Effective Hausdorff Dimension in General Metric Spaces

We introduce the concept of effective dimension for a wide class of metric spaces whose metric is not necessarily based on a measure. Effective dimension was defined by Lutz (Inf. Comput., 187(1), 49–79, 2003) for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses gambling, in particular the concept of gale and supergale, our extension of Hausdorff dimension to other metric spaces is also based on a supergale characterization of dimension, which in practice avoids an extra quantifier present in the classical definition of dimension that is based on Hausdorff measure and therefore allows effectivization for small time-bounds. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity, for which we extend the concept of Kolmogorov complexity to any metric space defining the Kolmogorov complexity of a point at a certain precision. Further research directions are indicated

Random Noise Increases Kolmogorov Complexity and Hausdorff Dimension

International audienceConsider a bit string x of length n and Kolmogorov complexity αn (for some α < 1). It is always possible to increase the complexity of x by changing a small fraction of bits in x [2]. What happens with the complexity of x when we randomly change each bit independently with some probability τ ? We prove that a linear increase in complexity happens with high probability, but this increase is smaller than in the case of arbitrary change considered in [2]. The amount of the increase depends on x (strings of the same complexity could behave differently). We give exact lower and upper bounds for this increase (with o(n) precision). The same technique is used to prove the results about the (effective Hausdorff) dimension of infinite sequences. We show that random change increases the dimension with probability 1, and provide an optimal lower bound for the dimension of the changed sequence. We also improve a result from [5] and show that for every sequence ω of dimension α there exists a strongly α-random sequence ω such that the Besicovitch distance between ω and ω is 0. The proofs use the combinatorial and probabilistic reformulations of complexity statements and the technique that goes back to Ahlswede, Gács and Körner [1]

Dimension Spectra of Lines

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim(a, b) is equal to the effective packing dimension Dim(a, b), then sp(L) contains a unit interval. We also show that, if the dimension dim(a, b) is at least one, then sp(L) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite

Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. 2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y) and Mdim(x:y)

Around Kolmogorov complexity: basic notions and results

Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author

The Dimensions of Individual Strings and Sequences

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0,1]. Sequences that are random (in the sense of Martin-Lof) have dimension 1, while sequences that are decidable, \Sigma^0_1, or \Pi^0_1 have dimension 0. It is shown that for every \Delta^0_2-computable real number \alpha in [0,1] there is a \Delta^0_2 sequence S such that \dim(S) = \alpha. A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit infimum of the dimensions of its prefixes. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit infimum of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number \beta in (0,1) is shown to have dimension \H(\beta), the binary entropy of \beta.Comment: 31 page

Projection Theorems Using Effective Dimension

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand\u27s projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand\u27s theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand\u27s theorem using the theory of computing