Abstract <p> To every finite word of a finite alphabet <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbf{A}\subseteq \mathbb{N}\)</EquationSource> </InlineEquation>, one can add a prefix <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(V\)</EquationSource> </InlineEquation> and a suffix <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(W\)</EquationSource> </InlineEquation> that are fixed finite words of the alphabet <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb{N}\)</EquationSource> </InlineEquation>. The words thus obtained are related to finite continued fraction expansions of certain numbers from <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((0,1)\cap \mathbb{Q}\)</EquationSource> </InlineEquation>. Their irreducible denominators that belong to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\([1,N]\)</EquationSource> </InlineEquation> form the set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathfrak{D}^{N}_{\mathbf{A},V,W}\)</EquationSource> </InlineEquation>. In the author’s previous work, it was shown that, under certain conditions on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathbf{A}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(Q\in\mathbb{N}\)</EquationSource> </InlineEquation>, and provided that the lengths of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(V\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(W\)</EquationSource> </InlineEquation> are not very large, the set <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\mathfrak{D}^{N}_{\mathbf{A},V,W}\)</EquationSource> </InlineEquation> contains almost all possible remainders modulo <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(Q\)</EquationSource> </InlineEquation>; moreover, the corresponding formula has power-law decay. In this paper, we obtain an analogous formula for an arbitrarily long suffix <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(W\)</EquationSource> </InlineEquation>. </p>

Remainders of Continuants with Large Fixed Suffixes

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发表日期：

2026年3月31日

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Igor Kan

摘要

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 \(W\) .