Subsequence Ergodic Theorems

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MSc Mathematics

Master Thesis

Subsequence Ergodic Theorems

Tristan Hands

10318925

30th August 2014

30EC

Supervisors:

Prof. dr. A.J. Homburg

Prof. dr. T. Eisner

Second Reader:

Dr. H. Peters

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Abstract

The word ergodic was first used by Boltzmann in 1898 to describe systems where the time and space average are equal. These systems are of importance in statistical mechanics and led von Neumann and Birkhoff to develop their celebrated classical ergodic theorems. It is however Furstenberg’s ergodic proof of Szemer´edi’s theorem and the resulting extension to the first proof of the multi-dimensional Szemer´edi theorem that has evoked an increased interest in the study of ergodic theorems of late. This interest has resulted in many researchers focussing their efforts on possible extensions of the classical ergodic theorems. In this thesis we discuss extensions of the classical ergodic theorems to subsequences of the integers and real numbers.

This thesis is organised as follows. In the first chapter some ergodic theory basics will be introduced, including the results of von Neumann and Birkhoff. In the second chapter we proceed to the subsequence results of von Neumann’s mean ergodic theorem, introducing the concept of mean good sequences. Developments of Weyl are studied including a proof that polynomial sequences are mean good. Certain growth criteria for mean good sequences are also studied. In the third and final chapter subsequence results of Birkhoff’s pointwise ergodic theorem are discussed. The concept of pointwise good sequences is introduced and two methods of Bourgain are developed to prove whether sequences are pointwise good or not, namely Bourgain’s oscillation method and Bourgain’s entropy method. The oscillation method will be used to prove Birkhoff’s classic ergodic theorem. The entropy method will be used to solve a problem that Bellow posed and to show that integer parts of pointwise good sequences remain pointwise good, following Lesigne’s proof.

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Acknowledgements

I would like to express my gratitude and offer special thanks to Professor Eisner for su-pervising this thesis. Professor Eisner went over and beyond the duties of a Supervisor by making time to supervise me after a move to Leipzig University and was always available to offer advice and assistance during my research. I would also like to thank Professor Homburg for his internal and administrative assistance, and valuable comments. I would finally like to thank the ema2sa scholarship programme. Without their financial support this research would not have been possible.

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Popular summary

To set the scene, let us consider a ball travelling in a circle, anticlockwise and at constant speed. Every point on the circle can be uniquely defined by its distance, anticlockwise along the circumference of the circle, from the starting point. At each point in time t, we use the notation dt _{to refer to this distance. In Figure 1 we see the position of a ball}

travelling at 2√10 seconds per a revolution at various times. The circle in the example has circumference equal to 1.

0s 1s 2s 3s ↘ 150s ↗ d0= 0 d1 = 0.158 d2= 0.316 d3= 0.474 d1₅₀= 0.717

Figure 1: Ball travelling at 2√10 seconds per a revolution in a circle of circumference 1. The next step in our construction is to do a measurement on the ball. We could for example measure the direction the ball is travelling in or the distance the ball is from another point. As the position of the ball is uniquely defined by dt_{, any measurement}

depending solely on the position of the ball is merely a function of this distance. So at each time t we can make a measurement which we call m(dt). One of the main questions

of ergodic theory is: what is the average value of all these measurements and does this limit exist if we take the average over all the measurements made at integer times? This problem was solved in 1931 by von Neumann and Birkhoff and in this thesis we consider an extension of the problem, namely whether these averages converge when measurements are taken at times other than all of the integers. For example if we consider our ball travelling in a circle and our measurement is that if the ball is within a distance of 0.5

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x Popular summary

from the staring point we measure a value 1 and if it is further we measure a value 0. Using the measure just described, Figure 2 displays the averages of the measurements if the measurements are taken at time log(n) and at time n2 _{for positive integers n.}

Figure 2: Average measurements when measurements are made at time n2 _{and log}(n).

One sees that if we have a lot of time on our hands, and do these measurements 5000 times, then the average measurements appear to converge when measurements are taken at times n2_{. The terminology in ergodic theory is that n}2 _{seems to be a good sequence.}

The graph of average measurements when taken at times log(n), however, moves between a value near 0 and a value near 1 without ever settling on any one of these points, or so it seems from the first 5000 measurements. We therefore suspect that log(n) is what ergodic theorists call a bad sequence. Only looking at a graph does not prove our suspicions and in this thesis we prove that n2 _{is indeed a good sequence and that log}(n) is a bad sequence.

Many other sequences will be considered throughout this thesis and interesting techniques will be developed to show which sequences are bad and which are good.

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Introduction

The word ergodic was first used by Boltzmann in 1898 to describe his ergodic hypothesis, that a system has time averages equal to space averages [9]. It was however shown that Boltzmann’s hypothesis in its original form was false and it was therefore modified to become the quasi-ergodic hypothesis. This was the problem that von Neumann was working on in 1931 and after coming across Koopman’s article [33] he had the inspiration needed to solve the hypothesis, via his mean ergodic theorem. Birkhoff learnt of the progress von Neumann had made and, whilst von Neumann faced delays translating his paper to English and incorporating comments from Stone, Birkhoff managed to publish a proof of his pointwise ergodic theorem. This resolved the quasi-ergodic hypothesis in December of 1931 [8], only one month prior to von Neumann’s publication [43]. If the reader is interested in the history and controversy surrounding the classical ergodic theorems then the historical article of Joseph D. Zund [49] gives a brief overview of all the events that took place surrounding the publication of the classical ergodic theorems and includes an original letter written by von Neumann during this period. It was this work done by Koopman, von Neumann and Birkhoff regarding the classical ergodic theorems that gave birth to what is known as ergodic theory today: the study of the long term behaviour of dynamical systems in a measure theoretic setting.

Since the classical ergodic theorems in 1931-1932, ergodic theory has continued to be an active field of research, with many abstractions of the classical theorems being proved. There has however been a surge in interest in ergodic theory due the elegant nature of ergodic theoretic proofs and the applications of ergodic theory to other areas of mathemat-ics, especially number theory and combinatorics. This started with Furstenberg’s ergodic theoretic proof of Szemer´edi’s theorem and the extension thereof to the first proof of the multi-dimensional Szemer´edi theorem in [25]. Due to this increased interest, many exten-sions of the classical ergodic theorems are currently being studied. This thesis considers extensions of the classical ergodic theorems by discussing the convergence of time averages along subsequences of the integers and real numbers.

This thesis is organised as follows. In the first chapter the basics of measure preserving transformations will be developed. The study of circle rotations and periodic transform-ations are then developed, as these examples come in useful throughout the thesis. The mean ergodic theorem is proved whilst the pointwise ergodic theorem is only stated, with

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2 Introduction

a proof being delayed until the final chapter. To end the first chapter continuous time dy-namical systems, known as flows, are introduced and the classical theorems are extended to these systems.

The second chapter addresses the question: for which subsequences (an) of the real

numbers do the averages

1 N N −1 ∑ n=0 f(Tan(⋅)) ₍₁₎

converge in L2(µ) for all dynamical systems (X, B, µ, T ) and functions f ∈ L2(µ)? Using

a result of Weyl, the above question is related to analytic number theory and using this result it is shown that polynomial sequences satisfy (1). Growth criteria which can be used to show whether a sequence(an) satisfies (1) or not are also developed. In particular it is

shown that if the growth of (an) is too fast or too slow then (1) does not hold. A growth

criterion, such that all sequences satisfying this criterion also satisfy (1) is developed as well. Some other results are proved in this chapter in order to provide an interesting comparison with the pointwise results of the third chapter.

The final chapter addresses the question: for which subsequences(an) of the real numbers

do the averages 1 N N −1 ∑ n=0 f(Tan(x)) ₍₂₎

converge for almost every x ∈ X for all dynamical systems (X, B, µ, T ) and functions f ∈ L2(µ)? Pointwise results are often much more challenging than their mean convergence

counterparts. This can be seen in the history of ergodic theory, where little progress was made in the area of subsequence results for pointwise convergence. This was until 1989, when Bourgain published his two seminal papers in this field, providing techniques which allowed one to find many sequences for which (2) does and does not hold. Bourgain was thereby able to solve numerous open problems in ergodic theory and give rise to several new results in the field. Bourgain’s two methods, the entropy method and oscillation method, will be the main results of this thesis and will be studied in the third chapter. The oscillation method will be used to prove Birkhoff’s pointwise ergodic theorem. One of Bourgain’s entropy criterion will be proved and one of Bourgain’s original consequences thereof will be included. To end the thesis a consequence of the entropy method developed by Lesigne will be shown, proving that if (2) holds for (an), then (2) holds for the integer

parts of (an) as well.

This thesis aims to be as self contained as possible. However, the basics of measure theory are assumed to be known and some topics from Gaussian processes and functional analysis are stated but not proved. This thesis hopes to make the beautiful results of subsequence ergodic theory accessible to the interested graduate student.

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Chapter 1 Classical ergodic theory

1.1 An introduction to measure preserving dynamical

systems

As stated in the introduction, ergodic theory is the study of dynamical systems in a measure theoretic stetting. In this section we give a brief introduction to dynamical systems, with some examples and properties that are useful in later chapters.

Definition 1.1.1. The quadruple (X, B, µ, T ) is called a measure preserving dynam-ical system, or dynamdynam-ical system for short, if (X, B, µ) is a probability space and T ∶ X → X a measurable map such that for every A ∈ B we have µ(T−1(A)) = µ(A).

