Nil Bohr-sets and Almost Automorphy of Higher Order

98 Pages·2016·0.86 MB·English

by  Wen Huang

MEMOIRS
of the
American Mathematical Society
Volume 241 • Number 1143 (fourth of 4 numbers) • May 2016
Nil Bohr-Sets and Almost
Automorphy of Higher Order
Wen Huang
Song Shao
Xiangdong Ye
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 241 • Number 1143 (fourth of 4 numbers) • May 2016
Nil Bohr-Sets and Almost
Automorphy of Higher Order
Wen Huang
Song Shao
Xiangdong Ye
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Names: Huang,Wen,1975–—Shao,Song,1976–—Ye,Xiangdong.
Title: Nil Bohr-sets and almost automorphy of higher order / Wen Huang, Song Shao, Xiang-
dongYe.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,2016. —Series: Mem-
oirsoftheAmericanMathematicalSociety,ISSN0065-9266;volume241,number1143—Includes
bibliographicalreferencesandindex.
Identifiers: LCCN2015050793—ISBN9781470418724(alk. paper)
Subjects: LCSH:Automorphicfunctions. —Fourieranalysis.
Classification: LCC QA353.A9 H83 2016 — DDC 515/.9–dc23 LC record available at http://
lccn.loc.gov/2015050793
DOI:http://dx.doi.org/10.1090/memo/1143
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Contents
Chapter 1. Introduction 1
1.1. Higher order Bohr problem 1
1.2. Higher order almost automorphy 4
1.3. Further questions 9
1.4. Organization of the paper 10
Chapter 2. Preliminaries 11
2.1. Basic notions 11
2.2. Bergelson-Host-Kra’ Theorem and the proof of Theorem A(2) 12
2.3. Equivalence of Problems B-I,II,III 13
Chapter 3. Nilsystems 17
3.1. Nilmanifolds and nilsystems 17
3.2. Nilpotent Matrix Lie Group 19
Chapter 4. Generalized polynomials 21
4.1. Definitions 21
4.2. Basic properties of generalized polynomials 22
Chapter 5. Nil Bohr -sets and generalized polynomials: Proof of Theorem B 27
0
5.1. Proof of Theorem B(1) 27
5.2. Proof of Theorem B(2) 32
Chapter 6. Generalized polynomials and recurrence sets: Proof of Theorem C 43
6.1. A special case and preparation 43
6.2. Proof of Theorem C 46
Chapter 7. Recurrence sets and regionally proximal relation of order d 55
7.1. Regionally proximal relation of order d 55
7.2. Nil Bohr -sets, Poincar´e sets and RP[d] 56
d 0
7.3. SG -sets and RP[d] 60
d
7.4. Cubic version of multiple recurrence sets and RP[d] 64
7.5. Conclusion 69
Chapter 8. d-step almost automorpy and recurrence sets 71
8.1. Definition of d-step almost automorpy 71
8.2. Characterization of d-step almost automorphy 72
Appendix A 75
A.1. The Ramsey properties 75
A.2. Compact Hausdorff systems 76
iii
iv CONTENTS
A.3. Intersective 78
Bibliography 81
Index 85
Abstract
Two closely related topics: higher order Bohr sets and higher order almost
automorphy are investigated in this paper. Both of them are relatedto nilsystems.
Inthefirstpart,theproblemwhichcanbeviewedasthehigherorderversionof
an old question concerning Bohr sets is studied: for any d ∈ N does the collection
of {n ∈ Z : S ∩ (S − n) ∩ ... ∩ (S − dn) (cid:5)= ∅} with S syndetic coincide with
that of Nil Bohr -sets? It is proved that Nil Bohr -sets could be characterized
d 0 d 0
via generalized polynomials, and applying this result one side of the problem is
answeredaffirmatively: foranyNil Bohr -setA, thereexistsasyndeticsetS such
d 0
that A ⊃ {n ∈ Z : S ∩(S −n)∩...∩(S −dn) (cid:5)= ∅}. Moreover, it is shown that
the answer of the other side of the problem can be deduced from some result by
Bergelson-Host-Kra if modulo a set with zero density.
In the second part, the notion of d-step almost automorphic systems with d∈
N∪{∞} is introduced and investigated, which is the generalization of the classical
almost automorphic ones. It is worth to mention that some results concerning
higherorderBohrsetswillbeappliedtotheinvestigation. Foraminimaltopological
dynamical system (X,T) it is shown that the condition x ∈ X is d-step almost
automorphic can be characterized via various subsets of Z including the dual sets
of d-step Poincar´e and Birkhoff recurrence sets, and Nil Bohr -sets. Moreover, it
d 0
turns out that the condition (x,y) ∈ X ×X is regionally proximal of order d can
also be characterized via various subsets of Z.
Received by the editor January 14, 2014 and, in revised form, April 8, 2014 and May 24,
2014.
ArticleelectronicallypublishedonDecember11,2015.
