Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equidistribution in shrinking sets for arithmetic spherical harmonics

Maximiliano Sanchez Garza

Abstract

We study a variant of the equidistribution of mass conjecture on the sphere posed by Böcherer, Sarnak, and Schulze-Pillot: quantum unique ergodicity in shrinking sets. Conditionally on the generalized Lindelöf 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.

Equidistribution in shrinking sets for arithmetic spherical harmonics

Abstract

We study a variant of the equidistribution of mass conjecture on the sphere posed by Böcherer, Sarnak, and Schulze-Pillot: quantum unique ergodicity in shrinking sets. Conditionally on the generalized Lindelöf 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 -Wasserstein distance and the spherical cap discrepancy between the involved measures.
Paper Structure (13 sections, 24 theorems, 126 equations)

This paper contains 13 sections, 24 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Quantum ergodicity on the sphere
  3. Preliminaries
  4. Spherical harmonics and automorphic representations of B(A)
  5. Watson--Ichino triple product formula
  6. Adèlic Watson--Ichino formula
  7. Computation of archimedean local constant
  8. Computation of nonarchimedean local constant
  9. Classical Watson--Ichino formula
  10. L-functions
  11. Selberg--Harish-Chandra transform
  12. 1-Wasserstein distance and the Berry--Esseen inequality
  13. Proofs

Key Result

Theorem 1.1

Let $\psi$ be a Hecke eigenfunction of Laplacian eigenvalue $\lambda_{\psi}^2 = \ell_{\psi}(\ell_{\psi}+1)$ for some integer $\ell_{\psi} \geq 0$, normalized such that $\langle \psi, \psi \rangle = 1$. Then, assuming GLH, for all $\varepsilon>0$ we have

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2: Weyl law
  • Proposition 2.3: Local Weyl law MinakshisundaramPleijelLocalWeylLaw
  • Theorem 2.4: IchinoFormula
  • Lemma 2.5
  • proof
  • ...and 26 more