Example 1.1.1. Let N ∈ Z+ _{and X}

N be the set XN = {0, . . . , N − 1}, µ a measure on 2X

such that µ({x}) = _N1 for all x∈ X and T the function T ∶ X → X, T (x) = x + 1 mod N, then the quadruple (XN, 2X, µ, T) is a dynamical system.

The system (XN, 2X, µ, T) from the above example is one of the simplest dynamical

systems and is known as the periodic system of N symbols. It should be clear to the reader that(XN, 2X, µ, T) is a dynamical system. Our next example is arguably the most

important and most studied dynamical system in ergodic theory and can be thought of as the continuous version of the periodic system in the previous example. For notational convenience we define the following operations on real numbers.

Definition 1.1.2. For every a∈ R let ⟨a⟩ refer to the fractional part of a and let [a] refer to the integer part.

Example 1.1.2. Let T = [0, 1) with the open intervals generating the open sets. Let B be the Borel σ-algebra on T, λ the Lebesgue measure on B and Tα ∶ X → X, Tα(x) = ⟨x + α⟩,

then (T, B, λ, T α) is a dynamical system for all α ∈ T.

Proof. From elementary measure theory we know that the Lebesgue measure λ defined on the unit circle is the Haar measure of the Borel σ-algebraB. The Haar measure is invariant under translations and we therefore have for all A ∈ B that λ(T−1

α (A)) = λ(A), showing

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4 1. Classical ergodic theory

The dynamical system in Example 1.1.2 above is called the circle rotation. If the reader is not acquainted with the Haar measure, it is a simple exercise to prove the above assertion directly by considering open intervals.

Let us introduce some notation before proceeding to our next result.

Definition 1.1.3. For any set X and subset A⊂ X the indicator function or charac-teristic function of A is

χA∶ X → {0, 1}, χA(x) = {

1 ∶ if x ∈ A, 0 ∶ otherwise.

In the following lemma we show a basic property of dynamical systems that is used in many proofs throughout the thesis.

Lemma 1.1.1. Let (X, B, µ, T ) be a dynamical system. Then for all 1 ≤ p < ∞ and f ∈ Lp(µ)

∫_Xf(T (x))dµ(x) = ∫

X

f(x)dµ(x).

Proof. Fix 1≤ p < ∞, from the definition of a dynamical system we have for all B ∈ B ∫_XχB(T (x))dµ(x) = ∫ X χT−_(B)(x)dµ(x) = µ(T−1_(B)) = µ(B) = ∫_XχB(x)dµ(x).

So the result holds for all indicator functions, and from the linearity of integration we have that the result holds for all simple functions. For all f ∈ Lp(µ) we can find a sequence of

simple functions(fn) such that fn→ f pointwise and ∣fN∣ ≤ ∣f∣. We can apply the bounded

convergence theorem to get

∫_Xf(T (x))dµ(x) = lim n→∞∫_Xfn(T (x))dµ(x) = lim_n→∞_∫ X fn(x)dµ(x) = ∫_Xf(x)dµ(x), completing the proof of the lemma.

Before proceeding to the classical results of von Neumann and Birkhoff we define an important class of dynamical systems.

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1.1 An introduction to measure preserving dynamical systems 5

Definition 1.1.4. Let (X, B, µ, T ) be a dynamical system. We call (X, B, µ, T ) ergodic if for all B∈ B such that B = T−1(B) we have µ(B) = 0 or 1.

It should be clear to the reader that the periodic dynamical systems from Example 1.1.1 are ergodic. Before proceeding to further examples, let us introduce some more terminology.

Definition 1.1.5. Given a dynamical system (X, B, µ, T ) the orbit or trajectory of the point x∈ X is the set {Tn_x∶ n ∈ Z+}.

In the following proposition and lemma we follow the proofs in Chapter 2.3 of Einsiedler and Ward’s book [21].

Proposition 1.1.1. The circle rotation (T, B, λ, Tα) is ergodic if and only if α ∈ R ∖ Q.

Proof. The claim is trivially true if α = 0. So let us consider α ∈ Q ∖ {0}. We can find a ∈ Z≠0, b ∈ Z+ such that α = a_b. It is clear that Tα is invertible, with inverse function

T−1

α (x) = ⟨x − α⟩. We also have Tα−b(x) = ⟨x −ab × b⟩ = x, so T −b

α is the identity. Define

Ai ∶= ( ai b , ai b + 1 2b) and A ∶= b−1 ⋃ i=0 Ai,

then we have by construction that T−1

α (A) = A. We also have

0< 1

2b ≤ λ(A) ≤ b 2b < 1. We have therefore found a set A∈ B such that T−1

α (A) = A and µ(A) ≠ 0 or 1 and hence

(Tα,B, λ, Tα) is not ergodic.

Now let us take α∈ R ∖ Q. First we will show that the orbit of any x ∈ T is dense in T. To show that the orbit of any x∈ T is dense in T, it is sufficient to show that the orbit of 0 is ε-dense in T for all ε ∈ R+_{, as the translation of a dense orbit remains dense. Let us fix}

ε∈ R+ _{and take n}∈ Z+ _{such that} 1

n < ε. We can partition T into n disjoint sets as follows:

T = n−1 ⋃ i=0 [i n, i+ 1 n ) . If we consider the set of n+1 points {Ti

α(0) ∶ 0 ≤ i ≤ n}, then from the pigeon hole principle

there exist 0 ≤ i < j ≤ n such that ⟨(j − i)α⟩ < 1_n < ε. As α is irrational we also have that 0< ⟨(j − i)α⟩. Combining these inequalities yields 0 < Tαj−i(0) < ε, showing that the orbit

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Let B∈ B such that Tα(B) = B. We want to show that

∫

T

∣χB(x) − µ(B)∣ dλ(x) = 0,

as then µ(B) = 0 or 1, which will complete this implication. From our assumption that Tα(B) = B we have for all n ∈ Z+

Tn

α(B) = B and Tαn(Bc) = Bc. (1.1)

Let us take ε∈ R+_{. As continuous functions on the unit circle C}(T) are dense in L1(λ),

we can find f ∈ C(T) such that ∫

T

∣f(x) − χB(x)∣ dλ(x) < ε.

This implies

∫_B∣f(x) − 1∣ dλ(x) + ∫_Bc∣f(x)∣ dλ(x) < ε.

We can use the above inequality to get that for any n∈ Z+

∫

T∣f(x + nα) − f(x)∣ dλ(x) = ∫B∣f(x + nα) − f(x)∣ dλ(x) + ∫Bc∣f(x + nα) − f(x)∣ dλ(x)

≤ ∫_B∣f(x + nα) − 1∣ dλ(x) + ∫_Bc∣f(x + nα)∣ dλ(x)

+ ∫_B∣f(x) − 1∣ dλ(x) + ∫_Bc∣f(x)∣ dλ(x)

= 2 ∫_B∣f(x) − 1∣ dλ(x) + 2 ∫_Bc∣f(x)∣ dλ(x) < 2ε,

where the equality in the last line above follows from (1.1) and Lemma 1.1.1. As f is continuous and nα is dense in T, we obtain from the above inequality that

∫

T

∣f(x + t) − f(x)∣ dλ(x) ≤ 2ε for all t∈ T. We can therefore use Fubini’s theorem to obtain

∫ T∣f(x) − ∫T f(t)dλ(t)∣ dλ(x) ≤ ∫ T∫T ∣f(x) − f(x + t)∣ dλ(x)dλ(t) ≤ 2ε. We also have ∫ T∣∫T f(t)dλ(t) − µ(B)∣ dλ(x) = ∫ T∣∫T f(t) − χB(t)dλ(t)∣ dλ(x) ≤ ∫ T∫T ∣f(t) − χB(t)∣ dλ(t)dλ(x) < ε.

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1.1 An introduction to measure preserving dynamical systems 7

We can combine the above two inequalities with our construction of f to get ∫ T ∣χB(x) − µ(B)∣ dλ(x) ≤ ∫ T ∣χB(x) − f(x)∣ dλ(x) + ∫ T∣f(x) − ∫T f(t)dλ(t)∣ dλ(x) + ∫ T∣∫T f(t)dλ(t) − µ(B)∣ dλ(x) < 4ε.

As ε was an arbitrary positive real number we have ∫

T

∣χB(x) − µ(B)∣ dx = 0,

completing the proof that µ(B) = 0 or 1. We therefore have (Tα,B, λ, Tα) is ergodic which

completes the irrational case and the proof of the proposition.

Let us end the chapter by proving some equivalent definitions of an ergodic dynamical system.

Lemma 1.1.2. For a dynamical system (X, B, µ, T ), the following properties are equival-ent.

1. (X, B, µ, T ) is ergodic.

2. For all B∈ B if µ(T−1_B△ B) = 0 then µ(B) = 0 or 1.

3. Fix 1 ≤ p < ∞, for all f ∈ Lp(µ) if f(T (x)) = f(x) almost everywhere, then f is

constant almost everywhere.

Proof. (1 ⇒ 2) Assume that (X, B, µ, T ) is ergodic and take A ∈ B such that µ(T−1_A△A) =

0. Define BN ∶= ∞ ⋃ n=N T−n(A) and B ∶= ∞ ⋂ N =0 BN.

We have from the construction that T−1(B) = ∞ ⋂ N =0 ∞ ⋃ n=N T−(n+1)(A) = ⋂∞ N =0 ∞ ⋃ n=N +1 T−(n)(A) = B

and from our hypothesis µ(B) = 0 or 1. So all that remains is to show that µ(B) = µ(A) to complete the proof. We have B0⊃ B1 ⊃ B2⊃ . . . is a nested chain and from our hypothesis

µ(A △ BN) ≤ µ( ∞

⋃

n=N

A△ T−n_{(A)) = 0.}

We therefore have µ(A) = µ(BN) for all N ∈ Z+. From the dominated convergence theorem

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(2 ⇒ 1) Assume that property (2) holds. Let A ∈ B such that T−1(A) = A. We therefore

have

µ(T−1(A) △ A) = µ(∅) = 0.

