Small scale equidistribution of Hecke eigenforms at infinity

TL;DR

This paper studies how Hecke eigenforms distribute at infinity across different scales, revealing a transition from non-equidistribution to equidistribution around the Planck scale, with detailed variance analysis.

Contribution

It demonstrates the scale-dependent equidistribution behavior of Hecke eigenforms and computes variance showing transition phenomena at half the Planck scale.

Findings

Eigenforms do not equidistribute below the Planck scale.

Eigenforms fully equidistribute above the Planck scale.

Variance analysis reveals transition at half the Planck scale.

Abstract

We investigate the equidistribution of Hecke eigenforms on sets that are shrinking towards infinity. We show that at scales finer than the Planck scale they do not equidistribute while at scales more coarse than the Planck scale they equidistribute on a full density subsequence of eigenforms. On a suitable set of test functions we compute the variance showing interesting transition behavior at half the Planck scale.