We prove that on a d-dimensional Riemannian manifold, the distance set of a Borel set E has a positive Lebesgue measure if <Equation ID="Equ31"> <EquationSource Format="TEX">$\begin{aligned} \dim _{\mathcal {H}}(E) > \frac{d}{2} + \frac{1}{4} + \frac{1 - (-1)^d}{8d}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mo>dim</mo> <mi mathvariant="script">H</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </msup> </mrow> <mrow> <mn>8</mn> <mi>d</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The key new ingredient is a natural local orthogonal projection bound on general Riemannian manifolds.

Falconer distance problem on Riemannian manifolds

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发表日期：

2025年11月17日

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Changbiao Jian,
Bochen Liu,
Yakun Xi

摘要

We prove that on a d-dimensional Riemannian manifold, the distance set of a Borel set E has a positive Lebesgue measure if $\begin{aligned} \dim _{\mathcal {H}}(E) > \frac{d}{2} + \frac{1}{4} + \frac{1 - (-1)^d}{8d}. \end{aligned}$ $\begin{matrix} {dim}_{H} (E) > \frac{d}{2} + \frac{1}{4} + \frac{1 - {(- 1)}^{d}}{8 d} . \end{matrix}$ The key new ingredient is a natural local orthogonal projection bound on general Riemannian manifolds.