So from our hypothesis µ(A) = 0 or 1, proving that (X, B, µ, T ) is ergodic.

(2 ⇒ 3) Now assume that property (2) holds and fix 1 ≤ p < ∞, take f ∈ Lp(µ) such that

f(T (x)) = f(x) almost everywhere and define Ak_n∶= {x ∈ X ∶ f(x) ∈ [k n, k+ 1 n )} . We then have µ(T−1(Ak_n) △ Ak_n) ≤ µ({x ∈ X ∶ f(T (x)) ≠ f(x)}) = 0,

where the last equality follows from our assumption that f(T (x)) = f(x) almost every-where. By construction we have that for each k∈ Z+_{, X is the following disjoint union}

X= ⊔

n∈Z

Ak_n.

As each set in the disjoint union has measure 0 or 1 there is exactly one k∶= k(n) for each n such that µ(Ak(n)n ) = 1. If we define

A= ∞ ⋃ n=1 Ak(n)n , then we have f(x) = lim n→∞ k(n) n

for all x ∈ A and µ(A) = 1 by construction. We therefore have f is constant almost everywhere, completing the proof of the implication.

(3 ⇒ 2) For our final implication assume property (3) holds and let A ∈ B such that µ(T−1(A) △ A) = 0. From this assumption we have that χ

A(T (x)) = χA(x) almost

every-where. From property (3) we have that χA is almost everywhere constant and hence

µ(A) = 0 or 1. Completing this implication and the proof of the lemma.

1.2 Classical ergodic theorems

We have developed sufficient tools in the previous section to tackle the classical ergodic theorems. We consider the mean ergodic theorem of von Neumann [43] first. The proof we follow is however not the original proof of von Neumann, but a shorter and neater version proved 20 years later by Riesz in [36].

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1.2 Classical ergodic theorems 9

Theorem 1.2.1 (Mean Ergodic Theorem). Let H be a Hilbert space and U ∶ H → H a unitary operator. Define H1 ∶= {h ∈ H ∶ Uh = h} and π ∶ H → H1 the orthogonal projection

onto H1. We then have for every h∈ H

1 N N −1 ∑ n=0 Unh→ π(h) (1.2) in the norm topology of H.

Proof. To start, let us take h∈ H1, then for all N ∈ Z+

1 N N −1 ∑ n=0 Unh= h = π(h). This shows that we get the convergence in (1.2) for all h∈ H1.

Now define H2 ∶= {Uh − h ∶ h ∈ H}. Let us show that H1 and H2 are orthogonal. Take

h∈ H1 and U g− g ∈ H2, then

⟨Ug − g, h⟩ = ⟨g, U−1_h_{⟩ − ⟨g, h⟩}

= ⟨g, h⟩ − ⟨g, h⟩ = 0.

So we have that π(h) = 0 for all h ∈ H2. We also have for all h∈ H2 there exists g∈ H such

that h= Ug − g. We therefore get 1 N N −1 ∑ n=0 Unh= 1 N N −1 ∑ n=0 (Un+1_g_{− U}n_g₎ = 1 NU N_g₋ 1 Ng→ 0 = π(h).

Combining the above results with the linearity of U , yields that the convergence in (1.2) holds for all h∈ H1+ H2. Consider h∈ H1+ H2, we are able to find (hk) ⊂ H1+ H2 such

that hk→ h. Note that because π is an orthogonal projection we have ∣∣π(h)∣∣ ≤ ∣∣h∣∣. For

any ε∈ R+ _{we can find k}∈ Z+ _{such that}

∣∣hk− h∣∣ ≤ ε

and as hk∈ H1+ H2 we can find an M ∈ Z+ such that for all N ≥ M

∥1 N N −1 ∑ n=0 Unhk− π(hk)∥ ≤ ε.

We can use the above inequalities to obtain that for all N ≥ M ∥ 1 N N −1 ∑ n=0 Un_h_{− π(h)∥ ≤ ∥} 1 N N −1 ∑ n=0 Un_(h k− h)∥ + ∥ 1 N N −1 ∑ n=0 Un_h k− π(hk)∥ + ∥π(hk) − π(h)∥ ≤ ∥ 1 N N −1 ∑ n=0 Unhk− π(hk)∥ + 2 ∥hk− h∥ ≤ 3ε.

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As ε above was arbitrary, we have the convergence in (1.2) holds for all h∈ H1+ H2. So if

we can show that H= H1+ H2, then the proof will be complete. Take h∈ H1+ H2 ⊥

. Note that U h− h ∈ H2 and U(U−1h) − U−1h= h − U−1h∈ H2. As h⊥ H2 we have

⟨Uh − h, Uh − h⟩ = ⟨Uh − h, Uh⟩ − ⟨Uh − h, h⟩ = ⟨h − U−1_{h, h}_{⟩ = 0.}

We therefore have U h− h = 0, but h ⊥ H1 and hence h= 0. We can therefore conclude that

H1+ H2= H and that (1.2) holds for all h ∈ H, concluding the proof of the theorem.

The mean ergodic theorem is for general Hilbert spaces, but the following corollary shows how the mean ergodic theorem is related to dynamical systems via the method of Koopman in [33].

Corollary 1.2.1. For all dynamical systems (X, B, µ, T ) and for all f ∈ L2(µ),

1 N N −1 ∑ n=0 f(Tn(⋅)) converges in the L2(µ)-norm as N → ∞.

Proof. We have that L2(µ) is a Hilbert space, so if we show that the operator (known as

the Koopman operator)

UT ∶ L2(µ) → L2(µ), UT(f(x)) = f(T (x))

is unitary, then the corollary follows from the mean ergodic theorem (Theorem 1.2.1). So all that remains is to show for all f, g∈ L2(µ)

⟨UTf, UTg⟩ = ⟨f, g⟩ .

We can use Lemma 1.1.1 to get

⟨UTf, UTg⟩ = ∫ X

f(T (x))g(T (x))dµ(x) = ∫_Xf(x)g(x)dµ(x)

= ⟨f, g⟩ .

We therefore have that UT is a unitary operator, completing the proof of the corollary.

Corollary 1.2.2. For all dynamical systems (X, B, µ, T ) and for all f ∈ L1(µ)

1 N N −1 ∑ n=0 f(Tn(⋅))

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1.2 Classical ergodic theorems 11

Proof. Let(X, B, µ, T ) be a dynamical system, from Corollary 1.2.1 and the fact that the ergodic averages are L∞(µ) contractions we have for all g ∈ L∞(µ)

1 N N −1 ∑ n=0 g(Tn_(⋅)) _(1.3)

converges in L2(µ) to g ∈ L∞(µ). As (X, B, µ) is a finite measure space we have that (1.3)

converges to g in L1(µ) as well. We have that L∞(µ) is dense in L1(µ) and hence for every

f ∈ L1(µ) and every ε ∈ R+ _{we can find g} ∈ L∞(µ) such that ∥f − g∥

1 < ε. Combining this

bound with convergence of (1.3) in L1(µ) yields that we can find an N ∈ Z+ _{such that for}

all N1, N2 > N ∥ 1 N1 N1−1 ∑ n=0 f(Tn(⋅)) − 1 N2 N2−1 ∑ n=0 f(Tn(⋅))∥ 1 ≤ ∥ 1 N1 N1−1 ∑ n=0 (f − g)(Tn_(⋅))∥ 1 + ∥ 1 N1 N1−1 ∑ n=0 (g)(Tn_{(⋅)) − g∥} 1 + ∥ 1 N2 N2−1 ∑ n=0 (f − g)(Tn_(⋅))∥ 1 + ∥ 1 N2 N2−1 ∑ n=0 (g)(Tn_{(⋅)) − g∥} 1 < 4ε.

This completes the convergence aspect of the claim. Let f ∈ L1(µ), we have

∥ 1 N N −1 ∑ n=0 f(Tn(T (⋅))) − 1 N N −1 ∑ n=0 f(Tn(⋅))∥ 1 ≤ 2 N ∥f∥1.

We can therefore conclude that if f is the L1(µ) limit of the ergodic averages of f in (1.3),

then f(T (x)) = f(x) for all x ∈ X, completing the proof of the corollary. Corollary 1.2.3. If (X, B, µ, T ) is ergodic, then for all f ∈ L1(µ)

1 N N −1 ∑ n=0 f(Tn(⋅)) → ∫ Xf(x)dµ(x) in the L1(µ)-norm.

Proof. We know from Corollary 1.2.2 that for all f ∈ L1(µ)

1 N N −1 ∑ n=0 f(Tn(⋅)) → f

in L1(µ), where f ∈ L1(µ) and f(T (x)) = T (x) for all x ∈ X. From Lemma 1.1.2 and our

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can combine this fact with Lemma 1.1.1 to get for almost every y∈ X ∣∫_Xf(x)dµ(x) − f(y)∣ = ∣ lim N →∞ 1 N N −1 ∑ n=0∫X f(Tn(x))dµ(x) − ∫ X f(x)dµ(x)∣ ≤ lim N →∞∫X∣ 1 N N −1 ∑ n=0 f(Tn(x)) − f(x)∣ dµ(x) = 0.

We therefore have f = ∫_Xf(x)dµ(x) in L1(µ), completing the proof of the corollary.

