I've already red this marvelous article on arithmetic progressions in primes. Among ingenious ideas presented to prove the Generalized Szemeredi Theorem, the notion of $\cal D F$ , defined by $$\mathbb{\cal D} F = \mathop{\mathbb{E}}\left(\prod_{\omega \in \{0,1\}^{k - 1} : \omega \neq {0^{k - 1}} }F(x + \omega.h)\bigg| h \in \mathbb{Z}_N ^{k - 1} \right)$$ had a prominent role in proving this part. However, I failed to capture the intuition behind this important definition. I believe there should be a definition like this in other areas(maybe in Functional analysis or ergodic theory). Have anyone seen such a definition before?
Here is a link to the proof (my question is from 6'th section): annals.math.princeton.edu/wp-content/uploads/annals-v167-n2-p03.pdf