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 ( x s ) of rational numbers which converges to x h ‐monotonically in the sense that h ( n )| x – x n | ≥ | x – x m | 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 Σ i ∈ℕ (1 – h ( i )) = ∞and the sum Σ 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 . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)