We obtain the following full characterization of constructive dimension in terms of algorithmic information content. For every sequence A, cdim(A)=liminf_n (K(A[0..n-1])/n. more >>>

The problem of predicting a sequence $x_1, x_2,.... $ where each $x_i$ belongs to a finite alphabet $A$ is considered. Each letter $x_{t+1}$ is predicted using information on the word $x_1, x_2, ...., x_t $ only. We use the game theoretical interpretation which can be traced to Laplace where there ... more >>>

We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in constructively given subsets of the Cantor ... more >>>

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $\beta$ is ... more >>>