Nil Bohr-sets and almost automorphy of higher order / Wen Huang, Song Shao and Xiangdong Ye.

Includes bibliographical references and index.

1. Introduction --
2. Preliminaries --
3. Nilsystems --
4. Generalized polynomials --
5. Nil Bohro-sets and generalized polynomials: c Proof of Theorem B --
6. Generalized polynomials and recurrence sets: Proof of Theorem C --
7. Recurrence sets and regionally proximal relation of order d --
8. d-step almost automorpy and recurrence sets --
Appendix.

Two closely related topics, higher order Bohr sets and higher order almost automorphy, are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can be viewed as the higher order version of an old question concerning Bohr sets is studied: for any $d\in \mathbb{N}$ does the collection of $\{n\in \mathbb{Z}: S\cap (S-n)\cap\ldots\cap (S-dn)\neq \emptyset\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets? In the second part, the notion of $d$-step almost automorphic systems with $d\in\mathbb{N}\cup\{\infty\}$ is introduced and investigated, which is the generalization of the classical almost automorphic ones.