Time-space tradeoffs in the counting hierarchy

We extend the lower bound techniques of [14], to the unbounded-error probabilistic model. A key step in the argument is a generalization of Nepomnjascii's theorem front the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspace-uniform TO circuits for iterated multiplication [9]. Here is an example of the sort of lower bounds that we obtain: we show that MAJ.MAJSAT is not contained in PrTiSp(n(1+o(1)), n(epsilon)) for any epsilon < 1. We also extend a lower bound of [14], from showing that SAT does not have uniform NC1 circuits of size n(1+o(1)), to a similar result for SAC(1) circuits.