This is the last result surrounding von Neumann’s ergodic theorem that we require for this thesis before moving onto the subsequence results. We will now state a version of Birkhoff’s pointwise ergodic theorem from his paper [8]. We will see throughout this thesis that the pointwise results are in general far more challenging to prove. We therefore delay proving Birkhoff’s pointwise ergodic theorem until Chapter 3, where we will illustrate the use of Bourgain’s oscillation method by proving Birkhoff’s pointwise ergodic theorem. Theorem 1.2.2 (Pointwise Ergodic Theorem). Let (X, B, µ, T ) be a dynamical system. For every f ∈ L2(µ) the ergodic averages

1 N N −1 ∑ n=0 f(Tn(x)) converge pointwise for almost every x∈ X.

1.3 Extension to flows

All the ergodic results from the previous section regard summation over the positive integers and iterations of a dynamical system. These are known as discrete time results. In this section we will consider the definitions and results from the previous sections extended to continuous time systems, known as flows.

Definition 1.3.1. A 1-parameter group is a continuous group homomorphism T ∶ R → G, where R is the usual additive group of real numbers and G is any topological group. We often write Tt= T (t) for notational convenience.

Definition 1.3.2. (X, B, µ, (Tt)) is a measure preserving flow, or flow for short, if

(X, B, µ) is a probability space and (Tt)

t≥0 is a 1-parameter group of dynamical systems,

with composition as the group operation, namely Ts_Tt= Ts+t _{for all s, t}≥ 0.

We can use Example 1.1.2 to get the following example of a flow.

Example 1.3.1. (T, B, λ, (Tt)), with Tt ∶ T → T, Tt(x) = ⟨x + t⟩ is a measure preserving

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1.3 Extension to flows 13

Using a simple technique we can extend the classical ergodic theorems from the previous section to account for flows as well. The technique used is well known by all ergodic theorists. An example of the technique used in the literature is by Bergelson, Leibman and Moreira in [7].

Proposition 1.3.1. Let (X, B, µ, (Tt)) be a measure preserving flow. For any f ∈ L2(µ)

1 N ∫

N 0

f(Tt(⋅))dt converges in the L2(µ)-norm as N → ∞.

Proof. Take f ∈ L2(µ) and define g(x) ∶= ∫1

0 f(Tt(x))dt. First let us show that g ∈ L2(µ).

This follows from Jensen’s inequality, Fubini’s theorem and Lemma 1.1.1 as follows: ∥g(⋅)∥2 2= ∫ X∣g(x)∣ 2_dµ_(x) = ∫_X∣∫₀1f(Tt(x))dt∣ 2 dµ(x) ≤ ∫_X∫₀1∣f(Tt_(x))∣2_dtdµ_(x) = ∫₀1∫_X∣f(Tt_(x))∣2_dµ_(x)dt = ∫₀1∫_X∣f(x)∣2dµ(x)dt = ∥f(⋅)∥2 2.

We can apply Corollary 1.2.1 to g to get

lim N →∞ 1 N ∫ N 0 f(T t_{(⋅))dt = lim} N →∞ 1 N [N ]−1 ∑ n=0 ∫ 1 0 f(T t+n_{(⋅))dt + lim} N →∞ 1 N ∫ N [N ]f(T t_(⋅))dt = lim N →∞ 1 N [N ]−1 ∑ n=0 g(Tn(⋅)) = g,

where the convergence being considered is in the L2(µ) norm. This completes the proof of

the proposition.

Using the same trick as above we can prove the extension of Birkhoff’s theorem to flows. To avoid repetition of the previous proof the proof is omitted.

Proposition 1.3.2. Let (X, B, µ, (Tt)) be a measure preserving flow. For any f ∈ L2(µ)

1 N ∫

N 0

f(Tt(x))dt converges pointwise for almost every x∈ X as N → ∞.

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1.4 Remarks on Chapter 1

This section aimed to give a brief introduction to the classical ergodic theorems. Knowledge of basic measure theory was assumed and if the reader is not acquainted with the topic, Bartle’s book [1] gives an accessible and comprehensive introduction to the field. For a more thorough grounding in classical ergodic theory, the reader is advised to make use of one of the many excellent books available, for example Einsiedler and Ward’s book [21].

The introduction states that different extensions of the classical ergodic theorems have been studied. This thesis only considers one of these extensions. Before proceeding to Chapter 2 we state two of the other main extensions of the classical ergodic theorem that have been the focus of much research. First in Furstenberg and Katznelson’s proof of Szemer´edi’s theorem [25] multiple ergodic averages of the form

1 N N −1 ∑ n=0 f1(Tn)⋯fk(Tkn)

were considered. The multiple ergodic averages can be generalised further to include dif-ferent dynamical systems and a large amount of research is still taking place in this field. Results in this field have been obtained by Host and Kra [28], Ziegler [48], Bergelson, Leibman and Lesigne [6], Tao [41] and many others.

Another extension of the classical ergodic theorems is to consider convergence of the ergodic averages with coefficients, namely for a sequence cn considering convergence of the

averages 1 N N −1 ∑ n=0 cnf(Tn).

Theorems of this type are known as weighted ergodic theorems and have received a large amount of attention, partly due to their use in proving return time results. Results in this field have been obtained by Bourgain, Furstenberg, Katznelson and Ornstein [18], Demeter, Lacey, Tao and Thiele [20], Rudolph [38], Host and Kra [29][30] and many others.

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Chapter 2 Subsequence ergodic theorems for

mean convergence

2.1 Introduction to subsequence theorems

In this and the following chapter we study natural extensions of the classical ergodic theorems of Chapter 1, namely what can be said about convergence of ergodic averages along subsequences of the positive integers and real numbers. The two main results from the previous chapter are the mean and pointwise ergodic theorems. In this chapter we will address the convergence of ergodic averages along subsequences for mean convergence only. Let us start with some definitions to set the scene for this chapter.

Definition 2.1.1. Let (an) ⊂ Z+ be an increasing sequence of positive integers and

(X, Σ, µ, T ) a measure preserving dynamical system. We say that (an) is mean good for

the dynamical system (X, Σ, µ, T ) if for all f ∈ L2(µ) the limit

lim N →∞ 1 N N −1 ∑ n=0 f(Tan(⋅)) = f(⋅)

exists in L2(µ). If the sequence (a

n) is mean good for all measure preserving systems, then

we say that (an) is universally mean good or just mean good for short.

We can also extend the definition to real sequences as follows.

Definition 2.1.2. Let (an) ⊂ R+ be an increasing sequence of positive real numbers and

(X, Σ, µ, (Tt)) a measure preserving flow. We say that (a

n) is mean good for the flow

(X, Σ, µ, (Tt)) if for all f ∈ L2(µ) the limit

lim N →∞ 1 N N −1 ∑ n=0 f(Tan(⋅)) = f(⋅)

exists in L2(µ). If the sequence (a

n) is mean good for all measure preserving flows, then

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16 2. Subsequence ergodic theorems for mean convergence

Definition 2.1.3. If (an) is an increasing sequence that is not mean good then we call

(an) mean bad.

Looking merely at the definitions provided above, it seems difficult to know where to start when trying to prove that a sequence is mean good. We know that the set of positive integers is mean good from Corollary 1.2.1 of the mean ergodic theorem, but before getting to some interesting examples we prove the most important result when dealing with subsequence theorems of mean convergence. This fundamental result is folklore within ergodic theory and shows the relation between mean good sequences and convergence of exponential sums. The result is a simple consequence of the spectral theorem. If the reader is unfamiliar with the spectral theorem, Halmos’s article [26] provides an intuitive explanation of the theorem. We will be dealing with Euler’s form of the unit circle many times throughout the rest of the thesis and we therefore define

e(x) = e2πix. (2.1) Proposition 2.1.1. Let (an) ⊂ Z+ be an increasing sequence of positive integers. The

sequence (an) is mean good if and only if, for every β ∈ R the limit

lim N →∞ 1 N N −1 ∑ n=0 e(anβ) = α(β) exists.

Proof. (⇐) First assume for every β ∈ R we have that the limit lim N →∞ 1 N N −1 ∑ n=0 e(anβ) = α(β) (2.2)

exists. Let (X, Σ, µ, T ) be a dynamical system, f ∈ L2(µ) and let µ

f be the spectral

measure of f . We then have from the spectral theorem that

∥ 1 N N −1 ∑ n=0 f(Tan(⋅)) − 1 M M −1 ∑ n=0 f(Tan(⋅))∥ 2 L2 = ∫ 1 0 ∣ 1 N N −1 ∑ n=0 e(anβ) − 1 M M −1 ∑ n=0 e(anβ)∣ 2 dµf(β).

It is easy to see that

∣ 1 N N −1 ∑ n=0 e(anβ) − 1 M M −1 ∑ n=0 e(anβ)∣ 2 ≤ (1 N N −1 ∑ n=0 ∣e(anβ)∣ + 1 M M −1 ∑ n=0 ∣e(anβ)∣) 2 ≤ 4. So we can combine the bounded convergence theorem with our assumption (2.2) to get

lim N,M →∞∫[0,1)∣ 1 N N −1 ∑ n=0 e(anβ) − 1 M M −1 ∑ n=0 e(anβ)∣ 2 dµf(β) = ∫_[0,1) lim N,M →∞∣ 1 N N −1 ∑ n=0 e(anβ) − 1 M M −1 ∑ n=0 e(anβ)∣ 2 dµf(β) = 0.

