Truncated pretentious distances of multiplicative arithmetic functions

Let $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{U}$ be a non-pretentious multiplicative function in the sense of Granville and Soundararajan. We study the convergence of the autocorrelation averages of $f$. In particular, we investigate the relationship between the local pretentious behaviour of $f$ on intervals of the form $[x^{\epsilon },x]$ and its deviation from $1$ on the primes. As an application, we exhibit a class of non-pretentious multiplicative functions for which Elliott’s conjecture in its original formulation holds, following up on a result by Klurman, Mangerel and Teräväinen.