On Average Time Hierarchies

For a time-constructible function T we give an explicit language LT which can be recognized in time T (n). We prove that any Turing machine that recognizes LT requires time close to T (n) for most inputs, thus forming an average time hierarchy. The existence of the average time hierarchy was known, but depended on languages de ned by complicated diagonalization. We also give simple proofs for the known stricter hierarchy for functions.