On combinatorial properties of nil-Bohr sets of integers and related problems

231 Pages·2017·1.241 MB·English

by  Jakub Konieczny

On combinatorial properties
of nil–Bohr sets of integers
and related problems
Jakub Konieczny
St Johns’s College
University of Oxford
A thesis submitted for the degree of
Doctor of Philosophy

Abstract
Author’s name: Jakub Konieczny.
(Under supervision of Ben Green.)
This thesis deals with five problems in additive combinatorics and ergodic
theory. A brief introduction to this general area and a summary of in-
cluded results is given in Chapter I.
InChapterII,weconsidersetsoftheform{n ∈ N | |p(n) mod 1| ≤ ε(n)},
0
where p is a polynomial and ε(n) ≥ 0. We obtain various conditions under
which any sufficiently large integer can be represented as a sum of 2 or 3
elements of a given set of this form.
In Chapter III, we study the class of weakly mixing sets of integers, and
prove that a certain class of polynomial equations can always be solved in
such a set.
In Chapter IV, we show that any nil–Bohr set contains a certain type
of additive pattern. Combined with earlier results of Host and Kra, this
leads to a partial combinatorial characterisation of nil–Bohr sets.
In Chapter V, we study the combinatorial properties of generalised poly-
nomials (expressions built from polynomials and the floor function). In
contrast with results of Bergelson and Leibman, we show that if the set
of integers where a given generalised polynomial takes a non-zero value
has asymptotic density 0, then it does not contain any IP set. This leads
to a partial characterisation of automatic sequences which are given by
generalised polynomial formulas.
InChapterVI,weestimatetheGowersnormsoftheThue-Morsesequence
and the Rudin-Shapiro sequence. This gives some of the simplest deter-
ministic examples of sequences with small Gowers norms of all orders.
Acknowledgements
The author wishes to express his gratitude to the following people:
• to Ben Green for an endless supply of problems and methods for solving them;
• to Jakub Byszewski for the numerous conversations which were productive and
enjoyable, and for the few which were both;
• to Tom Sanders, Roger Heath-Brown and David Conlon for comments on pre-
vious drafts of this thesis, especially Chapter II, and on writing in general;
• to Vitaly Bergelson for comments related to Chapters III and V, as well for his
highly contagious vitality;
• to Bryna Kra for comments related to Chapters II, III, IV;
• to Alexander Fish for comments related to Chapter III;
• to Dominik Kwietniak for comments related to Chapter V and life in general;
• toJean-PaulAlloucheandIngerH˚aland-KnutsonforcommentsrelatedtoChap-
ter V;
• to Tanja Eisner, Christian Mauduit, Clemens Mu¨llner, and Aihua Fan for com-
ments related to Chapter VI;
• to James Aaronson, Sean Eberhard, Sofia Lindqvist, Freddie Manners, Rudi
Mrazovi´c, Przemek Mazur and Aled Walker for many informal discussions;
• the organizers of the conference New developments around ×2 ×3 conjecture
and other classical problems in Ergodic Theory in Cieplice, Poland in May 2016;
• to National Science Centre in Poland, Clarendon Fund and St John’s College
Kendrew Fund for providing generous funding;
• last but not least, to all my family and friends and especially to Magda Rusac-
zonek, without whom this thesis would never have come to be.
