This paper shows that classical results about complexity classes invol
ving ''delayed diagonalization'' and ''gap languages'', such as Ladner
's Theorem and Schoning's Theorem and independence results of a kind n
oted by Schoning and Hartmanis, apply at very low levels of complexity
, indeed all the way down in Sipser's log-time hierarchy. This paper a
lso investigates refinements of Sipser's classes and notions of log-ti
me reductions, following on from recent work by Cai, Chen, and others.