Higher Order Oscillating Sequences, Affine Distal Flows on the $d$-Torus, and Sarnak's Conjecture

Yunping Jiang

Published 2016-12-13Version 1

In this paper, we give two precise definitions of a higher order oscillating sequence and show the importance of this concept in the study of Sarnak's conjecture. We prove that any higher order oscillating sequence of order $2$ is linearly disjoint from all affine distal flows on the $2$-torus. One consequence of this result is that any higher order oscillating sequence of order $2$ is linearly disjoint from all affine flows on the $2$-torus with zero topological entropy. In particular, this reconfirms Sarnak's conjecture for all affine flows on the $2$-torus with zero topological entropy. Furthermore, for $d>2$, we prove that any higher order oscillating sequence of order $d$ is linearly disjoint from all triangularizable affine distal flows on the $d$-torus. Thus it reconfirms Sarnak's conjecture for all triangularizable affine distal flows on the $d$-torus.