szemerédi's theorem via ergodic...

IntroductionIntroduction to Ergodic Theory

Furstenberg’s Correspondence Principle

Szemeredi’s Theorem via Ergodic Theory

Jonathan Hickman

March 6, 2012

Jonathan Hickman Szemeredi’s Theorem via Ergodic Theory

Arithmetic Progressionsvan der Waerden and Szemeredi’s Theorems

DefinitionAn arithmetic progression is a set {s + jb}k−1

j=0 ⊂ Z where k ∈ N,s ∈ Z and b ∈ N≥1. We call:

I s the start point

I b the step-size

I k the length

Example

For example{5, 11, 17, 23}

is an AP with start point 5, step-size 6 and length 4.

DefinitionAn arithmetic progression is a set {s + jb}k−1

j=0 ⊂ Z where k ∈ N,s ∈ Z and b ∈ N≥1. We call:

I s the start point

I b the step-size

I k the length

Example

For example{5, 11, 17, 23}

is an AP with start point 5, step-size 6 and length 4.

Fundamental Questions

QuestionWhen does a subset Λ ⊆ Z contain arithmetic progressions of everypossible length k?

Example

I Arithmetic sequence {s + kb}k∈Z, for some s ∈ Z and b ∈ N.

I Syndetic set? (Increasing sequence of integers such that thedifference between consecutive terms is bounded.)

I Randomly generated set of integers? Choose whether toinclude each integer independently at random, with probability0 < p < 1.

I Primes?

Fundamental Questions

QuestionWhen does a subset Λ ⊆ Z contain arithmetic progressions of everypossible length k?

Example

I Arithmetic sequence {s + kb}k∈Z, for some s ∈ Z and b ∈ N.

I Syndetic set? (Increasing sequence of integers such that thedifference between consecutive terms is bounded.)

I Randomly generated set of integers? Choose whether toinclude each integer independently at random, with probability0 < p < 1.

I Primes?

An Answer

In 1927 van der Waerden proved the following fundamentaltheorem:

Theorem (van der Waerden)

Suppose A1, . . . ,An ⊆ Z are disjoint sets which partition theintegers,

Z = A1 t A2 t · · · t An

Then there exists some l ∈ {1, . . . , n} such that Al containsarithmetic progressions of arbitrary length.

A more general approach is to consider the ‘density’ of a set ofintegers.

Example

Consider the set

{2k : k ∈ N0} = {1, 2, 4, 8, 16, 32, . . . }

it is easy to see this contains no arithmetic progressions of length> 2.

“The occurrence of integers belonging to this set becomes lessfrequent.”

|{2k : k ∈ N0} ∩ [1,N]|N

→ 0 as N →∞

A more general approach is to consider the ‘density’ of a set ofintegers.

Example

Consider the set

{2k : k ∈ N0} = {1, 2, 4, 8, 16, 32, . . . }

it is easy to see this contains no arithmetic progressions of length> 2.“The occurrence of integers belonging to this set becomes lessfrequent.”

|{2k : k ∈ N0} ∩ [1,N]|N

→ 0 as N →∞

DefinitionLet Λ ⊆ Z. We define the upper Banach density of Λ to be

dB(Λ) := lim supN−M→∞

|Λ ∩ [M,N)|N −M

I In 1936 Erdos and Turan [2] conjectured any subset of Z withpositive upper Banach density contains arithmeticprogressions of arbitrary length.

I Not until 1952 that the first non-trivial case, existence ofarithmetic progressions of length k = 3, was shown by Roth[5]. (Using a circle method).

I Szemeredi [6] then proved the case k = 4 in 1969 by applyingcombinatorial methods (Regularity Lemma).

I General case was proven, again by Szemerdi [7], in 1975.

Theorem (Szemeredi’s Theorem)

Let Λ ⊆ Z be a set of positive upper Banach density. Then Λcontains arithmetic progressions of arbitrary length.

Example

I For any a ∈ Z, b ∈ N the arithmetic sequenceΛ = {a + kb}k∈Z has density d(Λ) = 1

I A syndetic set has positive upper Banach density.

I Choose whether to include each integer independently atrandom, with probability 0 < p < 1. The resulting set almostsurely has positive upper Banach density.

I Primes?

Example

I For any a ∈ Z, b ∈ N the arithmetic sequenceΛ = {a + kb}k∈Z has density d(Λ) = 1

I A syndetic set has positive upper Banach density.

I Choose whether to include each integer independently atrandom, with probability 0 < p < 1. The resulting set almostsurely has positive upper Banach density.

I Primes?

Recall the Prime Number Theorem states:

limx→∞

π(x) log x

where π is the prime counting function

π(x) = |{p : p prime and p ≤ x}|

It follows the set of primes does not have positive upper Banachdensity. However,

Theorem (Green-Tao)

The primes contain arithmetic progressions of arbitrary length.

