On the query complexity for Showing Dense Model

A theorem of Green, Tao, and Ziegler can be stated as follows: if R is a pseudorandom distribution, and D is a dense distribution of R, then D can be modeled as a distribution M which is dense in uniform distribution such that D and M are indistinguishable. The reduction involved in the proof has exponential loss in the distinguishing probability. Reingold et al give a new proof of the theorem with polynomial loss in the distinguishing probability. In this paper, we are focus on query complexity for showing dense model, and then give a optimal bound of the query complexity. We also follow the connection between Impagliazzo’s Hardcore Theorem and Tao’s Regularity lemma, and obtain a proof of L2-norm version Hardcore Theorem via Regularity lemma.