Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$[\, \cdot \,]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mspace width="0.166667em" /> <mo>·</mo> <mspace width="0.166667em" /> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> be the floor function and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert x\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> denote the distance from x to the nearest integer. In this paper we show that whenever <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is irrational and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is real then for any fixed <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_IEq10.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{13}{14}<\gamma <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>13</mn> <mn>14</mn> </mfrac> <mo><</mo> <mi>γ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist infinitely many prime numbers p satisfying the inequality <Equation ID="Equ73"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_Equ73.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Vert \alpha p^2+\beta \Vert < p^{\frac{13-14\gamma }{29}+\varepsilon } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">‖</mo> <mi>α</mi> </mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> <mrow> <mo>+</mo> <mi>β</mi> <mo stretchy="false">‖</mo> <mo><</mo> </mrow> <msup> <mi>p</mi> <mrow> <mfrac> <mrow> <mn>13</mn> <mo>-</mo> <mn>14</mn> <mi>γ</mi> </mrow> <mn>29</mn> </mfrac> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and such that <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1210_Article_IEq11.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$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> </mrow> </math></EquationSource> </InlineEquation>.

On the distribution of $\alpha p^{2}$ modulo one over primes of the form $[n^{c}]$

期刊论文
发表日期：

2025年10月15日

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S. I. Dimitrov,
M. D. Lazarova

摘要

Let $[\, \cdot \,]$ $[\cdot]$ be the floor function and $\Vert x\Vert $ $‖ x ‖$ denote the distance from x to the nearest integer. In this paper we show that whenever $\alpha $ $α$ is irrational and $\beta $ $β$ is real then for any fixed $\frac{13}{14}<\gamma <1$ $\frac{13}{14} < γ < 1$ , there exist infinitely many prime numbers p satisfying the inequality $\begin{aligned} \Vert \alpha p^2+\beta \Vert < p^{\frac{13-14\gamma }{29}+\varepsilon } \end{aligned}$ $\begin{matrix} ‖ α p^{2} + β ‖ < p^{\frac{13 - 14 γ}{29} + ε} \end{matrix}$ and such that $p=[n^{1/\gamma }]$ $p = [n^{1 / γ}]$ .

On the distribution of \(\alpha p^{2}\) modulo one over primes of the form \([n^{c}]\)

摘要