Recall the Prime Number Theorem states:

limx→∞

π(x) log x

where π is the prime counting function

π(x) = |{p : p prime and p ≤ x}|

It follows the set of primes does not have positive upper Banachdensity. However,

Theorem (Green-Tao)

The primes contain arithmetic progressions of arbitrary length.

I General case was proven, again by Szemerdi [7], in 1975. Hisproof is long and difficult.

I Soon after Furstenberg noticed Szemeredi’s Theorem wouldfollow from a substantial generalisation of the classicalPoincare Recurrence Theorem from ergodic theory.

I He subsequently proved this generalisation, now known asFurstenberg’s Multiple Recurrence Theorem (Thus providingan alternative proof of Szemeredi’s Theorem).

I Also demonstrated a scheme, the Correspondence Principle,for relating problems in combinatorial number theory toproblems in ergodic theory [3].

Measure Preserving SystemsPoincare Recurrence

Ergodic Theory

Ergodic theory belongs to the broader discipline of the qualitativestudy of dynamical systems.

DefinitionA dynamical system essentially consists of a ‘state space’ X andtransformation T : X→ X.

Used to model physical systems (motion of particles); biological,economic or other real-world systems.

Ergodic Theory

Hamiltonian Mechanics

Let X ⊆ R6 and (x1, . . . , x6) ∈ X describe the position andmomentum of some particle (for example, a celestial body). Thelaw T : X → X governs the motion of the particle:

I Let x ∈ X be the initial state.

I Tx describes the position and momentum of the particle attime 1

I T 2x describes position and momentum of the particle at time2 ...and so on.

Ergodic Theory

One wishes to determine to the behaviour of X under iterations ofT .In ergodic theory the state space and transformation are endowedwith measure theoretic structure. This is natural in the contextof analysing the long-term average behaviour of systems [4].

Ergodic Theory

Used to model physical systems (motion of particles); biological,economic or other real-world systems.One wishes to determine to the behaviour of X under iterations ofT .

In ergodic theory the state space and transformation are endowedwith measure theoretic structure. This is natural in the contextof analysing the long-term average behaviour of systems [4].

Ergodic Theory

Used to model physical systems (motion of particles); biological,economic or other real-world systems.One wishes to determine to the behaviour of X under iterations ofT .In ergodic theory the state space and transformation are endowedwith measure theoretic structure.

This is natural in the contextof analysing the long-term average behaviour of systems [4].

Ergodic Theory

Used to model physical systems (motion of particles); biological,economic or other real-world systems.One wishes to determine to the behaviour of X under iterations ofT .In ergodic theory the state space and transformation are endowedwith measure theoretic structure. This is natural in the contextof analysing the long-term average behaviour of systems [4].

Basic Definitions

Definition

I Let (X ,B, µ) be a measure space. Suppose T : X → X ismeasurable and

µ(T−1A) = µ(A) for all A ∈ B

Then we say T is a measure preserving transformation on(X ,B, µ).

The time-evolution of the system is governed by laws whichdo not themselves change in time.

I The pair (X ,B, µ,T ) consisting of the space andtransformation is called a measure preserving system.

Basic Definitions

Definition

Then we say T is a measure preserving transformation on(X ,B, µ).The time-evolution of the system is governed by laws whichdo not themselves change in time.

Basic Definitions

Definition

Then we say T is a measure preserving transformation on(X ,B, µ).The time-evolution of the system is governed by laws whichdo not themselves change in time.

Rotations of the Circle

Example (Rotations of the Circle)

Consider the circle T := R/Z with the usual Lebesgue measure.

Then for any τ ∈ R the rotation map

Rτ : T→ T Rτ : x 7→ x + τ

is clearly measurable. It is easy to see the rotation map is alsomeasure preserving, i.e.

µ(R−1τ A) = µ(A) for all A ⊆ T measurable

Consider the circle T := R/Z with the usual Lebesgue measure.Then for any τ ∈ R the rotation map

Rτ : T→ T Rτ : x 7→ x + τ

is clearly measurable.

It is easy to see the rotation map is alsomeasure preserving, i.e.

Rτ : T→ T Rτ : x 7→ x + τ

is clearly measurable. It is easy to see the rotation map is alsomeasure preserving, i.e.

Bernoulli Shifts

Example (Bernoulli Shifts (on 2 Letters))

I X := {0, 1}Z be the space of all two-sided sequences in 0s and1s

I Give this the product topology induced from the discretetopology on {0, 1}

I Let B denote the corresponding Borel σ-algebra

Bernoulli Shifts

Bernoulli Shifts (Ctd.)

One defines a probability measure P on (X ,B) by first assigning a(fixed) probability to each cylinder set