Sarnak's Conjecture for Sequences of Almost Quadratic Word Growth

Abstract: We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many sign patterns which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $\kappa-1$-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon})$ many words of length $n$ where $t = \kappa(\kappa+1)/2$. We prove a variant of the $1$-Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension $< 1$.