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Orders of oscillation motivated by Sarnak's conjecture


Author: Yunping Jiang
Journal: Proc. Amer. Math. Soc. 147 (2019), 3075-3085
MSC (2010): Primary 37A35, 11K65; Secondary 37A25, 11N05
DOI: https://doi.org/10.1090/proc/14487
Published electronically: April 3, 2019
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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.


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Additional Information

Yunping Jiang
Affiliation: Queens College of the City University of New York, Flushing, New York 11367-1597; Department of Mathematics, Graduate School of the City University of New York, 365 Fifth Avenue, New York, New York 10016
Email: yunping.jiang@qc.cuny.edu

DOI: https://doi.org/10.1090/proc/14487
Received by editor(s): April 3, 2018
Received by editor(s) in revised form: July 28, 2018, September 13, 2018, and November 1, 2018
Published electronically: April 3, 2019
Additional Notes: This material is based upon work supported by the National Science Foundation. It was also partially supported by a collaboration grant from the Simons Foundation (grant number 523341) and PSC-CUNY awards and a grant from NSFC (grant number 11571122).
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society