  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 $<
1$.
