Zero Measure Spectrum for Multi-Frequency Schr\"odinger Operators

TL;DR

This paper demonstrates that for most two-dimensional torus translations, one can approximate quasi-periodic potentials with others that produce Schrödinger operators having Cantor spectrum of zero Lebesgue measure, extending potential applications in spectral theory.

Contribution

It establishes a method to approximate quasi-periodic potentials with zero-measure Cantor spectrum for Schrödinger operators on two-dimensional tori, extending to higher dimensions.

Findings

Almost every two-dimensional torus translation admits a coding satisfying Boshernitzan criterion.

Any quasi-periodic potential can be approximated by one with zero Lebesgue measure Cantor spectrum.

Framework proposed for extending results to higher-dimensional tori.

Abstract

Building on works of Berth\'e--Steiner--Thuswaldner and Fogg--Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schr\"odinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.