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<p>We prove that, for any <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_135_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(B \subset {\mathbb {R}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>⊂</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, the Cartesian product set <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_135_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(B \times B\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>×</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> determines <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_135_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega (|B|^{2+c})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>B</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mn>2</mn> <mo>+</mo> <mi>c</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> distinct angles.</p>

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A Lower Bound for the Number of Pinned Angles Determined by a Cartesian Product Set

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  • 发表日期：

    2025年3月7日

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  • Oliver Roche-Newton

摘要

We prove that, for any \(B \subset {\mathbb {R}}\) B ⊂ R , the Cartesian product set \(B \times B\) B × B determines \(\Omega (|B|^{2+c})\) Ω ( | B | 2 + c ) distinct angles.