A Perfect Set of Reals with Finite Self-Information

TL;DR

This paper constructs a perfect set of reals with finite mutual information, answering open questions and linking Kolmogorov complexity, effective dimensions, and computability properties.

Contribution

It provides a novel construction of a perfect Pi^0_1 set of reals with finite self-information, advancing understanding of Kolmogorov complexity and effective dimension.

Findings

Constructed a perfect Pi^0_1 set with finite self-information.

Showed the set can be low for effective Hausdorff and packing dimensions.

Extended the construction to a single perfect set satisfying complexity bounds for a class of functions.

Abstract

We examine a definition of the mutual information of two reals proposed by Levin. The mutual information is I(X:Y)=log(sum(2^{K(s)-K^X(s)+K(t)-K^Y(t)-K(s,t)}, (s,t) pairs of finite binary strings), where K is the prefix-free Kolmogorov complexity. A real X is said to have finite self-information if I(X:X) is finite. We give a construction for a perfect Pi^0_1 set of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals with K(s)<= K^{A}(s)+f(s)+c for a certain constant c and for any given Delta^0_2 f with a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Pi^0_1 set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfy K(s) <= K^A(s)+f(s)+c_f, where c_f is a constant that depends on f, for all f in a `nice' class of Delta^0_2 functions which includes all recursive orders.