# Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms

## TL;DR

This paper investigates the behavior of Maass forms and related automorphic objects in shrinking sets, establishing bounds on their norms and demonstrating quantum unique ergodicity under certain conditions, with implications for number theory and quantum chaos.

## Contribution

It provides new bounds on the $L^4$-norm of Eisenstein series and proves quantum unique ergodicity in shrinking sets for Maass forms, conditional on the generalized Lindelof hypothesis.

## Key findings

- Bounded the $L^4$-norm of Eisenstein series by $
oot{2}\log t_g$
- Quantum unique ergodicity holds for almost every shrinking ball above the Planck scale
- Equidistribution fails at or below the Planck scale for Maass eigenforms

## Abstract

We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$-norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in $\Gamma \backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindelof hypothesis for Maass eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Maass eigenforms need not hold at or below the Planck scale. Finally, we prove similar equidistribution results in shrinking sets for Heegner points and closed geodesics associated to ideal classes of quadratic fields.