Furstenberg Systems of Bounded Multiplicative Functions and Applications

Abstract

We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the Möbius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Teräväinen related to the odd order cases of the Chowla conjecture.

Value Patterns of Multiplicative Functions and Related Sequences

“…But this follows from Lemma 3.1 (noting that δ > c k > 1/2 and hence (1 − δ) + (1 − δ) + δ − 1 ≤ 1 + 2 min(δ, 1 − δ)). 12 Here it is essential that there are only three factors in the average considered here, so that the average is of "complexity one" and can thus be controlled by the Kronecker factor. The same is not true for the original average (2.14), but we will not need to directly pass to characteristic factors for that average.…”

Correlations of multiplicative functions along deterministic and independent sequences

“…The next result is a crucial element in the proof of Theorem 1.1 and follows by combining the structural result of [8,Theorem 1.5] with the disjointness statement of [7,Proposition 3.12]. Combining the previous two results we can now prove Theorem 3.1.…”

“…Proof strategy. To prove Theorem 1.1 we make essential use of a structural result from [7,8] for measure preserving systems (called Furstenberg systems) naturally associated with arbitrary multiplicative functions f 1 , . .…”

“…Some open problems. In [7,8] it was shown that any F-system of a collection of multiplicative functions f 1 , . .…”

“…In order to carry out the needed reduction we will make crucial use of certain identities satisfied by F-systems of multiplicative functions. They are based on work in [23] and [26] and are proved in [8,Theorem 3.8]. Henceforth, for d ∈ N we let P d := P ∩ (dZ + 1).…”