Some results on asymptotic versions of Mahler's problems

In this paper, we show the existence of a transcendental function $f\in \mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we demonstrate that certain subsets of algebraic numbers are exceptional sets of some transcendental function $f\in \mathbb{Z}\{z\}$ with almost all bounded coefficients.