The Green–Tao theorem

Step 1 (the obstruction). The von Mangoldt function $\Lambda(n)$ (equal to $\log p$ when $n=p^j$ and 0 otherwise) is the natural weight for detecting primes, with average size $\mathbb{E}[\Lambda] \approx 1$. But $\Lambda$ itself is not bounded, and $\mathcal{P}$ has density 0, so no classical density theorem applies to it directly.

Step 2 (a pseudorandom majorant). Using ideas from Goldston–Yıldırım-type sieve weights, Green and Tao construct a measure $\nu(n) \ge 0$ with $\mathbb{E}[\nu] \approx 1$ that dominates the primes, $\Lambda(n) \le K\nu(n)$ for a constant $K$, and that is pseudorandom: it satisfies precise linear-forms and correlation conditions modeled on what a genuinely random set of the same density would satisfy.

Step 3 (relative Szemerédi theorem). Green and Tao prove that any function $0 \le f \le \nu$ with positive relative density, $\mathbb{E}[f] \ge \delta$, still contains the expected density of $k$-term progressions, provided $\nu$ is pseudorandom. The proof decomposes $f$ into a bounded, structured part plus a part that is small in the Gowers uniformity norms relative to $\nu$; the uniform part contributes negligibly to the progression count by a generalized von Neumann theorem, so the structured part alone must already account for the expected progressions — exactly as in the classical hypergraph-regularity proof of Szemerédi's theorem, but relativized to $\nu$.

Step 4 (putting it together). Verifying that the Goldston–Yıldırım-type $\nu$ really is pseudorandom (checking the linear-forms and correlation conditions using standard prime-counting estimates) lets Step 3 apply to $f = \Lambda / K$, which has positive relative density since $\mathbb{E}[\Lambda] \approx 1$. This yields a positive relative density of $k$-term progressions weighted by $\Lambda$, and hence — after removing the negligible contribution from prime powers — an honest $k$-term progression of primes, for every $k$.