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

2023/02/23 by Backus, Aidan, Leng, James, Tao, Zhongkai
#28A80 #35B34 #81Q50 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2302.11708

Abstract

We prove a fractal uncertainty principle with exponent (d)/(2) - δ+ ε, ε > 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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