Fine Separation of Average-Time Complexity Classes

We extend Levin's definition of average polynomial time to arbitrary time-bounds in accordance with the following general principles: (1) It essentially agrees with Levin's notion when applied to polynomial time-bounds. (2) If a language L belongs to DTIME(T(n)) for some time-bound T(n), then every distributional problem $(L,\mu)$ is T on the $\mu$-average. (3) If L does not belong to DTIME(T(n)) almost everywhere, then no distributional problem $(L,\mu)$ is T on the $\mu$-average. We present hierarchy theorems for average-case complexity, for arbitrary time-bounds, that are as tight as the well-known Hartmanis--Stearns hierarchy theorem for deterministic complexity. As a consequence, for every time-bound T(n), there are distributional problems $(L,\mu)$ that can be solved using only a slight increase in time but that cannot be solved on the $\mu$-average in time T(n).