Abstract <p>A collection of finite sets <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="125" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{A_{1},A_{2},\ldots,A_{p}\}\)</EquationSource>  </InlineEquation> is said to be a double covering if each <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\in\cup_{k=1}^{p}A_{k}\)</EquationSource>  </InlineEquation> is included in exactly two sets of the collection. For fixed integers <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(l\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource>  </InlineEquation>, let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu_{l,p}\)</EquationSource>  </InlineEquation> be the number of equivalency classes of double coverings with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\#(A_{k})=l\)</EquationSource>  </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">\(k=1,2,\ldots,p\)</EquationSource>  </InlineEquation>. We characterize the asymptotic behavior of the quantity <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu_{l,p}\)</EquationSource>  </InlineEquation> as <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7229_Article_IEq9.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\to\infty\)</EquationSource>  </InlineEquation>. The results are applied to give an alternative approach to the Bonami–Kiener [<CitationRef CitationID="CR1">1</CitationRef>, <CitationRef CitationID="CR2">2</CitationRef>] hypercontraction inequality.</p>

Asymptotic Estimates for Double Coverings

期刊论文
发表日期：

2025年8月26日

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G. A. Karagulyan,
V. G. Karagulyan

摘要

Abstract

A collection of finite sets \(\{A_{1},A_{2},\ldots,A_{p}\}\) is said to be a double covering if each \(a\in\cup_{k=1}^{p}A_{k}\) is included in exactly two sets of the collection. For fixed integers \(l\) and \(p\) , let \(\mu_{l,p}\) be the number of equivalency classes of double coverings with \(\#(A_{k})=l\) , \(k=1,2,\ldots,p\) . We characterize the asymptotic behavior of the quantity \(\mu_{l,p}\) as \(p\to\infty\) . The results are applied to give an alternative approach to the Bonami–Kiener [1, 2] hypercontraction inequality.