Abel universal functions on the unit disc are the holomorphic functions whose dilates uniformly approximate every continuous function on every proper compact subset of the unit circle. This body of results settles two questions about them: their boundary behaviour, obtained by transferring known facts about Valiron functions and the MacLane class (divergent double-logarithmic growth integral, infinity as an asymptotic value, normality at no boundary point, every boundary point a Picard point, prescribed growth along sets meeting the circle in one point), and the behaviour of their Taylor partial sums outside the disc, which - contrary to the analogy with universal Taylor series - can diverge to infinity pointwise on any countable boundary set, almost everywhere on the circle and in the plane, and locally in capacity.
Stephane Charpentier, Myrto Manolaki and Konstantinos Maronikolakis (arXiv 2310.05611v1, math.CV, October 2023; MSC 30K15, 30B30, 30E10; supported by the Irish Research Council and by the French Mini…