Orders of oscillation motivated by Sarnak’s conjecture
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- by Yunping Jiang PDF
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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.References
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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
- MR Author ID: 238389
- Email: yunping.jiang@qc.cuny.edu
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3973908