## Structure of multicorrelation sequences with integer part polynomial iterates along primes

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- by Andreas Koutsogiannis, Anh N. Le, Joel Moreira and Florian K. Richter
- Proc. Amer. Math. Soc.
**149**(2021), 209-216 - DOI: https://doi.org/10.1090/proc/15185
- Published electronically: October 16, 2020
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## 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.## References

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## Bibliographic 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
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 209-216 - MSC (2010): Primary 37A45, 37A15; Secondary 11B30
- DOI: https://doi.org/10.1090/proc/15185
- MathSciNet review: 4172598