Chaika, JonDamanik, DavidFillman, JakeGohlke, Philipp2022-12-132022-12-132022Chaika, Jon, Damanik, David, Fillman, Jake, et al.. "Zero measure spectrum for multi-frequency Schrödinger operators." <i>Journal of Spectral Theory,</i> 12, no. 2 (2022) EMS Press: 573-590. https://doi.org/10.4171/jst/411.https://hdl.handle.net/1911/114080Building on works of Berthé–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 Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.engThis work is licensed under a CC BY 4.0 licenseJournal article7525219-10.4171-jst-411-printhttps://doi.org/10.4171/jst/411