H-monotonically Computable Real Numbers

Let h : N −→ Q be a computable function. A real number x is called h-monotonically computable (h-mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x h-monotonically in the sense that h(n)|x− xn| ≥ |x− xm| for all n and m > n. In this paper we investigate classes h-MC of h-mc real numbers for different computable functions h. Especially, for computable functions h : N −→ (0, 1)Q , we show that the class h-MC coincides with the classes of computable and semi-computable real numbers if and only if ∑ i∈N(1− h(i)) = ∞ and the sum ∑ i∈N(1− h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then h-MC = SC (the class of semi-computable reals) no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h-MC = c-MC . Finally, if h is monotone and unbounded, then h-MC contains all ω-mc real numbers which are g-mc for some computable function g.