DOI:http://dx.doi.org/10.1090/memo/1143
2010 MathematicsSubjectClassification. Primary: 37B05,22E25,05B10.
Key wordsand phrases. NilpotentLiegroup,nilsystem,Bohrset,Poincar´erecurrence,gen-
eralizedpolynomials,almostautomorphy,Birkhoffrecurrenceset,theregionallyproximalrelation
oforderd.
Huang,ShaoandYeweresupportedbytheNNSF(11225105,11371339,11431012,11571335).
The authors are affiliated with the Wu Wen-Tsun Key Laboratory of Mathematics,
USTC, Chinese Academy of Sciences and Department of Mathematics, University of Sci-
ence and Technology of China, Hefei, Anhui, 230026, People’s Republic of China. E-mails
are:[email protected];[email protected];[email protected].
(cid:2)c2015 American Mathematical Society
v

CHAPTER 1
Introduction
In this paper we study two closely related topics: higher order Bohr sets and
higher order almost automorphy. Both of them are related to nilsystems. In the
first part we investigate the higher order Bohr sets. Then in the second part we
study the higher order automorphy, and explain how these two topics are closely
related.
1.1. Higher order Bohr problem
A very old problem from combinatorial number theory and harmonic analysis,
rooted in the classical work of Bogoliuboff, Følner [17], Ellis-Keynes [16], and
Veech [52] is the following. Let S be a syndetic subset of the integers. Is the set
S−S aBohr neighborhoodof zeroinZ (alsocalledBohr -set)? For the equivalent
0
statements and results related to the problem in combinatorial number theory,
group theory and dynamical systems, see Glasner [26], Weiss [54], Katznelson
[42], Pestov [47], Boshernitzan-Glasner [11], Huang-Ye [41], Grivaux-Roginskaya
[32,33].
Bohr-sets are fundamentally abelian in nature. Nowadays it has become ap-
parent that higher order non-abelian Fourier analysis plays an important role both
incombinatorial numbertheoryandergodictheory. Relatedtothis, ahigher-order
version of Bohr sets, namely Nil Bohr -sets, was introduced in [35]. For the
0 d 0
recent results obtained by Bergelson-Furstenberg-Weiss and Host-Kra, see [4,35].
1.1.1. Nil Bohr-sets. There are several equivalent definitions for Bohr-sets.
Here is the one easy to understand: a subset A of Z is a Bohr-set if there exist
m∈N, α∈Tm, and a non-empty open set U ⊂Tm such that {n∈Z:nα∈U} is
contained in A; the set A is a Bohr -set if additionally 0∈U.
0
It is not hard to see that if (X,T) is a minimal equicontinuous system, x∈X
and U is a neighborhood of x, then N(x,U) =: {n ∈ Z : Tnx ∈ U} contains
S −S =: {a−b : a,b ∈ S} with S syndetic, i.e. with a bounded gap (S can be
chosenasN(x,U ),whereU ⊂U isanopenneighborhoodofx). Thisimpliesthat
1 1
if A is a Bohr -set, then A⊃S−S with S syndetic. The old question concerning
0
Bohr -sets is
0
Problem A-I: Let S be a syndetic subset of Z, is S−S a Bohr -set?
0
Note that Ellis-Keynes [16] proved that S−S+S−a is a Bohr -set for some
0
a∈S. Veechshowedthatitisatleast“almost”true[52]. Thatis,givenasyndetic
set S ⊂ Z, there is an N ⊂ Z with density zero such that (S−S)ΔN is a Bohr -
0
set. Kˇr´ıˇz [44]showed thatthereexistsasubset K ofZwithpositiveupper Banach
density such that K −K does not contains S−S for any syndetic subset S of Z.
1
2 1. INTRODUCTION
ThisimpliesthatProblemA-Ihasanegativeanswerifwereplaceasyndeticsubset
of Z by a subset of Z with positive upper Banach density.
A subset A of Z is a Nil Bohr -set if there exist a d-step nilsystem (X,T),
d 0
x ∈ X and an open neighborhood U of x such that N(x ,U) =: {n ∈ Z :
0 0 0
Tnx ∈ U} is contained in A. Denote by F the family1 consisting of all Nil
0 d,0 d
Bohr -sets. We can now formulate a higher order form of Problem A-I. We note
0
that
{n∈Z:S∩(S−n)∩...∩(S−dn)(cid:5)=∅}
can be viewed as the common differences of arithmetic progressions with length
d+1 appeared in the subset S. In fact, S∩(S−n)∩...∩(S−dn)(cid:5)=∅ if and only
if there is m∈S with
m,m+n,...,m+dn∈S.
Particularly, S−S ={n∈Z:S∩(S−n)(cid:5)=∅}.
Problem B-I: [Higher order form of Problem A-I] Let d∈N.
(1) For any Nil Bohr -set A, is it true that there is a syndetic subset S of Z
d 0
with A⊃{n∈Z:S∩(S−n)∩...∩(S−dn)(cid:5)=∅}?
