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 words 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 less than $1$ .

中文翻译：

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Sarnak 关于几乎二次词增长序列的猜想

我们证明了二次词增长序列的对数 Sarnak 猜想。特别是，我们证明了刘维尔函数至少具有二次方的符号模式。我们从一个变体中推导出主要定理，该变体将乘法函数和序列之间的相关性与以正对数密度出现的亚二次词的相关性联系起来。这使我们能够实际证明我们的乘法函数与亚二次词增长序列不局部相关。我们还证明了一个条件结果，表明如果 ($\卡帕-1$)-傅里叶均匀性猜想成立，则刘维尔函数与序列不相关$O(n^{t-\varepsilon })$许多长度的单词n在哪里$t = \kappa (\kappa +1)/2$. 我们证明了$1$-傅里叶均匀性猜想，其中频率被限制在任何一组小于$1$.