstract
To every finite word of a finite alphabet $$\mathbf{A}\subseteq \mathbb{N}$$ , one can add a prefix $$V$$ and a suffix $$W$$ that are fixed finite words of the alphabet $$\mathbb{N}$$ . The words thus obtained are related to finite continued fraction expansions of certain numbers from $$(0,1)\cap \mathbb{Q}$$ . Their irreducible denominators that belong to $$[1,N]$$ form the set $$\mathfrak{D}^{N}_{\mathbf{A},V,W}$$ . In the author’s previous work, it was shown that, under certain conditions on $$\mathbf{A}$$ and $$Q\in\mathbb{N}$$ , and provided that the lengths of $$V$$ and $$W$$ are not very large, the set $$\mathfrak{D}^{N}_{\mathbf{A},V,W}$$ contains almost all possible remainders modulo $$Q$$ ; moreover, the corresponding formula has power-law decay. In this paper, we obtain an analogous formula for an arbitrarily long suffix $$W$$ .
Fields of Study & Subjects
semigroups and automata theory
Cellular Automata and Applications
Analytic Number Theory Research
Publication details
- Journal / source
- Functional Analysis and Its Applications
- Publisher
- Pleiades Publishing Ltd
- Volume
- 60
- Issue
- 1
- Pages
- 14–33
- Publication date
- 1 March 2026
- Language
- EN