Orders of Oscillation Motivated by Sarnak's Conjecture, Part II

Yunping Jiang

Published 2022-01-21Version 1

I have investigated orders of oscillating sequences motivated by Sarnak's conjecture in~\cite{JPAMS} and proved that an oscillating sequence of order $d$ is linearly disjoint from affine distal flows on the $d$-torus. One of the consequences is that an oscillating sequence of order $d$ in the arithmetic sense is linearly disjoint from affine flows with zero topological entropy on the $d$-torus. In this paper, I will extend these results to polynomial skew products on the $d$-torus, that is, given a polynomial skew product on the $d$-torus, there is a positive integer $m$ such that any oscillating sequence of order $m$ is linearly disjoint from this polynomial skew product. In particular, when all polynomials depend only on the first variable, I have that an oscillating sequence of order $m=d+k-1$ is linearly disjoint from all polynomial skew products on the $d$-torus with polynomials of degree less than or equal to $k$. One of the consequences is the linear disjointness for flows which are automorphisms of the $d$-torus with absolute values of eigenvalues $1$ plus a polynomial vector and oscillating sequences of order $m$ in the arithmetic sense. Furthermore, I will prove that an oscillating sequence of order $d$ is linearly disjoint from minimal mean attractable and minimal quasi-discrete spectrum of order $d$ flows. Finally, I define and give some examples of Chowla sequences from our paper~\cite{AJ}.