2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2008)

Oct. 25, 2008 to Oct. 28, 2008

ISSN: 0272-5428

ISBN: 978-0-7695-3436-7

pp: 76-85

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2008.38

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 $D$ maybe 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'' ordistributions rather than sets.) The proof of this theorem is very general,and it applies to notions of pseudorandomness and indistinguishability definedin terms of any family of distinguishers with some mild closure properties.% \snote{added `with appropriate closure properties'}The proof proceeds via iterative partitioningand an energy increment argument, in the spirit of the proof of theweak Szemer\'edi regularity lemma. The ``reduction'' involved in the proofhas exponential complexity in the distinguishing probability.We present a new proof inspired by Nisan's proof of Impagliazzo's hardcoreset theorem. The reduction in our proof has polynomial complexity in thedistinguishing probability and provides a new characterization of thenotion of ``pseudoentropy'' of a distribution. A proof similar to ours hasalso been independently discovered by Gowers \cite{G08}.We also follow the connection between the two theorems and obtain a new proof ofImpagliazzo's hardcore set theorem via iterative partitioning andenergy increment. While our reduction has exponential complexity in someparameters, it has the advantage that the hardcore set is efficiently recognizable.

INDEX TERMS

pseudorandomness, additive combinatorics, regularity lemmas, pseudoentropy

CITATION

Salil Vadhan,
Omer Reingold,
Madhur Tulsiani,
Luca Trevisan,
"Dense Subsets of Pseudorandom Sets",

*2013 IEEE 54th Annual Symposium on Foundations of Computer Science*, vol. 00, no. , pp. 76-85, 2008, doi:10.1109/FOCS.2008.38SEARCH