On the divergence bounded computable real numbers

Abstract

For any function h : ℕ → ℕ, we call a real number x h-bounded computable (h-bc for short) if there is a computable sequence (xs) of rational numbers which converges to x such that, for any n ∈ ℕ, there are at most h(n) pairs of non-overlapped indices (i, j) with |xi - xj| ≥ 2-n. In this paper we investigate h-bc real numbers for various functions h. We will show a simple sufficient condition for class of functions such that the corresponding h-bc real numbers form a field. Then we prove a hierarchy theorem for h-bc real numbers. Besides we compare the semi-computability and weak computability with the h-bounded computability for special functions h. © Springer-Verlag Berlin Heidelberg 2003.