Global hypoellipticity for perturbations of complex vector fields on the torus

TL;DR

This paper investigates the conditions under which perturbations of complex vector fields on the torus lead to globally hypoelliptic operators, revealing that the set of such perturbations can be either discrete or dense, contrasting with measure-zero results.

Contribution

It applies Kröneckers approximation theorem to characterize the topological size of the set of constants making vector fields hypoelliptic, showing it can be either meager or dense.

Findings

The set of constants can be a discrete, meager subset of the real line.

The set can also be a dense G_delta subset of complex numbers.

This contrasts with previous results indicating measure-zero sets.

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.