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2.1 Introduction to subsequence theorems 17 We therefore have 1 N N −1 ∑ n=0 f(Tan(⋅))

converges in L2(µ). As this is true for all f ∈ L2(µ) and for all measure preserving systems

we have that(an) is mean good, completing the first implication.

(⇒) Now assume that there exists a β such that the limit lim N →∞ 1 N N −1 ∑ n=0 e(anβ) (2.3)

does not exist. Take T(x) = ⟨x + β⟩ which from Example 1.1.2 is a dynamical system on the unit circle (T, B, λ). Take f(x) = e(x) ∈ L2(λ) and we have by construction that

∥1 N N −1 ∑ n=0 f(Tan(⋅))∥ 2 = ∥e(⋅)∥2∣ 1 N N −1 ∑ n=0 e(anβ)∣ ,

which does not converge from assumption (2.3). This completes the second implication and the proof of the proposition.

The above proof addresses integer sequences. However, the same proof works for real sequences as well. One just has to consider the spectral measure over the set of real numbers instead of the unit circle. We now have a tool which makes it easy to give many examples and classes of mean good and mean bad sequences. What the above theorem is intuitively saying is that it is sufficient to only consider rotations on the unit circle under the exponential map in order to prove that a sequence is mean good. We now give a few of the simpler examples of mean good and mean bad sequences, delaying the more complicated examples for later in the chapter. The first set of examples are direct corollaries of Proposition 2.1.1.

Corollary 2.1.1. If (an) is mean good, then for any k ∈ R+ we have that (kan) is also

mean good.

From the mean ergodic theorem (Corollary 1.2.1) we have that (n) is mean good and we can combine this fact with Corollary 2.1.1 to get for any k∈ R+_that(kn) is mean good.

Another consequence of Proposition 2.1.1 is what is known as the translation invariance of mean good sequences.

Corollary 2.1.2. If (an) is mean good, then for any k ∈ R+ we have that (an+ k) is also

mean good.

The following result shows that there exists a mean good sequence and a bounded sequence such that their sum is mean bad.

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Proof. We know that (2n) is mean good from Corollary 2.1.1, now set bn∶= {

0 ∶ if ∃m ∈ Z+ _{such that 2}2m≤ n < 22m+1_,

1 ∶ if ∃m ∈ Z+ _{such that 2}2m+1≤ n < 22m+2_.

Take β =1₂ then by construction we have lim sup N →∞ 1 N N −1 ∑ n=0 e((2n + bn)β) = lim sup N →∞ 1 N N −1 ∑ n=0 e(bn 2) = 1 3 and lim inf N →∞ 1 N N −1 ∑ n=0 e((2n + bn)β) = lim inf N →∞ 1 N N −1 ∑ n=0 e(bn 2) = −1 3 . We therefore have that convergence of the exponential sum fails for β = 1

2 and from

Pro-position 2.1.1 we have (kn + bn) is mean bad.

We now show how the mean good property can be related to equidistribution on the unit circle. This follows from Weyl’s famous result which shows the equivalence of equidistri-bution on the unit circle and the convergence of exponential sums. Let us first define equidistribution on the unit circle.

Definition 2.1.4. A sequence(an) of real numbers is said to be equidistributed on the

unit circle or equidistributed mod 1 if for any a< b ∈ T we have lim N →∞ N −1 ∑ n=0 χ[a,b](⟨an⟩) = b − a.

Tying together the definition of equidistribution and the mean good property will allow us to use the substantial results in the study of equidistribution to develop many examples of mean good sequences. The following proposition, known as Weyl’s criterion and proved by Weyl in [44], allows us to do just that.

Proposition 2.1.2 (Weyl’s Criterion). A sequence(an) on the unit circle is equidistributed

if and only if for all k∈ Z≠0

lim N →∞ 1 N N −1 ∑ n=0 e(kan) = 0.

Proof. (⇒) Let λ be the Lebesgue measure on the unit circle and assume that (an) is

equidistributed on the unit circle. We then have

lim N →∞ 1 N N −1 ∑ n=0 χ[a,b](an) = b − a = λ(χ[a,b](⋅)).

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2.1 Introduction to subsequence theorems 19

As the measure of a function is linear, we have for any linear combination of characteristic functions (any simple function) f that

lim N →∞ 1 N N −1 ∑ n=0 f(an) = λ(f(⋅)).

Any bounded continuous function on the unit circle can be uniformly approximated by simple functions. Integration respects uniform approximation and e(kx) is continuous and bounded by 1. So we have for all k∈ Z≠0

lim N →∞ 1 N N −1 ∑ n=0 e(kan) = λ(e(k⋅)) = 0,

completing the first implication.

(⇐) Assume now that for all k ∈ Z ∖ {0} lim N →∞ 1 N N −1 ∑ n=0 e(kan) = 0 = λ(e(k⋅)). We also have lim N →∞ 1 N N −1 ∑ n=0 e(0an) = 1 = λ(e(0⋅)).

From the linearity of measures we get for any f(x) = ∑M_m=0bme(mx) that

lim N →∞ 1 N N −1 ∑ n=0 f(an) = λ(f(⋅)).

From the Stone-Weierstrass theorem we have that any bounded continuous function f can be approximated with functions of the form above. We therefore have for any continuous f lim N →∞ 1 N N −1 ∑ n=0 f(an) = λ(f(⋅)).

Let a < b be on the unit circle, then for any ε ∈ R+ _{we can find a bounded continuous}

function f such that f ≤ χ[a,b] and ∣∣f(⋅) − χ[a,b](⋅)∣∣1< ε. We therefore have

lim inf n→∞ 1 N N −1 ∑ n=0

χ[a,b](an) ≥ lim inf n→∞ 1 N N −1 ∑ n=0 f(an) = λ(f(⋅)) ≥ λ(χ[a,b](⋅)) − ε.

The same argument can be done to yield that for any ε∈ R+

lim sup n→∞ 1 N N −1 ∑ n=0 χ[a,b](an) ≤ λ(χ[a,b](⋅)) + ε.

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Given that this was shown for any ε∈ R+ _{we can conclude}

lim N →∞ N −1 ∑ n=0 χ[a,b](an) = b − a = λ(χ[a,b](⋅)).

This is exactly the definition for equidistribution, thereby completing the second implica-tion and proof of this proposiimplica-tion.

We finish this section by answering the question of how mean bad a sequence can be. Namely for a sequence (an) to be mean bad, from Proposition 2.1.1, we only require one

β∈ R such that 1 N N −1 ∑ n=0 e(anβ) (2.4)

does not converge. A logical question then arises as to how bad a sequence can be. Is it, for instance, possible to find a sequence such that convergence of (2.4) fails for all β ∈ [0, 1)? The answer is negative shown by the following famous theorem of Weyl, also proved in his seminal paper [44], on the equidistribution of sequences on the unit circle. This theorem not only shows that for every strictly increasing integer sequence (an) we

can find a β∈ [0, 1) such that (2.4) converges, it also shows that the set of all such β is of full measure in [0, 1).

Theorem 2.1.1. If (an) is a strictly increasing sequence of positive integers, then for

almost every β ∈ [0, 1) we have

lim N →∞ 1 N N −1 ∑ n=0 e(anβ) = 0.

Proof. First let λ be the Lebesgue measure on the unit circle and note that the functions e(anx) form an orthonormal set in L2(λ). We therefore have

∫ 1 0 ∣ N −1 ∑ n=0 e(anx)∣ 2 dλ(x) = ∣∣e(a0x) + ⋅ ⋅ ⋅ + e(aN −1x)∣∣22= N. This implies ∫ 1 0 ∣ 1 N N −1 ∑ n=0 e(anx)∣ 2 dλ(x) = 1 N. (2.5)

As N was arbitrary we can substitute N2 _{for N in (2.5). We can then sum over N and}

apply Fubini’s theorem to interchange the integral and summation, which yields

∫ 1 0 ⎛ ⎝ ∞ ∑ N =1 RRRRR RRRRR R 1 N2 N2₋₁ ∑ n=0 e(anx)RRRRRRRRRR R 2 ⎞ ⎠dλ(x) = ∞ ∑ N =1 1 N2 = π2 6 . (2.6) However we have that if

lim N →∞ 1 N2 N2₋₁ ∑ n=0 e(anx) ≠ 0,

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2.2 Mean good sequences 21 then ∞ ∑ N =1 ∣ 1 N2 N2₋₁ ∑ n=0 e(anx)∣2= ∞. (2.7)

But the integral in (2.6) is finite, the property in (2.7) can therefore only occur on a set of measure zero. From this we can conclude

lim N →∞ 1 N2 N2₋₁ ∑ n=0 e(anβ) = 0 (2.8)

for almost every β∈ [0, 1). We can use (2.8) to get that for almost every β ∈ [0, 1) lim sup N →∞ ∣ 1 N N −1 ∑ n=0 e(anβ)∣ ≤ lim sup N →∞ RRRRR RRRRR RRRRR 1 [√N]2 [ √ N ]2−1 ∑ n=0 e(anβ) + 1 [√N]2 N −1 ∑ n=[√N ]2 e(anβ) RRRRR RRRRR RRRRR ≤ lim sup N →∞ ⎛ ⎜⎜ ⎝ RRRRR RRRRR RRRRR 1 [√N]2 [ √ N ]2−1 ∑ n=0 e(anβ) RRRRR RRRRR RRRRR+ 1 [√N]2 N −1 ∑ n=[√N ]2 ∣e(anβ)∣ ⎞ ⎟⎟ ⎠ ≤ lim sup N →∞ ⎛ ⎜⎜ ⎝ RRRRR RRRRR RRRRR 1 [√N]2 [ √ N ]2−1 ∑ n=0 e(anβ) RRRRR RRRRR RRRRR+ 2[√N] + 1 [√N]2 ⎞ ⎟⎟ ⎠ = 0.

