Assume that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is an irrational number, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq2.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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$c>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are real numbers. The corresponding Beatty sequence and Piatetski–Shapiro sequence are defined as<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_Equa.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="394" /> </MediaObject> <EquationSource Format="TEX">$\mathcal{B}_{\alpha,\beta}:= \{\lfloor\alpha n+\beta\rfloor: n\in\mathbb{N}\} \,\,{\rm and}\,\, \mathcal{N}^c:= \{\lfloor n^c\rfloor: n\in\mathbb{N}\},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo>⌊</mo> <mi>α</mi> <mi>n</mi> <mo>+</mo> <mi>β</mi> <mo>⌋</mo> </mrow> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi mathvariant="normal">and</mi> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <msup> <mrow> <mi mathvariant="script">N</mi> </mrow> <mi>c</mi> </msup> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo>⌊</mo> <msup> <mi>n</mi> <mi>c</mi> </msup> <mo>⌋</mo> </mrow> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>respectively. Here, the symbol <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\lfloor y\rfloor$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mi>y</mi> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the largest integer not exceeding y. Let p be a prime, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma=c^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <msup> <mi>c</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$F_{\alpha,\beta,c}(p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the least quadratic non-residue in the intersection of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal{B}_{\alpha,\beta}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal{N}^c$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">N</mi> </mrow> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$1<c<8/7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>c</mi> <mo><</mo> <mn>8</mn> <mo stretchy="false">/</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, we obtain <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq10.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="196" /> </InlineMediaObject> <EquationSource Format="TEX">$F_{\alpha,\beta,c}(p)\ll_c p^{1/((6\gamma-5)4\sqrt{e})+\varepsilon}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>≪</mo> <mi>c</mi> </msub> <msup> <mi>p</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mi>γ</mi> <mo>-</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mn>4</mn> <msqrt> <mi>e</mi> </msqrt> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. As <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$c\rightarrow1^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, our result tends to the Burgess bound <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1537_Article_IEq12.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$p^{1/(4\sqrt{e})+\varepsilon}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>4</mn> <msqrt> <mi>e</mi> </msqrt> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>.

On the least non-residue in the intersection of a Piatetski–Shapiro sequence and a Beatty sequence

期刊论文
发表日期：

2025年6月4日

查看全文

M. Jing

摘要

Assume that $\alpha>1$ $α > 1$ is an irrational number, $\beta$ $β$ and $c>1$ $c > 1$ are real numbers. The corresponding Beatty sequence and Piatetski–Shapiro sequence are defined as $\mathcal{B}_{\alpha,\beta}:= \{\lfloor\alpha n+\beta\rfloor: n\in\mathbb{N}\} \,\,{\rm and}\,\, \mathcal{N}^c:= \{\lfloor n^c\rfloor: n\in\mathbb{N}\},$ $B_{α, β} : = {⌊ α n + β ⌋ : n \in N} and N^{c} : = {⌊ n^{c} ⌋ : n \in N},$ respectively. Here, the symbol $\lfloor y\rfloor$ $⌊ y ⌋$ denotes the largest integer not exceeding y. Let p be a prime, $\gamma=c^{-1}$ $γ = c^{- 1}$ , and let $F_{\alpha,\beta,c}(p)$ $F_{α, β, c} (p)$ be the least quadratic non-residue in the intersection of $\mathcal{B}_{\alpha,\beta}$ $B_{α, β}$ and $\mathcal{N}^c$ $N^{c}$ . For $1<c<8/7$ $1 < c < 8 / 7$ , we obtain $F_{\alpha,\beta,c}(p)\ll_c p^{1/((6\gamma-5)4\sqrt{e})+\varepsilon}$ $F_{α, β, c} (p) ≪_{c} p^{1 / ((6 γ - 5) 4 \sqrt{e}) + ε}$ . As $c\rightarrow1^{+}$ $c \to 1^{+}$ , our result tends to the Burgess bound $p^{1/(4\sqrt{e})+\varepsilon}$ $p^{1 / (4 \sqrt{e}) + ε}$ .