# An introduction to fractal uncertainty principle

## TL;DR

This paper reviews recent advances in the fractal uncertainty principle, highlighting its implications for quantum chaos, eigenfunction bounds, and spectral gaps on hyperbolic surfaces.

## Contribution

It synthesizes recent developments and applications of the fractal uncertainty principle in quantum chaos and spectral theory.

## Key findings

- Lower bounds on eigenfunction mass on negatively curved surfaces
- Existence of spectral gaps on convex co-compact hyperbolic surfaces
- Connections between fractal sets and quantum spectral properties

## Abstract

Fractal uncertainty principle states that no function can be localized in both position and frequency near a fractal set. This article provides a review of recent developments on the fractal uncertainty principle and of their applications to quantum chaos, including lower bounds on mass of eigenfunctions on negatively curved surfaces and spectral gaps on convex co-compact hyperbolic surfaces.