REGAININGLY APPROXIMABLE NUMBERS AND SETS
Peter H. Hertling, RUPERT HÖLZL, Philip Janicki · Journal of Symbolic Logic · 2024
Abstract We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n n}$ for infinitely many n . Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.