On the defect ("signed area") of toral Laplace eigenfunctions and exponential sums

Язык доклада – русский
This talk is based on a joint work with P. Kurlberg and N. Yesha. The defect (also known as "signed area") of a real-valued function defined on a two-dimensional domain is the difference between its positive and negative regions. We are interested in the defect of toral Laplace eigenfunctions (exponential sums) restricted to Planck-scale shrinking subdomains ("shrinking balls"). It is proved that, under a flatness assumption on the exponential sums, the defect asymptotically vanishes on the set of balls centres of almost full measure, for a generic sequence of energy levels. To establish our results we start from Bourgain's de-randomization technique, borrow the Integral-Geometric sandwich from Nazarov-Sodin, and also invoke other techniques.