# Sarnak's Conjecture for Sequences of Almost Quadratic Word Growth

## TL;DR

This paper proves the logarithmic Sarnak conjecture for sequences with subquadratic word growth, demonstrating that the Liouville function exhibits complex sign patterns and does not locally correlate with such sequences, under certain conjectural conditions.

## Contribution

It establishes the logarithmic Sarnak conjecture for sequences of subquadratic word growth and introduces a conditional result linking Fourier uniformity conjectures to the behavior of the Liouville function.

## Key findings

- Liouville function has at least quadratically many sign patterns
- Sequences with subquadratic word growth do not locally correlate with multiplicative functions
- Conditional results depend on Fourier uniformity conjectures

## 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 $< 1$.