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Quantitative FUP and spectral gap for quasi-Fuchsian group

Long Jin, An Zhang, Hong Zhang

TL;DR

This work provides a quantitative higher-dimensional fractal uncertainty principle (FUP) with an explicit exponent $β(ν,d)$, showing that $L^2$ functions Fourier localized near a fractal set cannot concentrate on a spatial fractal, with $β$ governed by porosity on lines and balls. By connecting porosity to damping via Cohen–Han–Schlag frameworks, the authors derive explicit FUP bounds and carry these through to obtain an explicit essential spectral gap for convex cocompact hyperbolic 3-manifolds arising from quasi-Fuchsian groups; for dimension two in the Fourier-analytic setting this yields a gap parameter $β= frac12 β(ν)$ linked to the limit-set porosity. The results extend Jin–Zhang (2020) to higher dimensions and refine Tao’s spectral-gap results (2025) by making the exponent explicit in terms of geometric data such as the porosity constant and the limit-set regularity, while handling the circle (Fuchsian) case separately via separation of variables. The approach provides sharp resolvent estimates in a spectral strip and has implications for open quantum chaos and semiclassical analysis on hyperbolic manifolds, making the connection between fractal geometry and spectral theory quantitatively precise.

Abstract

We derive an explicit formula for the exponent $β$ in the higher-dimensional fractal uncertainty principle (FUP) established by Cohen 2023, quantifying its dependence on the porosity parameter $ν$ of the Fourier support. This quantitative version of FUP yields an explicit essential spectral gap for convex co-compact hyperbolic 3-manifolds arising from quasi-Fuchsian groups, thereby refining the result of Tao 2025. Our result extends the earlier work of Jin-Zhang 2020 to higher dimensions.

Quantitative FUP and spectral gap for quasi-Fuchsian group

TL;DR

This work provides a quantitative higher-dimensional fractal uncertainty principle (FUP) with an explicit exponent , showing that functions Fourier localized near a fractal set cannot concentrate on a spatial fractal, with governed by porosity on lines and balls. By connecting porosity to damping via Cohen–Han–Schlag frameworks, the authors derive explicit FUP bounds and carry these through to obtain an explicit essential spectral gap for convex cocompact hyperbolic 3-manifolds arising from quasi-Fuchsian groups; for dimension two in the Fourier-analytic setting this yields a gap parameter linked to the limit-set porosity. The results extend Jin–Zhang (2020) to higher dimensions and refine Tao’s spectral-gap results (2025) by making the exponent explicit in terms of geometric data such as the porosity constant and the limit-set regularity, while handling the circle (Fuchsian) case separately via separation of variables. The approach provides sharp resolvent estimates in a spectral strip and has implications for open quantum chaos and semiclassical analysis on hyperbolic manifolds, making the connection between fractal geometry and spectral theory quantitatively precise.

Abstract

We derive an explicit formula for the exponent in the higher-dimensional fractal uncertainty principle (FUP) established by Cohen 2023, quantifying its dependence on the porosity parameter of the Fourier support. This quantitative version of FUP yields an explicit essential spectral gap for convex co-compact hyperbolic 3-manifolds arising from quasi-Fuchsian groups, thereby refining the result of Tao 2025. Our result extends the earlier work of Jin-Zhang 2020 to higher dimensions.
Paper Structure (13 sections, 21 theorems, 149 equations, 2 figures)

This paper contains 13 sections, 21 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. A rough review of the literature
  4. Structure of the paper
  5. Quantitative FUP in higher dimensions
  6. From damping condition to FUP
  7. From porosity to damping condition
  8. Quantitative FUP
  9. Spectral gap for quasi-Fuchsian 3-manifold
  10. Line porosity condition of $X$
  11. Proof of Theorem \ref{['thm:Spectral']}
  12. FUP for circles and the resonances for Fuchsian groups
  13. Some useful lemmas

Key Result

Theorem 1.1

Let $d\ge 1$, $\nu\in (0, 1/3),\;h\in (0,1/100)$. Suppose (1) $\mathbf X\subset[-1,1]^d$ is $\nu$-porous on balls from scales $h$ to $1$. (2) $\mathbf Y\subset [-h^{-1},h^{-1}]^d$ is $\nu$-porous on lines from scales $1$ to $h^{-1}$. Then there exists some constant $C=C(\nu,d)>0$ independent of $h$ holds for any $f\in L^2(\mathbb R^d)$ with $\mathop{\mathrm{supp}}\nolimits \hat{f}\subset \mathbf

Figures (2)

  • Figure 1: The primary example in our analysis is the celebrated Koch snowflake. This set is $(\log_3 4)$-regular and clearly satisfies the three-point condition.
  • Figure 2: Illustration of our argument in the proof of Proposition \ref{['prop:Xis-line-porous']}. The horizontal dashed gray line is $L$, extended from the segment $I$ passing roughly between $q_1$ and $q_3$. Each $q_i$ lies in $X$, while each $p_i$ lies on $I$. The point $q$ belongs to $X$ but is far from $L$. The red ball $B$, together with the points $q_1$, $q_2$, $q_3$, and $q_5$, are all $(\nu r)$-very close to $L$. The point $q_4$ is moderately close to $q_2$ but not moderately-very close to $L$. The point $p_5$ is the projection infimum of the arc $[q_4,q_3]$ and lies on the boundary of $B$, which is a hole in $X$.

Theorems & Definitions (44)

  • Theorem 1.1: Quantitative FUP
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1: Porosity
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5: Damping functions
  • ...and 34 more