On Elliott's conjecture and applications

Source abstract

Abstract Let be a multiplicative function. Under the merely necessary assumption that is nonpretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts , the two‐point correlation tends to 0 along a set of of full upper logarithmic density. We also show that the same result holds for the ‐point correlations if is odd and is a real‐valued nonpretentious function. Previously, the vanishing of correlations was known only under stronger nonpretentiousness hypotheses on by the works of Tao, and Tao and the third author. We derive several applications, including: A classification of ‐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 with a given (unique) Furstenberg system, answering a question of Lemańczyk. A density version of the Erdős discrepancy theorem of Tao.