Distribution symmetry of toral eigenfunctions

TL;DR

This paper investigates the value distribution symmetry of eigenfunctions on tori, confirming conjectures in 2D, providing counterexamples in higher dimensions, and proving related symmetry theorems for specific trigonometric polynomials.

Contribution

It confirms the symmetry conjecture for 2D toral eigenfunctions, presents counterexamples in higher dimensions, and proves a new symmetry theorem for certain trigonometric polynomials.

Findings

Symmetry conjecture holds strongly on 2D torus.

Counterexample exists for higher-dimensional tori.

Proved a new distribution symmetry theorem for specific trigonometric polynomials.

Abstract

In this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we provide a counterexample for higher dimensional tori, which relies on a computer assisted argument. Moreover we prove a theorem on the distribution symmetry of a certain class of trigonometric polynomials that might be of independent interest.