Fractal uncertainty for discrete 2D Cantor sets

TL;DR

This paper establishes a fractal uncertainty principle for discrete 2D Cantor sets based on their geometric structure, linking it to number theory and quantum map applications.

Contribution

It proves a necessary and sufficient condition for fractal uncertainty in discrete Cantor sets, connecting geometric and number-theoretic properties.

Findings

A Cantor set has fractal uncertainty iff it lacks orthogonal line pairs.

The proof uses a quantitative form of Lang's conjecture in number theory.

Applications to open quantum maps are demonstrated.

Abstract

We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of Lang's conjecture in number theory due to Ruppert and Beukers & Smyth. Our theorem answers a question of Dyatlov and has applications to open quantum maps.