A Perfect Set of Reals with Finite Self-Information (1210.7779v1)
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)-KX(s)+K(t)-KY(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 Pi0_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 Delta0_2 f with a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Pi0_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) <= KA(s)+f(s)+c_f, where c_f is a constant that depends on f, for all f in a `nice' class of Delta0_2 functions which includes all recursive orders.