Structure of multicorrelation sequences with integer part polynomial iterates along primes
Authors:
Andreas Koutsogiannis, Anh N. Le, Joel Moreira and Florian K. Richter
Journal:
Proc. Amer. Math. Soc. 149 (2021), 209-216
MSC (2010):
Primary 37A45, 37A15; Secondary 11B30
DOI:
https://doi.org/10.1090/proc/15185
Published electronically:
October 16, 2020
MathSciNet review:
4172598
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let $T$ be a measure-preserving $\mathbb {Z}^\ell$-action on the probability space $(X,{\mathcal B},\mu ),$ let $q_1,\dots ,q_m\colon \mathbb {R}\to \mathbb {R}^\ell$ be vector polynomials, and let $f_0,\dots ,f_m \in L^\infty (X)$. For any $\epsilon > 0$ and multicorrelation sequences of the form $\alpha (n) =\int _Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu$ we show that there exists a nil- sequence $\psi$ for which $\lim _{N - M \to \infty } \frac {1}{N-M} \sum _{n=M}^{N-1} |\alpha (n) - \psi (n)| \leq \epsilon$ and $\lim _{N \to \infty } \frac {1}{\pi (N)} \sum _{p \in \mathbb {P}\cap [1,N]} |\alpha (p) - \psi (p)| \leq \epsilon .$ This result simultaneously generalizes previous results of Frantzikinakis and the authors.
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Additional Information
Andreas Koutsogiannis
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio
MR Author ID:
974679
Email:
koutsogiannis.1@osu.edu
Anh N. Le
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois
ORCID:
0000-0003-4928-4932
Email:
anhle@math.northwestern.edu
Joel Moreira
Affiliation:
Mathematics Institute, University of Warwick, Coventry, United Kingdom
MR Author ID:
1091663
Email:
joel.moreira@warwick.ac.uk
Florian K. Richter
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois
MR Author ID:
1147216
Email:
fkr@northwestern.edu
Keywords:
Multicorrelation sequences,
nilsequences,
integer part polynomials,
prime numbers
Received by editor(s):
April 24, 2020
Published electronically:
October 16, 2020
Additional Notes:
The fourth author was supported by the National Science Foundation under grant number DMS 1901453.
Communicated by:
Katrin Gelfert
Article copyright:
© Copyright 2020
American Mathematical Society