Contents
I Introduction 3
I.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I.1.1 Additive combinatorics . . . . . . . . . . . . . . . . . . . . . . 3
I.1.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
I.1.3 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
I.1.4 Ergodic theory in additive combinatorics . . . . . . . . . . . . 7
I.1.5 IP sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I.1.6 Nilsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
I.1.7 Nil-Bohr sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
I.1.8 Filtered groups and polynomial sequences . . . . . . . . . . . 13
I.1.9 Mal’cev coordinates and generalised polynomials . . . . . . . . 16
I.1.10 Equidistribution . . . . . . . . . . . . . . . . . . . . . . . . . . 19
I.1.11 Higher order Fourier analysis . . . . . . . . . . . . . . . . . . 21
I.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
II Nil–Bohr type sets as bases for the positive integers 27
II.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
II.2 Non-bases of order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
II.2.1 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . 32
II.2.2 Quadratic irrationals . . . . . . . . . . . . . . . . . . . . . . . 34
II.2.3 Badly approximable reals . . . . . . . . . . . . . . . . . . . . . 36
II.2.4 Generic reals . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
II.3 Bases and almost bases of order 2 . . . . . . . . . . . . . . . . . . . . 42
II.3.1 Equidistribution and quantitative rationality . . . . . . . . . . 42
II.3.2 Almost bases of order 2 . . . . . . . . . . . . . . . . . . . . . 45
II.3.3 Exceptional values of α . . . . . . . . . . . . . . . . . . . . . . 47
II.4 Threshold for being a basis of order 2 . . . . . . . . . . . . . . . . . . 50
II.4.1 Quadratic irrationals . . . . . . . . . . . . . . . . . . . . . . . 52
i
II.4.2 The algorithmic approach . . . . . . . . . . . . . . . . . . . . 55
II.5 Bases of order 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
II.5.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
II.5.2 Minor arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
II.5.3 Major arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
II.5.4 Main contribution . . . . . . . . . . . . . . . . . . . . . . . . . 62
II.6 Higher degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
II.6.1 Bases of order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 64
II.6.2 Non-bases of order 2 . . . . . . . . . . . . . . . . . . . . . . . 67
IIIWeakly mixing sets and polynomial equations 69
III.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
III.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
III.3 Uniform ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . . . 75
III.3.1 Outline and initial reductions . . . . . . . . . . . . . . . . . . 75
III.3.2 Uniform convergence for linear polynomials . . . . . . . . . . . 77
III.4 PET induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
III.4.1 Definitions and basic properties . . . . . . . . . . . . . . . . . 80
III.4.2 Uniform convergence in higher degrees . . . . . . . . . . . . . 82
III.5 Doubly polynomial averages . . . . . . . . . . . . . . . . . . . . . . . 83
III.5.1 Initial reductions . . . . . . . . . . . . . . . . . . . . . . . . . 83
III.5.2 Polynomial Følner averages . . . . . . . . . . . . . . . . . . . 85
III.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
IVCombinatorial characterisation of nil–Bohr sets of integers 91
IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
IV.2 Polynomial maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
IV.2.1 Main results reformulated . . . . . . . . . . . . . . . . . . . . 94
IV.2.2 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
IV.2.3 VIP-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
IV.2.4 Host-Kra cube groups . . . . . . . . . . . . . . . . . . . . . . 98
IV.2.5 Host-Kra cubes and nilmanifolds . . . . . . . . . . . . . . . . 100
IV.3 S -sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
k
IV.3.1 IP sets revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 102
IV.3.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 103
IV.3.3 Asymptotic subsequences . . . . . . . . . . . . . . . . . . . . . 108
IV.3.4 Stable sequences . . . . . . . . . . . . . . . . . . . . . . . . . 110
ii
IV.3.5 Stable polynomials . . . . . . . . . . . . . . . . . . . . . . . . 114
IV.4 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
IV.4.1 Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
IV.4.2 Case d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
IV.5 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
IV.5.1 Robust version and induction . . . . . . . . . . . . . . . . . . 120
IV.5.2 Reduction to an abelian problem . . . . . . . . . . . . . . . . 122
IV.5.3 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . 123
IV.6 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
IV.6.1 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
IV.6.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
IV.6.3 Final step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
IV.6.4 Proof of Theorem IV.1.1 . . . . . . . . . . . . . . . . . . . . . 131
V Automatic sequences and generalised polynomials 132
V.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
V.2 Automatic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
