Let P be a <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-separated <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$(\delta , s, C_P)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <msub> <mi>C</mi> <mi>P</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-set of points in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$B(0, 1)\subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\Pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Π</mi> </math></EquationSource> </InlineEquation> be a <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-separated <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$(\delta , t, C_\Pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <msub> <mi>C</mi> <mi mathvariant="normal">Π</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-set of hyperplanes intersecting B(0, 1) in <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. Define <Equation ID="Equ6"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_Equ6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="318" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} I_{C\delta }(P, \Pi )=\#\{(p, \pi )\in P\times \Pi :p\in \pi (C\delta )\}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>I</mi> <mrow> <mi>C</mi> <mi>δ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>#</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi>P</mi> <mo>×</mo> <mi mathvariant="normal">Π</mi> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi (C\delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$C\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> neighborhood of the hyperplane <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq12.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>. Suppose that <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq13.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$s, t\ge \frac{d+1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mfrac> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, then we have <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2098_Article_IEq14.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="146" /> </InlineMediaObject> <EquationSource Format="TEX">$I_{C\delta }(P, \Pi )\lesssim \delta |P||\Pi |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mrow> <mi>C</mi> <mi>δ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">)</mo> </mrow> <mo>≲</mo> <mrow> <mi>δ</mi> <mo stretchy="false">|</mo> <mi>P</mi> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The main ingredient in our argument is a measure theoretic result due to Eswarathansan, Iosevich, and Taylor (2011) which was proved by using Sobolev bounds for generalized Radon transforms. Our result is essentially sharp, a construction will be provided and discussed in the last section.

A Discretized Point-Hyperplane Incidence Bound in $\mathbb {R}^d$

期刊论文
发表日期：

2025年9月4日

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Thang Pham,
Chun-Yen Shen,
Nguyen Pham Minh Tri

摘要

Let P be a $\delta $ $δ$ -separated $(\delta , s, C_P)$ $(δ, s, C_{P})$ -set of points in $B(0, 1)\subset \mathbb {R}^d$ $B (0, 1) \subset R^{d}$ and $\Pi $ $Π$ be a $\delta $ $δ$ -separated $(\delta , t, C_\Pi )$ $(δ, t, C_{Π})$ -set of hyperplanes intersecting B(0, 1) in $\mathbb {R}^d$ $R^{d}$ . Define $\begin{aligned} I_{C\delta }(P, \Pi )=\#\{(p, \pi )\in P\times \Pi :p\in \pi (C\delta )\}, \end{aligned}$ $\begin{matrix} I_{C δ} (P, Π) = # {(p, π) \in P \times Π : p \in π (C δ)}, \end{matrix}$ where $\pi (C\delta )$ $π (C δ)$ denotes the $C\delta $ $C δ$ neighborhood of the hyperplane $\pi $ $π$ . Suppose that $s, t\ge \frac{d+1}{2}$ $s, t \geq \frac{d + 1}{2}$ , then we have $I_{C\delta }(P, \Pi )\lesssim \delta |P||\Pi |$ $I_{C δ} (P, Π) ≲ δ | P | | Π |$ . The main ingredient in our argument is a measure theoretic result due to Eswarathansan, Iosevich, and Taylor (2011) which was proved by using Sobolev bounds for generalized Radon transforms. Our result is essentially sharp, a construction will be provided and discussed in the last section.

A Discretized Point-Hyperplane Incidence Bound in \(\mathbb {R}^d\)

摘要