Orders of oscillation motivated by Sarnak’s conjecture

Abstract:

In view of Sarnak’s conjecture in number theory, we investigate orders of oscillating sequences. For oscillating sequences (of order $1$), we have proved that they are linearly disjoint from all MMA and MMLA flows. We define oscillating sequences of order $d$ and oscillating sequences of order $d$ in the arithmetic sense for $d\geq 2$ in this paper. Moreover, we prove that oscillating sequences of order $d$ are linearly disjoint from all affine distal flows as well as all nonlinear affine distal flows with Diophantine translations on the $d$-torus. We prove that oscillating sequences of order $d$ in the arithmetic sense are linearly disjoint from all nonlinear distal flows with rational translations on the $d$-torus, too. Furthermore, the linear disjointness of oscillating sequences of order $d$ in the arithmetic sense from other affine flows with zero topological entropy as well as associated nonlinear flows with Diophantine translations on the $d$-torus can be treated as a consequence of the main result in this paper. One of the consequences is that Sanark’s conjecture holds for all the flows discussed in this paper.