This shows convergence of the exponential sum for almost all β ∈ [0, 1) and completes the proof of the proposition.

2.2 Mean good sequences

Our aim in this section is to show some examples of mean good sequences that will allow one to gain a feel for the structure of the class of all mean good sequences. Some tools will be developed to aid in the study of mean good sequences and representative examples will be displayed in order to highlight techniques applied in this area. Some of the examples shown in this section were chosen to highlight the differences and similarities between mean good and pointwise good sequences, the latter being addressed in the following chapter.

Our first result highlights the stability of mean good sequences. We saw in Example 2.1.1 that by perturbing a mean good sequence by a bounded sequence we can get a mean bad sequence. The following proposition however, shows that if we perturb a mean good sequence by a sequence converging to zero, then the sequence remains good. This is an interesting result as one will see in the next chapter that the same is not true for pointwise convergence.

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Proposition 2.2.1. If(an) is a mean good sequence of real numbers and (bn) is a sequence

of real numbers such that bn→ 0, then (an+ bn) is mean good.

Proof. If (an) is mean good, then from Proposition 2.1.1 we have for every β ∈ R there

exist an α(β) ∈ R such that

lim N →∞ 1 N N −1 ∑ n=0 e(anβ) = α(β). (2.9)

We then have that

∣ lim N →∞ 1 N N −1 ∑ n=0 e((an+ bn)β) − α(β)∣ = lim N →∞∣ 1 N N −1 ∑ n=0 e(anβ) (e(bnβ) − 1)∣ ≤ lim N →∞ 1 N N −1 ∑ n=0 ∣e(bnβ) − 1∣ = 0.

The last equality follows from the property that every convergent sequence is Cesaro con-vergent to the same limit. We therefore have from Proposition 2.1.1 that(an+ bn) is mean

good.

The following theorem will provide several interesting examples of mean good sequences. It shows that we can classify a large class of mean good sequences by their rate of growth. There is a large amount of research in this area, with an important result being a necessary and sufficient growth condition for mean good sequences in Hardy fields obtained in Bosh-ernitzan and Wierdl’s article [13]. We will also see some classifications of bad sequences via their growth rates in the next section. The following result has an elementary proof and is attributed to Fej´er by Boshernitzan in [10]. The proof provided is an adaptation of Boshernitzan’s proof in [10]. Before we start we introduce a definition for notational convenience.

Definition 2.2.1. Let (an) be a sequence of real numbers, define ∆an∶= an+1− an.

Theorem 2.2.1 (Fej´er’s Theorem). If (an) ⊂ R is a strictly increasing sequence of real

numbers, with (∆an) a monotonically decreasing sequence, such that ∆an→ 0 and n∆an→

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2.2 Mean good sequences 23

Proof. Assume that the criteria of the hypothesis holds, then we have for every k∈ Z+_that

2πk N ∣ N −1 ∑ n=0 e(ank)∣ = 1 N ∣ N −1 ∑ n=0 (2πike(ank) + e(ank) ∆an −e(an+1k) ∆an+1 ) +e(aNk) ∆aN −e(a0k) ∆a0 ∣ ≤ 1 N ∣ N −1 ∑ n=0 (2πike(ank) + e(ank) ∆an − e(an+1k) ∆an )∣ + 1 N ∣ N −1 ∑ n=0 (e(an+1k) ∆an −e(an+1k) ∆an+1 )∣ + 1 N ∣ 1 ∆aN ∣ + 1 N ∣ 1 ∆a0 ∣ ≤ 1 N ∣ N −1 ∑ n=0 ( 1 ∆an ∣2πik∆an+ 1 − e(∆ank)∣)∣ + 1 N N −1 ∑ n=0 ∣ 1 ∆an+1 − 1 ∆an∣ + 1 N ∣ 1 ∆aN∣ + 1 N ∣ 1 ∆a0∣ .

Using a trick by replacing the sum of the first term above by an integral yields 2πk N ∣ N −1 ∑ n=0 e(ank)∣ ≤ 1 N N −1 ∑ n=0 (4π2 ∆an∣∫ k∆an 0 (k∆an− x)e(x)dx∣) + 1 N N −1 ∑ n=0 ( 1 ∆an+1 − 1 ∆an ) + 1 N ∣ 1 ∆aN ∣ + 1 N ∣ 1 ∆a0 ∣ ≤ 1 N N −1 ∑ n=0 (4π2 ∆an∫ k∆an 0 ∣k∆an− x∣ dx) + 1 N ∣ 2 ∆aN ∣ + 1 N ∣ 1 ∆a0 ∣ = 1 N N −1 ∑ n=0 (2π2_k2_∆a2 n) + 1 N ∣ 2 ∆aN ∣ + 1 N ∣ 1 ∆a0 ∣ → 0 as N → ∞.

Thus from Weyl’s criterion (Proposition 2.1.2) we have that(an) is equidistributed modulo

1.

Corollary 2.2.1. If f ∶ R+ → R is a strictly increasing differentiable function such that

f′(x) → 0 is decreasing and xf′(x) → ∞, then (f(n))

n∈Z+ is equidistributed on the unit

circle.

Corollary 2.2.2. If (an) ⊂ R is a strictly increasing sequence of real numbers, with (∆an)

a monotonically decreasing sequence, such that ∆an→ 0 and n∆an→ ∞, then (an) is mean

good.

Proof. Let (an) ⊂ R+ be a strictly increasing sequence of real numbers, with (∆an) a

monotonically decreasing sequence, such that ∆an → 0 and n∆an → ∞. Then for any

β ∈ (0, 1] we have (∆βan) is a monotonically decreasing sequence with ∆βan → 0 and

n∆βan→ ∞. We therefore have from Theorem 2.2.1 that (βan) is equidistributed on the

unit circle. We can apply Proposition 2.1.2 to get for every k∈ Z ∖ {0} and every β ∈ (0, 1] 1 N N −1 ∑ n=0 e(anβk) → 0. (2.10)

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We therefore have for any β ∈ R that 1 N N −1 ∑ n=0 e(anβ) (2.11)

converges and applying Proposition 2.1.1 yields that (an) is mean good.

From the above corollary we get the following examples of mean good sequences. Example 2.2.1. If (an) is any of the following sequences

(nα_{) , (n}α_logβ_{(n)) , (n}α_log_(nβ_{)) or (log}γ_(n))

for any α∈ (0, 1), β > 0 and γ > 1, then (an) is mean good.

These examples are all clearly not integer sequences, one can therefore wonder whether ([an]) is also mean good. Unfortunately, the general result that for all mean good sequences

(an), ([an]) is also mean good, does not hold. This will be proven in the next section. It

can however be shown that if (βan) is equidistributed for all β ≠ 0 then both (an) and

([an]) are mean good sequences. A more general version of this result was proved by

Niederreiter in [35]. The version dealt with here is however that of Boshernitzan, Kolesnik, Quas and Wierdl in [12].

Proposition 2.2.2. If (an) is an increasing sequence of real numbers and for all β ≠ 0

we have (βan) is equidistributed on the unit circle, then (an) and ([an]) are mean good

sequences.

Proof. Let λ be the usual Lebesgue measure on the unit circle and assume that for all β≠ 0, (anβ) mod 1 is equidistributed on the unit circle. The fact that (an) is mean good

follows trivially from Weyl’s criterion (Proposition 2.1.2), as for any β ≠ 0 we have 1 N N −1 ∑ n=0 e(anβ) → 0. (2.12)

Let us show that ([an]) is mean good. As we are dealing with an integer sequence it is

trivial that 1 N N −1 ∑ n=0 e([an]β) → 1

for all β ∈ Z. All that remains is to show that the exponential sums converge for all β ∈ R ∖ Z. Take ε ∈ (0,1₂) and choose M ∈ Z+ _{to be large enough such that if} ∣x − y∣ < 2

M

then ∣e(xβ) − e(yβ)∣ < ε₂. Finally define δ ∶= _Mε and the functions fm(x) ∶ T → T with

m= 0, . . . , M − 1 by fm(x) = ⎧⎪⎪⎪ ⎪⎪ ⎨⎪⎪ ⎪⎪⎪⎩ 1 ∶ x ∈ (_Mm + δ,m+1_M − δ), −1 2δx+ m+1 2δM + 1 2 ∶ x ∈ [ m+1 M − δ, m+1 M + δ], 1 2δx− m 2δM + 1 2 ∶ x ∈ [ m M − δ, m M + δ], 0 ∶ otherwise.

We have defined the functions fm to be piecewise linear continuous functions of the

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1 1

x fm(x)

Before we continue let us note the following facts about the functions fm we have just

constructed.

• By construction we have that ∑M −1

m=0fm(⋅) = 1.

• If fm(an) > 0 then an− [an] ∈ [_Mm − δ,m+1_M + δ], which implies that an− [an] −_Mm ∈

[−δ, 1

M + δ]. We therefore have ∣an− [an] − m M∣ < δ +

1 M <

2

M. But from the definition

of M this means ∣e([an]β) − e((an−_Mm)β)∣ < ₂ε.

• From the Stone-Weierstrass theorem there exists an L ∈ Z+ _{such that}

∣∣ ∑L−1

l=−L+1bm,le(l⋅) − fm(⋅)∣∣2 < _2Mε .

