Simultaneous zero-free approximation and universal optimal polynomial approximants

Let E be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space H² for which the partial sums of its Taylor series approximate any continuous function on E. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in H² instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to the Dirichlet space D and are expected for more general Dirichlet-type spaces.