Orders of Oscillation Motivated by Sarnak's Conjecture--Part II
Yunping Jiang
TL;DR
This work extends Sarnak-type linear disjointness to nonlinear dynamics and higher-order oscillations. It proves that oscillating sequences of order $m=d+k-1$ are disjoint from simple polynomial skew products of degree $k$ on $\mathbb{T}^d$, and that oscillating sequences of order $d$ are disjoint from MMA and MQDS$(d)$ systems, while introducing the generalized linear disjointness framework and the notion of multi-linearly disjoint sequences. It also constructs examples of multi-linearly disjoint sequences, notably of the form $c_n=e^{2\pi i(\alpha \beta^{n} g(\beta))}$ with $g\in C^{2}_{+}((1,\infty))$, showing these can have positive entropy while remaining multi-linearly disjoint. Overall, the paper broadens the landscape of arithmetic-dynamical disjointness to nonlinear, higher-order, and multi-linear settings, offering tools and constructions relevant to Chowla-type phenomena and beyond.
Abstract
This work is a continuation of [13]. We study the linear disjointness between higher-order oscillating sequences and nonlinear dynamical systems. Specifically, we prove that any oscillating sequence of order $m=d+k-1$ and any simple polynomial skew product of degree $k$ on the $d$-Euclidean space are linearly disjoint. Additionally, we demonstrate that any oscillating sequence of order $d$ and any minimal mean attractable and minimal quasi-discrete spectrum dynamical system of order $d$ are linearly disjoint. Finally, we introduce multi-linearly disjoint sequences and construct examples of such sequences.