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

## TL;DR

This paper extends the fractal uncertainty principle to higher dimensions using Dolgopyat's method, providing new spectral gap results for hyperbolic manifolds with Zariski dense groups.

## Contribution

It generalizes Dolgopyat's method to higher dimensions for fractal sets, establishing a quantitative spectral gap for the Laplacian on hyperbolic manifolds.

## Key findings

- Proved a fractal uncertainty principle with explicit exponent in higher dimensions.
- Established a spectral gap for Laplacians on convex cocompact hyperbolic manifolds.
- Extended Dolgopyat's method beyond the one-dimensional case.

## Abstract

We prove a fractal uncertainty principle with exponent $\frac{d}{2} - \delta + \varepsilon$, $\varepsilon > 0$, for Ahlfors--David regular subsets of $\mathbb R^d$ with dimension $\delta$ 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.