For <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$0<w<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>w</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and any integer <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$t\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$E_{w,t}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mi>w</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be the least integer d such that for every set A of nonnegative integers with the lower asymptotic density <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\underline{d}(A)=w$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mi>d</mi> <mo>̲</mo> </munder> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation>, the set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$(k+t)A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> contains an infinite arithmetic progression with difference at most d, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$k=\lceil w^{-1}\rceil $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mo>⌈</mo> <msup> <mi>w</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>⌉</mo> </mrow> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$t=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, Chen-Yang-Zhao have given the asymptotic behavior of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$E_{w,t}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mi>w</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. In this paper, we generalize their result and prove the asymptotic behavior of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$E_{w,t}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <mi>w</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1199_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="192" /> </InlineMediaObject> <EquationSource Format="TEX">$t=O(k\exp (-1.5c\sqrt{\log k}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>exp</mo> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1.5</mn> <mi>c</mi> <msqrt> <mrow> <mo>log</mo> <mi>k</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where c is a positive integer.

Infinite arithmetic progressions in h-fold sumsets

期刊论文
发表日期：

2025年9月16日

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Ya-Li Li,
Yu-Zhu Zhao

摘要

For $0<w<1$ $0 < w < 1$ and any integer $t\ge 1$ $t \geq 1$ , let $E_{w,t}$ $E_{w, t}$ be the least integer d such that for every set A of nonnegative integers with the lower asymptotic density $\underline{d}(A)=w$ $\underset{̲}{d} (A) = w$ , the set $(k+t)A$ $(k + t) A$ contains an infinite arithmetic progression with difference at most d, where $k=\lceil w^{-1}\rceil $ $k = ⌈ w^{- 1} ⌉$ . When $t=1$ $t = 1$ , Chen-Yang-Zhao have given the asymptotic behavior of $E_{w,t}$ $E_{w, t}$ . In this paper, we generalize their result and prove the asymptotic behavior of $E_{w,t}$ $E_{w, t}$ for $t=O(k\exp (-1.5c\sqrt{\log k}))$ $t = O (k exp (- 1.5 c \sqrt{log k}))$ , where c is a positive integer.