Equidistribution in shrinking sets for arithmetic spherical harmonics

TL;DR

This paper investigates quantum unique ergodicity on shrinking spherical caps, establishing conditions under which equidistribution holds and providing explicit bounds, assuming the generalized Lindel"of hypothesis.

Contribution

It proves quantum unique ergodicity for shrinking caps larger than the Planck scale under the generalized Lindel"of hypothesis and offers explicit bounds on measure discrepancies.

Findings

Quantum unique ergodicity holds on caps larger than the Planck scale.

Explicit bounds are provided for Wasserstein distance and discrepancy.

Results are conditional on the generalized Lindel"of hypothesis.

Abstract

We study a variant of the equidistribution of mass conjecture on the sphere posed by B\"ocherer, Sarnak, and Schulze-Pillot: quantum unique ergodicity in shrinking sets. Conditionally on the generalized Lindel\"of hypothesis, we show that quantum unique ergodicity holds on every shrinking spherical cap whose radius is considerably larger than the Planck scale, and that it holds on almost every shrinking spherical cap whose radius is larger than the Planck scale. Additionally, conditionally on GLH, we provide explicit upper bounds for the $1$-Wasserstein distance and the spherical cap discrepancy between the involved measures.