Green–Tao Möbius and nilsequences conjecture

Let $s\geqslant 1$. An $s$-step nilmanifold is a quotient $G/\Gamma=(G/\Gamma,d_{G/\Gamma})$ with smooth metric, and an $s$-step nilsequence is a sequence of the form $(F(g^n x))_{n\in[N]}$. Let $\mu$ denote the Möbius function. Green–Tao Möbius and nilsequences conjecture. For every real $A>0$,

$$\left|\mathbb{E}_{n\leqslant N}\mu(n)F(g^n x)\right|\ll_{A,M,G/\Gamma,s}\log^{-A}N.$$

Thus the Möbius function has negligible correlation with every bounded $s$-step nilsequence of fixed Lipschitz complexity. The conjecture was open in general in the paper; the case $s=2$ was subsequently settled.