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

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 $d\geq 2$.