Characterize attainable Kolmogorov-complexity profiles

Determine whether every function satisfying properties (P1)–(P4) is equal, up to an additive O(log n) term (or at least an additive o(n) term), to the complexity profile g_x(r) of some binary string x of length n.

Background

The paper defines the complexity profile of a binary string x by g_x(r)=max{C(y): y is within Hamming distance r of x}. It proves four universal properties of this profile: monotonicity, its initial value and eventual size, an upper growth bound, and a lower growth implication. The authors also show that the minimal and maximal functions compatible with these properties are attained, up to O(log n), for all radii simultaneously.

The unresolved problem is whether these necessary properties are also sufficient to characterize all attainable profiles. The authors note that their pointwise realization theorem controls one radius, and that certain constructions realize an entire initial segment, but that combining different growth rates at different scales remains difficult.