Counting the Changes of Random ${\Delta^0_2}$ Sets

Abstract

Consider a Martin-Löf random ${\Delta^0_2}$ set Z. We give lower bounds for the number of changes of $Z_s \upharpoonright n$ for computable approximations of Z. We show that each nonempty ${\Pi^0_1}$ class has a low member Z with a computable approximation that changes only o(2ⁿ) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that $Z_s\upharpoonright n$ changes more than c 2ⁿ times.