In this paper we derive several results which generalise the constructive
dimension of (sets of) infinite strings to the case of exact dimension. We
start with proving a martingale characterisation of exact Hausdorff
dimension. Then using semi-computable super-martingales we introduce the
notion of exact constructive dimension ... more >>>

The Kolmogorov complexity function of an infinite word $\xi$ maps a natural
number to the complexity $K(\xi|n)$ of the $n$-length prefix of $\xi$. We
investigate the maximally achievable complexity function if $\xi$ is taken
from a constructively describable set of infinite words. Here we are
interested ... more >>>

The present paper generalises results by Tadaki [12] and Calude et al. [1] on oscillation-free partially random infinite strings. Moreover, it shows that oscillation-free partial Chaitin randomness can be separated from scillation-free partial strong Martin-L\"of randomness by $\Pi_{1}^{0}$-definable sets of infinite strings.

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The present paper generalises results by Lutz and Ryabko. We prove a
martingale characterisation of exact Hausdorff dimension. On this base we
introduce the notion of exact constructive dimension of (sets of) infinite
strings.

Furthermore, we generalise Ryabko's result on the Hausdorff dimension of the
... more >>>