Erdős introduced the quantity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_997_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$S=T\sum ^T_{i=1}|X_i|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mi>T</mi> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_997_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$X_1,\dots , X_T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>T</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> are arithmetic progressions that cover the squares up to N. He conjectured that S is close to N, i.e. the square numbers cannot be covered “economically” by arithmetic progressions. Sárközy confirmed this conjecture and proved that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_997_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$S\ge cN/\log ^2N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>≥</mo> <mi>c</mi> <mi>N</mi> <mo stretchy="false">/</mo> <msup> <mo>log</mo> <mn>2</mn> </msup> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper we extend this to shrinking polynomials and so-called <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_997_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\{X_i\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> quasi progressions.

Covering shrinking polynomials by quasi progressions

期刊论文
发表日期：

2025年1月31日

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Norbert Hegyvári

摘要

Erdős introduced the quantity $S=T\sum ^T_{i=1}|X_i|$ $S = T \sum_{i = 1}^{T} | X_{i} |$ , where $X_1,\dots , X_T$ $X_{1}, \dots, X_{T}$ are arithmetic progressions that cover the squares up to N. He conjectured that S is close to N, i.e. the square numbers cannot be covered “economically” by arithmetic progressions. Sárközy confirmed this conjecture and proved that $S\ge cN/\log ^2N$ $S \geq c N / {log}^{2} N$ . In this paper we extend this to shrinking polynomials and so-called $\{X_i\}$ ${X_{i}}$ quasi progressions.