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

TL;DR

This paper introduces the concept of higher order oscillating sequences and demonstrates their disjointness from affine distal flows on tori, providing new evidence supporting Sarnak's conjecture for these systems.

Contribution

It defines higher order oscillating sequences and proves their linear disjointness from affine distal flows on tori, confirming Sarnak's conjecture in these settings.

Findings

Higher order oscillating sequences are linearly disjoint from affine distal flows on the $d$-torus.

Higher order oscillating sequences of order 2 are disjoint from affine flows with zero entropy on the 2-torus.

The results support Sarnak's conjecture for all affine flows with zero entropy on the 2-torus and affine distal flows on higher-dimensional tori.

Abstract

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 $d$ is linearly disjoint from all affine distal flows on the $d$-torus for all $d\geq 2$. 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 and for all affine distal flows on the $d$-torus for all $d\geq 2$.