<p>Assume that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1, \lambda _2, \lambda _3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are non-zero real numbers, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1/\lambda _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is an irrational number. Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> be a well-spaced sequence, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For any given positive integer <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we give an upper bound of the number of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\upsilon \in \mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>υ</mi> <mo>∈</mo> <mi mathvariant="script">V</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\upsilon \le X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>υ</mi> <mo>≤</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> for which the inequality <Equation ID="Equ39"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_Equ39.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="227" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \left| \lambda _1p_1+\lambda _2p_2^2+\lambda _3p_3^k-\upsilon \right| <{\upsilon }^{-\delta } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfenced close="|" open="|"> <msub> <mi>λ</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <msubsup> <mi>p</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>λ</mi> <mn>3</mn> </msub> <msubsup> <mi>p</mi> <mn>3</mn> <mi>k</mi> </msubsup> <mo>-</mo> <mi>υ</mi> </mfenced> <mo><</mo> <msup> <mrow> <mi>υ</mi> </mrow> <mrow> <mo>-</mo> <mi>δ</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>has no solution in primes <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1010_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1, p_2, p_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>

Diophantine approximation with mixed powers of primes

期刊论文
发表日期：

2025年2月16日

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Linzhu Fu,
Liqun Hu,
Xuan Long

摘要

Assume that $\lambda _1, \lambda _2, \lambda _3$ $λ_{1}, λ_{2}, λ_{3}$ are non-zero real numbers, where $\lambda _1/\lambda _2$ $λ_{1} / λ_{2}$ is an irrational number. Let $\mathcal {V}$ $V$ be a well-spaced sequence, and $\delta >0$ $δ > 0$ . For any given positive integer $k\ge 3$ $k \geq 3$ , we give an upper bound of the number of $\upsilon \in \mathcal {V}$ $υ \in V$ with $\upsilon \le X$ $υ \leq X$ for which the inequality $\begin{aligned} \left| \lambda _1p_1+\lambda _2p_2^2+\lambda _3p_3^k-\upsilon \right| <{\upsilon }^{-\delta } \end{aligned}$ $\begin{matrix} |λ_{1} p_{1} + λ_{2} p_{2}^{2} + λ_{3} p_{3}^{k} - υ| < υ^{- δ} \end{matrix}$ has no solution in primes $p_1, p_2, p_3$ $p_{1}, p_{2}, p_{3}$ .