If we take β ∈ R ∖ Z and define αl∶= l + β ≠ 0 for any integer l, we get

∣1 N N −1 ∑ n=0 e([an]β)∣ = ∣ 1 N N −1 ∑ n=0 M −1 ∑ m=0 fm(an)e([an]β)∣ < ε 2+ ∣ 1 N N −1 ∑ n=0 M −1 ∑ m=0 fm(an)e ((an− m M) β)∣ < ε + ∣1 N N −1 ∑ n=0 M −1 ∑ m=0 L−1 ∑ l=0 bm,le(lan)e ((an− m M) β)∣ = ε + ∣M −1∑ m=0 L−1 ∑ l=0 bm,l 1 N N −1 ∑ n=0 e(lan)e ((an− m M) β)∣ ≤ ε +M −1∑ m=0 L−1 ∑ l=0 (∣bm,l∣ ∣ 1 N N −1 ∑ n=0 e(anαl)∣) → ε as N → ∞.

Where the last limit follows from (2.12). As this holds for any ε∈ (0,1₂) we have for any β∈ R ∖ Z 1 N N −1 ∑ n=0 e([an]β) → 0.

We have therefore shown that the limit of the exponential sum exists for all β ≠ 0 and we can apply Proposition 2.1.1 to get that ([an]) is a mean good sequence.

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This proposition yields that the integer part of all the mean good sequences in Example 2.2.1 are also mean good, giving us the following example.

Example 2.2.2. If (an) is any of the following sequences

([nα_{]) , ([n}α_logβ_{(n)]) , ([n}α_log_(nβ_{)]) or ([log}γ_(n)])

for any α∈ (0, 1), β > 0 and γ > 1, then (an) is mean good.

To conclude this section, we will prove that for any polynomial p(x) ∈ Z[x] with positive coefficients, the sequence (p(n)) is mean good. It should be clear from Corollary 2.1.1 that this result will immediately imply that (p(n)) is mean good for any p(x) ∈ Q[x] with positive coefficients. This is a good example to consider as it highlights the use of one of the standard machines in showing that a sequence is mean good. This result is a consequence of another of Weyl’s equidistribution theorems, this time for polynomials on the unit circle, and was also included in his seminal paper [44]. Furstenberg proved a more general result in 1981 in [24] which applies to operators on general Hilbert spaces. For our purpose it is sufficient to only prove the result for ergodic averages of L2_-functions.

The standard machine used here is to consider the convergence of the exponential sums in Proposition 2.1.1 for irrational and rational β separately. We first show that the con-vergence of exponential sums holds for irrational β by proving (βp(n)) is equidistributed on the unit circle for all irrational β and applying Weyl’s criterion (Proposition 2.1.2). We then show convergence for rational β using periodicity modulo q. Combining these two results yields convergence for all β and hence(p(n)) is mean good. Let us start by proving that for all irrational β we have (βp(n)) is equidistributed on the unit circle. The proof is an inductive argument using van der Corput’s difference theorem [42] for the induction step. Let us now prove the induction base which is referred to as Weyl’s equidistribution theorem.

Theorem 2.2.2 (Weyl’s Equidistribution Theorem). For all β ∈ R ∖ Q and α ∈ R the sequence (βn + α) is equidistributed on the unit circle.

Proof. Fix β an irrational real number and take α ∈ R. Then we have for all k ≠ 0 and N ∈ Z+ _{that e}(kβN) ≠ 1. This implies that

∣ 1 N N −1 ∑ n=0 e(k(βn + α))∣ = 1 N ∣ N −1 ∑ n=0 e(kβ)n∣ = 1 N ∣ 1− e(kβ)N 1− e(kβ) ∣ ≤ 2 N ∣(1 − e(kβ)) −1 ∣ → 0.

From Weyl’s criterion (Proposition 2.1.2) we have (βn + α) is equidistributed on the unit circle, completing the proof of this proposition.

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We have proved our induction base, so let’s prove the tool used for the induction step, namely the van der Corput difference theorem [42].

Theorem 2.2.3 (Van der Corput Difference Theorem). Let (an) be a sequence of real

numbers. If for all h∈ Z+ _{the sequence} (a

n+h− an) is equidistributed on the unit circle then

(an) is also equidistributed on the unit circle.

Proof. Take ε∈ R, k ∈ Z≠0and H > 1_ε. Assume that for every h∈ Z+the sequence(an+h−an)

is equidistributed on the unit circle. We will now use Weyl’s criterion (Proposition 2.1.2) to prove that (an) is also equidistributed on the unit circle.

∣ 1 N N −1 ∑ n=0 e(kan)∣ 2 ≤ ∣ 1 H H ∑ h=1 1 N N −1 ∑ n=0 e(kan+h)∣ 2 + ∣ 1 N H ∑ n=0 e(kan)∣ 2 + ∣ 1 N N +H ∑ n=H e(kan)∣ 2 ≤ ∣ 1 H H ∑ h=1 1 N N −1 ∑ n=0 e(kan+h)∣ 2 +2H N .

We can take N to be large enough such that 2H_N < ε, and we get ∣1 N N −1 ∑ n=0 e(kan)∣ 2 < 1 N N −1 ∑ n=0 ∣ 1 H H ∑ h=1 e(kan+h)∣ 2 + ε = 1 N N −1 ∑ n=0 1 H2 H ∑ h1=1 H ∑ h2=1 e(kan+h1− kan+h2) + ε ≤ 1 H + 1 H2 H ∑ h1=1 H ∑ h2=1 ∣ 1 N N −1+h2 ∑ n=h2 e(kan+h1−h2 − kan)∣ + ε.

We have by construction that _H1 < ε. We also have by assumption that for all h1 ≠ h2,

(an+h1−h2 − an) is equidistributed. Applying Weyl’s criterion (Proposition 2.1.2) we can

take N large enough such that

∣ 1 N N −1+h2 ∑ n=h2 e(kan+h1−h2− kan)∣ < ε.

Combining all these facts yields

∣1 N N −1 ∑ n=0 e(kan)∣ 2 < 3ε.

As ε is any positive real number we get convergence to zero. We can therefore apply Weyl’s criterion (Proposition 2.1.2) to get (an) is equidistributed on the unit circle, completing

the proof of the theorem.

We can now combine the induction hypothesis and induction step to prove Weyl’s equidistribution theorem for polynomials [44].

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Theorem 2.2.4. Let p(x) be a polynomial with leading coefficient irrational, then (p(n)) is equidistributed on the unit circle.

Proof. We will prove this claim by induction on the degree of the polynomial. From Theorem 2.2.2 we have(p(n)) is equidistributed on the unit circle if the degree of p(x) is 1 and the leading coefficient is irrational. Now assume the theorem holds for all polynomials of degree less than k and take p(x) to be a polynomial of degree k with leading coefficient α irrational. We have for any h∈ Z+ _{that the polynomial p}(x + h) − p(x) is a polynomial of

degree k− 1, with leading coefficient kαh, which is irrational. We can therefore apply the induction hypothesis to get (p(n + h) − p(n)) is equidistributed on the unit circle. As this holds for any h∈ Z+_{, we can apply Theorem 2.2.3 to get that}(p(n)) is equidistributed on

the unit circle. This completes the induction step and shows that for all polynomials p(x), with leading coefficient irrational,(p(n)) is equidistributed on the unit circle.

We can now proceed to the case of rational β by proving the following proposition.

Proposition 2.2.3. Let (p(n)) be a polynomial and β ∈ Q then the exponential sum 1 N N −1 ∑ n=0 e(p(n)β) (2.13) converges.

Proof. Let p(x) ∈ Z[X] be a polynomial with positive coefficients and take β ∈ (0, 1]. Without loss of generality assume that p(x) has zero constant term and β ≠ 0. As β is rational we can find a, b ∈ Z+ _{such that β} = a

b. Note that as the polynomial has integer

coefficients and constant term equal to zero we have p(bn+r) = p(r) mod b for all n, r ∈ Z+_.

But then lim N →∞ 1 N N −1 ∑ n=0 e(p(n)a b) = 1 b b−1 ∑ n=0 e(p(n)a b) .

This shows that convergence also holds for rational β and completes the proof of the proposition.

We can now combine Theorem 2.2.4 and Proposition 2.2.3 to get our main, and final, result for this section.

Proposition 2.2.4. Let p(x) ∈ Z[x] be a polynomial with positive integer coefficients, then the sequence (p(n)) is mean good.

Proof. Let p(x) ∈ Z[x], then from Theorem 2.2.4 we have for all irrational β that (βp(n)) is equidistributed on the unit circle. Applying Weyl’s criterion (Proposition 2.1.2) yields for all irrational β

1 N N −1 ∑ n=0 e(p(n)β) → 0.

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2.3 Mean bad sequences 29

We can also apply Proposition 2.2.3 to get for all rational β 1 N N −1 ∑ n=0 e(p(n)β)

converges. From Proposition 2.1.1 we have that(p(n)) is mean good, thereby completing the proof of the proposition.

2.3 Mean bad sequences

To finish off this chapter we will discuss some interesting examples of mean bad sequences. To start, let us construct a mean good sequence of positive real numbers (an) such that

([an]) is mean bad. This result was discussed by Lesigne in [34], where the integer parts

of good sequences were considered. The construction used by Lesigne is a similar idea to the one provided here.

Example 2.3.1. There exists an increasing sequence (an) of real numbers, such that (an)

is mean good whilst ([an]) is mean bad.

Proof. Define an= n + (1 − cn) where

cn∶= { 1

n ∶ if ∃m ∈ Z

+ _{such that 2}2m≤ n < 22m+1_,

0 ∶ if ∃m ∈ Z+ _{such that 2}2m+1≤ n < 22m+2_.

