# Dense Subsets of Pseudorandom Sets

Omer Reingold, Luca
Trevisan, Madhur Tulsiani, and Salil Vadhan

### Abstract

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 Szemerédi 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. A proof similar to ours has also been independently discovered by
Gowers.

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
efficiently recognizable.

### Versions

*Electronic Colloquium on Computational Complexity*, Technical
Report TR08-045. [pdf][official
ECCC page]
- “New Proofs of the
Green-Tao-Ziegler Dense Model Theorem: An Exposition,” [pdf][arXiv:0806.0381v1]
- To appear in
*Proceedings of the 49th Annual IEEE
Symposium onFoundations of Computer Science (FOCS `08).* IEEE, 26-28
October 2008. [pdf]

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