New Proofs of the Green–Tao–Ziegler Dense Model Theorem: An Exposition

Green, Tao and Ziegler [GT, TZ] prove “Dense Model Theorems” of the following form: if R is a (possibly very sparse) pseudorandom subset of set X , and D is a dense subset of R, then D may be modeled by a set M whose density inside X is approximately the same as the density of D in R. More generally, they show that a function that is majorized by a pseudorandom measure can be written as a sum of a bounded function having the same expectation plus a function that is “indistinguishable from zero.” This theorem plays a key role in the proof of the Green–Tao Theorem [GT] that the primes contain arbitrarily long arithmetic progressions. In this note, we present a new proof of the Green–Tao–Ziegler Dense Model Theorem, which was discovered independently by ourselves [RTTV] and Gowers [Gow]. Our presentation follows the argument in [RTTV] (which in turn was inspired by Nisan’s proof of the Impagliazzo Hardcore Set Theorem [Imp]), but is translated to the original notation of Green, Tao, and Ziegler. We refer to our full paper [RTTV] for variants of the result with connections and applications to computational complexity theory, and to Gowers’ paper [Gow] for applications of the proof technique to “decomposition, “structure,” and “transference” theorems in arithmetic and extremal combinatorics (as well as a broader survey of such theorems). 1 The Green-Tao-Ziegler Theorem Let X be a finite universe. We use the notation Ex∈X f(x) := 1 |X| ∑ x∈X f(x). For two functions f, g : X → R we define their inner product as 〈f, g〉 := E x∈X f(x)g(x) Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel. [email protected]. Research supported by US-Israel Binational Science Foundation grant 2006060. Computer Science Division, U.C. Berkeley. [email protected]. Work partly done while visiting Princeton University and the IAS. This material is based upon work supported by the National Science Foundation under grants CCF-0515231 and CCF-0729137 and by the US-Israel Binational Science Foundation under grant 2006060. Computer Science Division, U.C. Berkeley. [email protected] Work partly done while visiting Princeton University. This material is based upon work supported by the National Science Foundation under grants CCF0515231 and CCF-0729137 and by the US-Israel Binational Science Foundation under grant 2006060. School of Engineering and Applied Sciences, Harvard University. [email protected]. Work done during a visit to U.C. Berkeley, supported by the Miller Foundation for Basic Research in Science, a Guggenheim Fellowship, US-Israel Binational Science Foundation grant 2006060, and the Office of Naval Research grant N00014-04-1-0478.