Global hypoellipticity for perturbations of complex vector fields on the torus

Abstract

We apply Kr\"{o}necker's approximation theorem to measure (in a topological sense) a set of constants which turn a vector field into a non-globally hypoelliptic operator. We present situations in which this set is a discrete enumerable (hence, meager) subset of the real line, and we also show that this set may be a dense $\mathcal{G}_\delta$ subset of the complex numbers (hence, nonmeager), which produces a contrast to a known result stating that this set has null Lebesgue measure.