Asymptotics of the proportion of stochastic objects

Determine the asymptotic behavior, as k tends to infinity, of N(k,α,β)/2^k, where N(k,α,β) denotes the number of (α,β)-stochastic finite objects of complexity at most k, including regimes such as α,β=εk and α,β=√k.

Background

The appendix reproduces problems posed by Andrey Kolmogorov in a 1981 seminar. It defines N(k,α,β) as the number of (α,β)-stochastic objects whose complexity is at most k and asks for the asymptotics of the normalized quantity N(k,α,β)/2^k as k grows. The paper notes that Kolmogorov’s definition may have intended plain rather than prefix complexity and may contain a typographical error: N(k,α,β) may have been intended to count nonstochastic rather than stochastic objects. Thus, the unresolved asymptotic problem concerns the interpretation and behavior of this proportion under the relevant complexity convention.