  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.