From Corollary 1.2.1 of the mean ergodic theorem and the translation invariance property (Proposition 2.1.2) we have (n + 1) is mean good. We also have from Proposition 2.2.1 that a mean good sequence added to a sequence converging to zero is also mean good. We have by construction that cn → 0 and we therefore get (n + 1 − cn) = (an) is mean good.

In Proposition 2.1.1 we constructed a 0− 1 sequence (bn) such that n + bn is mean bad.

However we have by construction that [an] = n + bn, where bn is exactly the 0− 1 sequence

in Proposition 2.2.1. We therefore have that([an]) is mean bad, which completes the proof

of the proposition.

The following results further highlight the effect that the growth rate of a sequence has on the mean good property. The first theorem, proved by Bellow in [2], shows that very fast growing sequences, known as lacunary sequences, are mean bad.

Definition 2.3.1. A positive sequence (an) is called lacunary if there exists λ > 1 such

that an+1

an ≥ λ for all n ∈ Z

+_.

In Rosenblatt and Wierdl’s survey paper [37] the following proposition and lemma are left as exercises.

Lemma 2.3.1. Let (fn) and (gn) be sequences of continuous functions on (T, B, µ). If

there exist dense sets, A, B∈ T and a, b ∈ T, a > b such that for all x1∈ A, x2∈ B we have

lim infN →∞fN(x1) > a and lim supN →∞gN(x2) < b, then there exists a dense set S ⊂ T such

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Proof. Let us define the sets An ∶= {x ∈ T ∶ fn(x) > a} and Bm ∶= {x ∈ T ∶ gm(x) < b}.

From the continuity of fn and gm we have that An and Bm are open sets for all n, m∈ Z+.

We therefore have for all N, M ∈ Z+ _that ⋃∞

n=NAn and ⋃∞m=MBn are open. We also have

that ⋃∞

n=NAn ⊃ A and ⋃∞m=MBn ⊃ B and hence ⋃n=N∞ An and ⋃∞m=MBn are open dense

subsets of T. We can therefore apply the Baire category theorem to get that any countable intersection of open dense sets is dense. Hence, the set

S ∶= ∞ ⋂ N =1 ∞ ⋂ M =1 (⋃∞ n=N An) ∩ ( ∞ ⋃ m=M Bm) = ⋂∞ N =1 ∞ ⋃ n=N ∞ ⋂ M =1 ∞ ⋃ m=M (An∩ Bm) = {x ∈ T ∶ lim sup N →∞

fN(x) > a and lim inf

N →∞ gN(x) < b}

is dense in T, which completes the proof of the lemma.

Theorem 2.3.1. If(an) is a lacunary sequence of positive real numbers, then (an) is mean

bad.

Proof. Let(an) be a lacunary sequence of positive real numbers, then there exists a λ > 1

such that an+1

an ≥ λ for all n ∈ Z

+_{. As} ( n

log(2λn)) is an unbounded strictly increasing sequence,

we can find q∈ Z>3 such that _log(2λq)q > _{log λ}2 . This is equivalent to

log λ log(2λq) >

2

q. (2.14)

Let us use the above construction to prove that for a dense set A⊂ T, we have lim inf N →∞ 1 N N −1 ∑ n=0 χ_(0,1 q)(⟨anβ⟩) > log λ log(2λq) (2.15) for all β ∈ A. Let m ∈ Z+ _{be the smallest positive integer such that λ}m ≥ 2q. We therefore

have that λm < 2λq, which is equivalent to 1 m >

log λ

log(2λq). Now define the sequence (bn) by

bn∶= amn. If we find an N ∈ Z+ such that for all n≥ N we have ⟨bnβ⟩ ∈ (0,1_q), then

lim inf N →∞ 1 N N −1 ∑ n=0 χ_(0,1 q)(⟨anβ⟩) ≥ 1 m > log λ log(2λq).

So it is sufficient to show for each β in a dense set that there exists an Nβ ∈ Z+, with

⟨bnβ⟩ ∈ (0,1_q) for all n ≥ Nβ. Let γ ∈ T, ε ∈ R+, U ∶= B(γ, ε) (the open ε-ball centred at γ)

and An∶= [0,_b_n1_q) + _b1_nZ+. It is then clear that⟨bnβ⟩ ∈ (0,1_q) for all n ≥ Nβ is equivalent to

β ∈ ⋂n≥NβAn. Using this construction it is sufficient to show that there exists an N ∈ Z

+

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2.3 Mean bad sequences 31

Let N ∈ Z+ _{be large enough such that we can find an integer α}

N with BN ∶= [α_bN N, αN bN + 1 bNq] ⊂ U. We also have bn+1 bn ≥ λ

m ≥ 2q by construction. So for any n, α

n ∈ Z+ we can

choose αn+1 to be the smallest positive integer such that bn+1_b_n αn ≤ αn+1. This implies

αn+1≤ bn+1_b_n αn+ 1, which in turn implies

αn+1 bn+1 + 1 bn+1q ≤ αn bn + 1 bn+1 + 1 bn+1q ≤ αn bn + 1 2bnq + 1 2bnq2 ≤ αn bn + 1 bnq . We therefore have [αn+1 bn+1, αn+1 bn+1 + 1 bn+1q] ⊂ [ αn bn, αn bn + 1

bnq]. We have constructed a descending

chain of closed sets (BN ⊃ BN +1⊃ BN +2 ⊃ . . . ) and applying Cantor’s intersection property

yields ⋂n>NBn ≠ ∅. As BN ⊂ U and by construction Bn+1 ⊂ An for all n ≥ N, we have

U∩ ⋂n>NAn≠ ∅. This completes the proof of (2.15).

From Theorem 2.1.1 we have for almost every β∈ T that ⟨anβ⟩ is equidistributed on T.

We therefore have there exists a set B⊂ T of full measure, and hence dense, such that for any δ∈ (0,1₄−_2q1) lim N →∞ 1 N N −1 ∑ n=0 χ_(−2δ−1 2q, 3 2q+2δ)(⟨anβ⟩) = 2 q+ 4δ (2.16) for all β ∈ B. For any δ ∈ (0,1₄ −_2q1) we can also define continuous functions fδ and gδ on

T, such that ∣∣fδ(⋅)∣∣∞≤ 1, ∣∣gδ(⋅)∣∣∞≤ 1 and

fδ(β) = { 1 ∶ β ∈ [0,1_q], 0 ∶ β ∈ [1_q + δ, 1 − δ], gδ(β) = { 1 ∶ β ∈ [−δ − _2q1,_2q3 + δ], 0 ∶ β ∈ [_2q3 + 2δ, 1 − 2δ −_2q1 ]. From the above construction fδ ≥ χ(0,1_q) and gδ ≤ χ(−2δ−1

2q, 3

2q+2δ) and we therefore have for

all β1 ∈ A, β2 ∈ B and ε ∈ R+ that

lim inf N →∞ 1 N N −1 ∑ n=0 fδ(⟨anβ1⟩) > log λ log(2λq) and lim sup N →∞ 1 N N −1 ∑ n=0 gδ(⟨anβ2⟩) < 2 q + 4δ + ε. From inequality (2.14) we can find γ∈ R+_{such that} log λ

log(2λq) > 2

q+γ, and as δ, ε are arbitrary

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can use this construction and the above inequalities to apply Lemma 2.3.1, which yields a dense set S⊂ T such that for any β ∈ S

lim sup N →∞ 1 N N −1 ∑ n=0 fδ(⟨anβ⟩) > 2 q + γ and lim inf N →∞ 1 N N −1 ∑ n=0 gδ(⟨anβ⟩) < 2 q+ 4δ + ε. We also have by construction that χ_(−δ,1

q+δ)≥ fδ and χ(−δ− 1 2q,

3

2q+δ)≤ gδ, hence for all β∈ S

lim sup N →∞ 1 N N −1 ∑ n=0 χ_(−δ,1 q+δ)(⟨anβ⟩) > 2 q + γ and lim inf N →∞ 1 N N −1 ∑ n=0 χ_(−δ−1 2q, 3 2q+δ)(⟨anβ⟩) < 2 q + 4δ + ε.

If we take β∈ S from the above limits, we can find two subsequences of the positive integers (rn) and (sn) such that for all x ∈ (0,_2q1)

1 rn rn−1 ∑ n=0 χ_(−δ,3 2q+δ)(⟨x + anβ⟩) > 2 q + γ and 1 sn sn−1 ∑ n=0 χ_(−δ,3 2q+δ)(⟨x + anβ⟩) < 2 q + 4δ + ε

for all n ∈ Z+_{. Let T be the circle rotation by some β} ∈ S. We can combine our two

sequences to form a sequence of positive integers (tn), namely t2n = rn and t2n+1= sn, such

that for all x∈ (0,_2q1 )

1 t2k t2k−1 ∑ n=0 χ_(−δ,3 2q+δ)(T an(x)) > 2 q + γ and 1 t2k+1 t2k+1−1 ∑ n=0 χ_(−δ,3 2q+δ)(T an(x)) < 2 q + 4δ + ε. We therefore have for all k∈ Z+

∥ 1 t2k t2k−1 ∑ n=0 χ_(−δ,3 2q+δ)(T an(⋅)) − 1 t2k+1 t2k+1−1 ∑ n=0 χ_(−δ,3 2q+δ)(T an(⋅))∥ 2 ≥ ζ2 2q > 0.

Therefore there exists a dynamical system such that the ergodic averages do not converge along (an). This proves that the lacunary sequence (an) is mean bad and completes the

Subsequence Ergodic Theorems

MSc Mathematics

Master Thesis