Reflection symmetries of almost periodic functions

TL;DR

This paper investigates the reflection symmetries of almost periodic functions, establishing bounds on symmetry sets and implications for spectral properties of associated Schrödinger operators, highlighting differences between limit periodic and non-limit periodic cases.

Contribution

It provides new bounds on symmetry measures for non-limit periodic almost periodic functions and links these to spectral criteria for Schrödinger operators, revealing cases where the criterion applies universally.

Findings

Upper bound on Haar measure of symmetric elements in the hull

Jitomirskaya-Simon criterion applies only on a measure-zero set in the non-limit periodic case

Examples where the criterion applies to all hull elements in limit periodic functions

Abstract

We study global reflection symmetries of almost periodic functions. In the non-limit periodic case, we establish an upper bound on the Haar measure of the set of those elements in the hull which are almost symmetric about the origin. As an application of this result we prove that in the non-limit periodic case, the criterion of Jitomirskaya and Simon ensuring absence of eigenvalues for almost periodic Schr\"odinger operators is only applicable on a set of zero Haar measure. We complement this by giving examples of limit periodic functions where the Jitomirskaya-Simon criterion can be applied to every element of the hull.