  Green, Tao and Ziegler 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 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 and Gowers. We refer
to our full paper for variants of the result with connections and applications
to computational complexity theory, and to Gowers' paper 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).
