PP-lowness and a simple definition of AWPP

We show that the counting classes AWPP and APP [FFKL], [L] are more robust than previously thought. Our results identify a sufficient condition for a language to be low for PP, and we show that this condition is at least as weak as other previously studied criteria. We extend a result of Kobler et al. by proving that all sparse co-C=P languages are in APP, and are thus PP-low. Our results also imply that AWPP subset of or equal to APP, and thus APP contains many other established subclasses of PP-low, thereby reducing several different lowness results to membership in APP. We also show that AWPP and APP are Sigma(2)(0)-definable classes. Some of our results are reminiscent of amplifying certainty in probabilistic computation.