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arXiv · 2302.11708

The fractal uncertainty principle via Dolgopyat's method in higher dimensions

Aidan Backus·James Leng·Zhongkai Tao
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Abstract

We prove a fractal uncertainty principle with exponent $\frac{d}{2} - δ+ \varepsilon$, $\varepsilon > 0$, for Ahlfors--David regular subsets of $\mathbb R^d$ with dimension $δ$ which satisfy a suitable "nonorthogonality condition". This generalizes the application of Dolgopyat's method by Dyatlov--Jin (arXiv:1702.03619) to prove the same result in the special case $d = 1$. As a corollary, we get a quantitative spectral gap for the Laplacian on convex cocompact hyperbolic manifolds of arbitrary dimension with Zariski dense fundamental groups.

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Aidan Backus, James Leng, Zhongkai Tao. 2023-10-09. The fractal uncertainty principle via Dolgopyat's method in higher dimensions. https://doi.org/10.2140/apde.2025.18.1769

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