On Elliott's conjecture and applications

Abstract

Let f:N→D$f:\mathbb {N}\rightarrow \mathbb {D}$ be a multiplicative function. Under the merely necessary assumption that f$f$ is nonpretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts h1,h2$h_1,h_2$ , the two‐point correlation 1x∑n⩽xf(n+h1)f¯(n+h2)$$\begin{equation*} \frac{1}{x}\sum _{n\leqslant x}{f(n+h_1)\overline{f}(n+h_2)} \end{equation*}$$tends to 0 along a set of x∈N$x\in \mathbb {N}$ of full upper logarithmic density. We also show that the same result holds for the k$k$ ‐point correlations 1x∑n⩽xf(n+h1)⋯f(n+hk)$$\begin{equation*} \frac{1}{x}\sum _{n\leqslant x}{f(n+h_1)\cdots f(n+h_k)} \end{equation*}$$if k$k$ is odd and f$f$ is a real‐valued nonpretentious function. Previously, the vanishing of correlations was known only under stronger nonpretentiousness hypotheses on f$f$ by the works of Tao, and Tao and the third author. We derive several applications, including: A classification of ±1$\pm 1$ ‐valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. A proof that a class of “Liouville‐like” functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. Constructing examples of multiplicative f:N→{−1,0,1}$f:\mathbb {N}\rightarrow \lbrace -1,0,1\rbrace$ with a given (unique) Furstenberg system, answering a question of Lemańczyk. A density version of the Erdős discrepancy theorem of Tao.