Sparse systems of functions and quasi-analytic classes

We provide a new characterization of quasi-analyticity of Denjoy–Carleman classes, related to Wetzel’s Problem. We also completely resolve which Denjoy–Carleman classes carry sparse systems: if the Continuum Hypothesis (CH) holds, all Denjoy–Carleman classes carry sparse systems; but if CH fails, a Denjoy–Carleman class carries a sparse system if and only if it is not quasi-analytic. As corollaries, we extend results of Bajpai–Velleman (2016) and Cody–Cox–Lee (2023) about non-existence of “anonymous predictors” for real functions.