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Abstract <p> For an infinite set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathscr A\subset\mathbb N\)</EquationSource> </InlineEquation>, let the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\rho(x,y;\mathscr A)\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(x\in\mathbb N\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(y+1\in\mathbb N\)</EquationSource> </InlineEquation>, be defined as the length of a maximal interval <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((\alpha,\beta)\subset(y,x+y)\)</EquationSource> </InlineEquation> disjoint from <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathscr A\)</EquationSource> </InlineEquation>. We study the function <Equation ID="Equi"> <EquationSource Format="TEX">\(\rho_*(x;\mathscr A)=\inf_{y+1\in\mathbb N}\rho(x,y;\mathscr A).\)</EquationSource> </Equation> In particular, we determine the order of this function for the set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathscr A\)</EquationSource> </InlineEquation> of square-free numbers. </p>

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On the Local Distribution of Elements of Subsets of the Set of Positive Integers

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • S. V. Konyagin

摘要

Abstract

For an infinite set \(\mathscr A\subset\mathbb N\) , let the function \(\rho(x,y;\mathscr A)\) , where \(x\in\mathbb N\) and \(y+1\in\mathbb N\) , be defined as the length of a maximal interval \((\alpha,\beta)\subset(y,x+y)\) disjoint from \(\mathscr A\) . We study the function \(\rho_*(x;\mathscr A)=\inf_{y+1\in\mathbb N}\rho(x,y;\mathscr A).\) In particular, we determine the order of this function for the set \(\mathscr A\) of square-free numbers.