Recently N. Hindman and D. Strauss (Topology Proc. 61, 49–76 (2023)) proved if A be a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$u\times v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>×</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation> matrix with rational entries with the property that for every <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$n\in \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> there exists <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\vec {z}\in \mathbb {Z}^{v}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>z</mi> <mo stretchy="false">→</mo> </mover> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>v</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$A\vec {z}=\overline{n}\in \mathbb {Z}^{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mover accent="true"> <mi>z</mi> <mo stretchy="false">→</mo> </mover> <mo>=</mo> <mover> <mi>n</mi> <mo>¯</mo> </mover> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>u</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>n</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> is the vector in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}^{u}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>u</mi> </msup> </math></EquationSource> </InlineEquation> where every entry is n, then for a J-set <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$B\subseteq \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>⊆</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> we have <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="149" /> </InlineMediaObject> <EquationSource Format="TEX">$\left\{ \vec {x}\in \mathbb {Z}^{v}:A\vec {x}\in B^{u}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>v</mi> </msup> <mo>:</mo> <mi>A</mi> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo>∈</mo> <msup> <mi>B</mi> <mi>u</mi> </msup> </mfenced> </math></EquationSource> </InlineEquation> is a J-set in <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10504_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}^{v}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>v</mi> </msup> </math></EquationSource> </InlineEquation>. We obtain a similar result for CR-sets.

A concept of largeness of combinatorially rich sets

期刊论文
发表日期：

2025年2月18日

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Pintu Debnath

摘要

Recently N. Hindman and D. Strauss (Topology Proc. 61, 49–76 (2023)) proved if A be a $u\times v$ $u \times v$ matrix with rational entries with the property that for every $n\in \mathbb {Z}$ $n \in Z$ there exists $\vec {z}\in \mathbb {Z}^{v}$ $\vec{z} \in Z^{v}$ such that $A\vec {z}=\overline{n}\in \mathbb {Z}^{u}$ $A \vec{z} = \bar{n} \in Z^{u}$ , where $\overline{n}$ $\bar{n}$ is the vector in $\mathbb {Z}^{u}$ $Z^{u}$ where every entry is n, then for a J-set $B\subseteq \mathbb {Z}$ $B \subseteq Z$ we have $\left\{ \vec {x}\in \mathbb {Z}^{v}:A\vec {x}\in B^{u}\right\} $ $\{\vec{x} \in Z^{v} : A \vec{x} \in B^{u}\}$ is a J-set in $\mathbb {Z}^{v}$ $Z^{v}$ . We obtain a similar result for CR-sets.