V.3 Density 1 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
V.3.1 Polynomial sequences . . . . . . . . . . . . . . . . . . . . . . . 142
V.3.2 Generalised polynomials . . . . . . . . . . . . . . . . . . . . . 144
V.4 Sparse sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
V.4.1 Arid sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
V.4.2 Proof strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
V.4.3 Comments and applications . . . . . . . . . . . . . . . . . . . 151
V.5 Sparse generalised polynomials . . . . . . . . . . . . . . . . . . . . . . 152
V.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
V.5.2 Initial reductions . . . . . . . . . . . . . . . . . . . . . . . . . 154
V.5.3 Fractional parts and limits . . . . . . . . . . . . . . . . . . . . 155
V.5.4 Fractional parts of polynomials . . . . . . . . . . . . . . . . . 157
V.5.5 Group generated by fractional parts . . . . . . . . . . . . . . . 160
V.6 Sparse automatic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
V.6.1 Density of symbols . . . . . . . . . . . . . . . . . . . . . . . . 162
V.6.2 Dichotomy for sparse automatic sets . . . . . . . . . . . . . . 163
V.6.3 IP rich automatic sets . . . . . . . . . . . . . . . . . . . . . . 166
V.6.4 Proof of Theorem V.1.6 . . . . . . . . . . . . . . . . . . . . . 167
V.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
iii
V.7.1 Small fractional parts . . . . . . . . . . . . . . . . . . . . . . . 171
V.7.2 IP rich sequences . . . . . . . . . . . . . . . . . . . . . . . . . 175
V.7.3 Very sparse sequences . . . . . . . . . . . . . . . . . . . . . . 178
V.8 Exponential sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 180
V.8.1 Automaticity of recursive sequences . . . . . . . . . . . . . . . 181
V.8.2 Exponentially sparse generalised polynomial sets . . . . . . . . 183
V.8.3 Quadratic Pisot numbers . . . . . . . . . . . . . . . . . . . . . 184
V.8.4 Cubic Pisot numbers . . . . . . . . . . . . . . . . . . . . . . . 186
V.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
V.9.1 Small fractional parts . . . . . . . . . . . . . . . . . . . . . . . 189
V.9.2 Exponential sequences . . . . . . . . . . . . . . . . . . . . . . 190
V.9.3 Morphic words . . . . . . . . . . . . . . . . . . . . . . . . . . 191
V.9.4 Regular sequences . . . . . . . . . . . . . . . . . . . . . . . . . 192
VIUniformity of automatic sequences 193
VI.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
VI.2 Thue-Morse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
VI.3 Rudin-Shapiro sequence . . . . . . . . . . . . . . . . . . . . . . . . . 201
VI.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
A Continued fractions 208
A.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
A.2 Ergodic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
A.3 Good rational approximations . . . . . . . . . . . . . . . . . . . . . . 210
B Ultrafilters and limits 212
Bibliography 214
iv
Notation
Below we list some of the notation used in this thesis, some of which is not entirely
standard.
N, N : the sets of positive integers and non-negative integers, respectively;
0
Z, Q, R, C,...: integers, rational numbers, real numbers, complex numbers, etc.;
F, F : the family of non-empty (resp. all) subsets of N;
∅
T: the 1-dimensional torus R/Z;
[N]: the initial interval {0,1,2,...,N −1} of N ;
0
Φ : the Iverson bracket, equal 1 if Φ is a true sentence and 0 otherwise;
(cid:74) (cid:75)
(cid:98)x(cid:99),(cid:100)x(cid:101),(cid:104)(cid:104)x(cid:105)(cid:105): thebestintegerapproximationofx ∈ Rbyintegerfrombelow, above
and in absolute value (cid:98)x+1/2(cid:99), respectively;
{x},(cid:107)x(cid:107): fractional part x − (cid:98)x(cid:99) and absolute fractional part |x−(cid:104)(cid:104)x(cid:105)(cid:105)| of x ∈ R
respectively, also used for x ∈ T;
1 : the characteristic sequence x ∈ X of the set X;
X
(cid:74) (cid:75)
e(x): the function e2πix, x ∈ R or x ∈ T;
ν (x): the p-adic valuation of x ∈ Q, ν (pta) = t if p (cid:45) a,b;
p p b
degp, lcp: the degree and the leading coefficient of a polynomial p ∈ R[x];
A+B: the sumset {a+b | a ∈ A, b ∈ B} of two sets A and B;
kA: the k-fold sumset A+A+···+A (k times).
1
We use the symbol E borrowed from probability to denote averages:
E 1 (cid:88)
f(x) := f(x).
|X|
x∈X x∈X
Weusestandardasymptoticnotation. WewriteX (cid:28) Y orX = O(Y)if|X| ≤ cY
for a universal constant c > 0. If c is allowed to depend on a parameter p, we denote
it by a subscript: X (cid:28) Y or X = O (Y). Conversely, we write X (cid:29) Y of X = Ω(Y)
p p
if X ≥ cY. If X (cid:28) Y (cid:28) X, we write X = Θ(Y) or X ∼ Y.
Similarly, for a variable v, we write X = o (Y) if |X| ≤ c(v)Y where c(v) → 0
v→z
as v → z. If v and z are clear from the context, we suppress the subscript v → z. If
c is allowed to depend on a parameter p, we denote this by subscript: X = o or
p;v→z
X = o (Y). Finally, we write X = ω(Y) if X > c(v)Y where c(v) → ∞ as v → z.
p
We often write symbols O(·),Ω(·),Θ(·),o(·),ω(·) to denote unspecified functions
with the appropriate growth.
For a set A ⊂ N, we define various notions of density. We list the ones which will
be useful in this document:
|A∩[n]|
d(A) = lim (natural/asymptotic density, if exists),
n→∞ n
|A∩[n]|
d(A) = limsup (upper natural density),
n
n→∞
|A∩[m,m+n)|
d∗(A) = limsupsup (upper Banach density).
n
n→∞ m
2