Pseudorandomness in computer science and in additive combinatorics

Abstract

Dedicated to Endre Szemeré;di on the occasion of his 70th birthday Notions of pseudorandomness and (implicitly) of indistinguishability arise in several key results in additive combinatorics. In this expository paper, we show how several results can be translated from the analytic language of norms, decompositions, and transference to the computer science language of indistinguishability, simulability and pseudoentropy. Some of these results, once so reformulated, can be given “computer science proofs” which are quantitatively better in some respects, and which have some applications. We discuss variants of the Szeméredi regularity lemma for graphs; the inverse theorems for the Gowers uniformity norms; a key step in the “transference” results of Green, Tao and Ziegler; and various “decomposition” results.