A theorem of Green, Tao, and Ziegler can be stated (roughly)
as follows: if R is a pseudorandom set, and D is a dense subset of R,
then D may
be modeled by a set M that is dense in the entire domain such that D and
M are indistinguishable. (The precise statement refers to``measures'' or
distributions rather than sets.) The proof of this theorem is very
general, and it applies to notions of pseudorandomness and
indistinguishability defined in terms of any family of distinguishers
with some mild closure properties.
The proof proceeds via iterative partitioning and an energy increment
argument, in the spirit of the proof of the weak Szemeredi regularity
lemma. The ``reduction'' involved in the proof has exponential
complexity in the distinguishing probability.
We present a new proof inspired by Nisan's proof of Impagliazzo's
hardcore set theorem. The reduction in our proof has polynomial
complexity in the distinguishing probability and provides a new
characterization of the notion of ``pseudoentropy'' of a distribution.
We also follow the connection between the two theorems and obtain a new
proof of Impagliazzo's hardcore set theorem via iterative partitioning
and energy increment. While our reduction has exponential complexity in
some parameters, it has the advantage that the hardcore set is