Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$[\, x\,]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mspace width="0.166667em" /> <mi>x</mi> <mspace width="0.166667em" /> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> denote the integer part of a real number x, and let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert x\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> represent the distance from x to the nearest integer. We establish that for any irrational number <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and any real number <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>, and for any fixed <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{99}{100}<\gamma <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>99</mn> <mn>100</mn> </mfrac> <mo><</mo> <mi>γ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist infinitely many prime numbers p for which the inequality <Equation ID="Equ88"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_Equ88.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="164" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Vert \alpha p^4+\beta \Vert <p^{\frac{99-100\gamma }{199}+\varepsilon } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">‖</mo> <mi>α</mi> </mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> <mrow> <mo>+</mo> <mi>β</mi> <mo stretchy="false">‖</mo> <mo><</mo> </mrow> <msup> <mi>p</mi> <mrow> <mfrac> <mrow> <mn>99</mn> <mo>-</mo> <mn>100</mn> <mi>γ</mi> </mrow> <mn>199</mn> </mfrac> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>holds, where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1195_Article_IEq9.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$p=[n^{1/\gamma }].$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>n</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>γ</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

On the distribution of $\alpha p^4$ modulo one over a thin set of primes

期刊论文
发表日期：

2025年8月30日

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S. I. Dimitrov,
M. D. Lazarova

摘要

Let $[\, x\,]$ $[x]$ denote the integer part of a real number x, and let $\Vert x\Vert $ $‖ x ‖$ represent the distance from x to the nearest integer. We establish that for any irrational number $\alpha $ $α$ and any real number $\beta $ $β$ , and for any fixed $\frac{99}{100}<\gamma <1$ $\frac{99}{100} < γ < 1$ , there exist infinitely many prime numbers p for which the inequality $\begin{aligned} \Vert \alpha p^4+\beta \Vert <p^{\frac{99-100\gamma }{199}+\varepsilon } \end{aligned}$ $\begin{matrix} ‖ α p^{4} + β ‖ < p^{\frac{99 - 100 γ}{199} + ε} \end{matrix}$ holds, where $p=[n^{1/\gamma }].$ $p = [n^{1 / γ}] .$

On the distribution of \(\alpha p^4\) modulo one over a thin set of primes

摘要