(2) For any syndeticsubset S ofZ, is{n∈Z:S∩(S−n)∩...∩(S−dn)(cid:5)=∅}
a Nil Bohr -set?
d 0
1.1.2. Dynamical version of the higher order Bohr problem. Some-
timescombinatorialquestionscanbetranslatedintodynamicalonesbytheFursten-
berg correspondence principle, see Section 2.3.1. Using this principle, it can be
shown that Problem A-I is equivalent to the following version:
Problem A-II: For any minimal system (X,T) and any nonempty open subset U
of X, is the set {n∈Z:U ∩T−nU (cid:5)=∅} a Bohr -set?
0
Similarly, Problem B-I has its dynamical version:
Problem B-II: [Dynamical version of Problem B-I] Let d∈N.
(1) ForanyNil Bohr -setA,isittruethatthereareaminimalsystem(X,T)
d 0
and a non-empty open subset U of X with
A⊃{n∈Z:U ∩T−nU ∩...∩T−dnU (cid:5)=∅}?
(2) For any minimal system (X,T) and any non-empty open subset U of X,
is it true that {n∈Z:U∩T−nU∩...∩T−dnU (cid:5)=∅} is a Nil Bohr -set?
d 0
In the next section, we will give the third version of Problem B via recurrence
sets. The equivalence of three versions will be shown in Chapter 2.
1A collection F of subsets of Z (or N) is a family if it is hereditary upward, i.e. F1 ⊂ F2
and F1 ∈ F imply F2 ∈ F. Any nonempty collection A of subsets of Z generates a family
F(A):={F ⊂Z:F ⊃AforsomeA∈A}.
1.1. HIGHER ORDER BOHR PROBLEM 3
1.1.3. MainresultsonthehigherorderBohrproblem. Weaimtostudy
Problem B-I or its dynamical version Problem B-II. We will show that Problem
B-II(1) has an affirmative answer, and Problem B-II(2) has a positive answer if
ignoring a set with zero density. Namely, we will show
Theorem A: Let d∈N.
(1) If A⊂Z is a Nil Bohr -set, then there exist a minimal d-step nilsystem
d 0
(X,T) and a nonempty open subset U of X with
A⊃{n∈Z:U ∩T−nU ∩...∩T−dnU (cid:5)=∅}.
(2) For any minimal system (X,T) and any non-empty open subset U of X,
I = {n ∈ Z : U ∩T−nU ∩...∩T−dnU (cid:5)= ∅} is almost a Nil Bohr -set,
d 0
i.e. there is M ⊂ Z with zero upper Banach density such that IΔM is a
Nil Bohr -set
d 0
As we said before for d=1 Theorem A(1) can be easily proved. To show The-
orem A(1) in the general case, we need to investigate the properties of F . It is
d,0
interestingthatintheprocesstodothis,generalizedpolynomials(see§4.1foradef-
inition) appear naturally. Generalized polynomials have been studied extensively,
seefor exampletheremarkable paperby BergelsonandLeibman[6] andreferences
therein. Afterfinishing this paper we even find that it also plays an important role
in the recent work by Green, Tao and Ziegler [31]. In fact the special generalized
polynomials defined in this paper are closely related to the nilcharacters defined
there. We remark that Theorem A(2) was first proved by Veech in the case d= 1
[52], and its proof will be presented in Section 2.2.
Let F (resp. F ) be the family generated by the sets of forms
GPd SGPd
(cid:2)k
{n∈Z:P (n) (mod Z)∈(−(cid:3) ,(cid:3) )},
i i i
i=1
where k ∈ N, P ,...,P are generalized polynomials (resp. special generalized
1 k
polynomials) of degree ≤ d, and (cid:3) > 0. For the precise definitions see Chapter 4.
i
We remark that one can in fact show that F =F (Theorem 4.2.11).
GPd SGPd
The following theorem illustrates the relation between Nil Bohr -sets and the
d 0
sets defined above using generalized polynomials.
Theorem B: Let d∈N. Then F =F .
d,0 GPd
When d = 1, we have F = F . This is the result of Katznelson [42],
1,0 SGP1
since F is generated by sets of forms ∩k {n ∈ Z : na (mod Z) ∈ (−(cid:3) ,(cid:3) )}
SGP1 i=1 i i i
with k ∈N, a ∈R and (cid:3) >0.
i i
Theorem A(1) follows from Theorem B and the following result:
Theorem C: Let d∈N. If A∈F , then there exist a minimal d-step nilsystem
GPd
(X,T) and a nonempty open subset U of X such that
A⊃{n∈Z:U ∩T−nU ∩...∩T−dnU (cid:5)=∅}.
The proof of Theorem B is divided into two parts, namely
Theorem B(1): F ⊂F and
d,0 GPd
Theorem B(2): F ⊃F .
d,0 GPd