Nil Bohr-Sets and Almost Automorphy of Higher Order.
Huang, Wen.
Nil Bohr-Sets and Almost Automorphy of Higher Order. - 1st ed. - 1 online resource (98 pages) - Memoirs of the American Mathematical Society Series ; v.241 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Higher order Bohr problem -- 1.2. Higher order almost automorphy -- 1.3. Further questions -- 1.4. Organization of the paper -- Chapter 2. Preliminaries -- 2.1. Basic notions -- 2.2. Bergelson-Host-Kra' Theorem and the proof of Theorem A(2) -- 2.3. Equivalence of Problems B-I,II,III -- Chapter 3. Nilsystems -- 3.1. Nilmanifolds and nilsystems -- 3.2. Nilpotent Matrix Lie Group -- Chapter 4. Generalized polynomials -- 4.1. Definitions -- 4.2. Basic properties of generalized polynomials -- Chapter 5. Nil Bohr₀-sets and generalized polynomials: Proof of Theorem B -- 5.1. Proof of Theorem B(1) -- 5.2. Proof of Theorem B(2) -- Chapter 6. Generalized polynomials and recurrence sets: Proof of Theorem C -- 6.1. A special case and preparation -- 6.2. Proof of Theorem C -- Chapter 7. Recurrence sets and regionally proximal relation of order -- 7.1. Regionally proximal relation of order -- 7.2. Nil_ Bohr₀-sets, Poincaré sets and \RP^ -- 7.3. _-sets and \RP^ -- 7.4. Cubic version of multiple recurrence sets and \RP^ -- 7.5. Conclusion -- Chapter 8. -step almost automorpy and recurrence sets -- 8.1. Definition of -step almost automorpy -- 8.2. Characterization of -step almost automorphy -- Appendix A. -- A.1. The Ramsey properties -- A.2. Compact Hausdorff systems -- A.3. Intersective -- Bibliography -- Index -- Back Cover.
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 does the collection of \: S\cap (S-n)\cap\ldots\cap (S-dn)
eq \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\cup\ is introduced and investigated, which is the generalization of the classical almost automorphic ones.
9781470428792
Automorphic functions.
Fourier analysis.
Electronic books.
QA353.A9 H83 2015
515/.9
Nil Bohr-Sets and Almost Automorphy of Higher Order. - 1st ed. - 1 online resource (98 pages) - Memoirs of the American Mathematical Society Series ; v.241 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Higher order Bohr problem -- 1.2. Higher order almost automorphy -- 1.3. Further questions -- 1.4. Organization of the paper -- Chapter 2. Preliminaries -- 2.1. Basic notions -- 2.2. Bergelson-Host-Kra' Theorem and the proof of Theorem A(2) -- 2.3. Equivalence of Problems B-I,II,III -- Chapter 3. Nilsystems -- 3.1. Nilmanifolds and nilsystems -- 3.2. Nilpotent Matrix Lie Group -- Chapter 4. Generalized polynomials -- 4.1. Definitions -- 4.2. Basic properties of generalized polynomials -- Chapter 5. Nil Bohr₀-sets and generalized polynomials: Proof of Theorem B -- 5.1. Proof of Theorem B(1) -- 5.2. Proof of Theorem B(2) -- Chapter 6. Generalized polynomials and recurrence sets: Proof of Theorem C -- 6.1. A special case and preparation -- 6.2. Proof of Theorem C -- Chapter 7. Recurrence sets and regionally proximal relation of order -- 7.1. Regionally proximal relation of order -- 7.2. Nil_ Bohr₀-sets, Poincaré sets and \RP^ -- 7.3. _-sets and \RP^ -- 7.4. Cubic version of multiple recurrence sets and \RP^ -- 7.5. Conclusion -- Chapter 8. -step almost automorpy and recurrence sets -- 8.1. Definition of -step almost automorpy -- 8.2. Characterization of -step almost automorphy -- Appendix A. -- A.1. The Ramsey properties -- A.2. Compact Hausdorff systems -- A.3. Intersective -- Bibliography -- Index -- Back Cover.
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 does the collection of \: S\cap (S-n)\cap\ldots\cap (S-dn)
eq \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\cup\ is introduced and investigated, which is the generalization of the classical almost automorphic ones.
9781470428792
Automorphic functions.
Fourier analysis.
Electronic books.
QA353.A9 H83 2015
515/.9