__
TR11-038 | 10th March 2011 14:32
__

#### On the query complexity for Showing Dense Model

**Abstract:**
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 $L_{2}$-norm version Hardcore Theorem via Regularity lemma.