The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

We study the asymptotic behaviour of higher order correlations≤X/d g1(n + ah1)· · · gk (n + ahk ) a function of the parameters a and d, where g1, . . . , gk are bounded multiplicative functions, h1, . . . , hk integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations vanish for almost all X if g1 · · · gk does not (weakly) pretend to be a twisted Dirichlet n 7→ χ (n)n, and behave asymptotically like a multiple of d− itχ (a) otherwise. This extends earlier work on the structure of logarithmically averaged correlations, in which the d parameter is out and one can set t = 0. Among other things, the result enables us to establish special cases the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we the k-point Chowla conjecture En≤X λ(n + h1)· · · λ(n + hk ) = o(1) for k odd or equal to 2 for scales X outside of a set of zero logarithmic density