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

Understand

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.