Proof mining in ergodic theory - a survey · Ergodic Theory Analysis of Mean Ergodic Theorem Analysis of Szemeredi’s Theorem Proof Mining - Metatheorems Based on G¨odel’s (’Dialectica’)

IntroductionErgodic Theory

Analysis of Mean Ergodic TheoremAnalysis of Szemeredi’s Theorem

Proof mining in ergodic theory - a survey

Philipp GerhardyDepartment of Mathematics

University of Oslo

Ramsey Theory in Logic, Combinatorics and Complexity,Bertinoro 25.-30.10.2009

Philipp Gerhardy Proof mining in ergodic theory - a survey



Introduction

Ergodic Theory

Analysis of Mean Ergodic Theorem

Analysis of Szemeredi’s Theorem




Proof Mining

The idea of proof mining is to use proof theoretic techniques toextract additional information from sufficiently formal proofs, inparticular from existence proofs in mathematics.

Additional information can be:

I Quantitative - algorithms, bounds.

I Qualitative - uniformities, weakening of premises.




Proof Mining

The main technique in proof mining are proof interpretations:

Given a formal system – a language, constants, axioms and rules –we want to give computational interpretations of:

I constants by some computational constants or terms,

I axioms by suitable terms realizing existential quantifiers,

I derivation rules by rules for combining realizers.

Then we can give computational interpretations of formal proofsand the theorems they prove.




Proof Mining - Metatheorems

Based on Godel’s (’Dialectica’) functional interpretation, one maydevelop general logical metatheorems that describe classes oftheorems and proofs from which additional information may beextracted.

These metatheorems both give a-priori criteria for when and whatkind of information – bounds, uniformities, etc. – may beextracted, as well as describing an algorithm for the extraction.

These metatheorems cover theories for intutionistic and classicalarithmetic, but extend to full classical analysis and also abstractmetric spaces, normed linear spaces, Hilbert spaces, etc.




Proof Mining - Example

The principle of convergence for bounded monotone sequences –short: PCM – says the following:

Every bounded monotone sequence of real numbers converges.

More formally, this can be expressed as:

∀(an)n∈IN∀b ∈ IN∀ε > 0∃n ∈ IN∀m1,m2 > n(∀k(ak ≤ ak+1 ≤ b) → |am1 − am2 | ≤ ε

).

Can we compute a rate of convergence?




Proof Mining - Example

It is well known that there exist computable bounded monotonesequences of rational numbers that do not converge to acomputable limit, i.e. there is no computable rate of convergence.

However, PCM is classically equivalent to:

∀(an)n∈IN∀b ∈ IN∀ε > 0∀M : IN → IN∃n ∈ IN∀m1,m2 ∈ [n,M(n)](∀k(ak ≤ ak+1 ≤ b) → |am1 − am2 | ≤ ε

),

which is the Dialectica transform of PCM. This version is alsoknown as the no-counterexample version of PCM and thisweakened form of convergence has been called local stability.

Here, a bound on n in the parameters ε, b,M is computed easily.




Ergodic Theory

Ergodic theory studies the long-time and limit behaviour ofdynamical systems.

Let (X ,B, µ) be a (finite) measure space, let T : X → X be ameasure-preserving transformation – together: a measurepreserving system – and let f ∈ L1(X ,B, µ).

Then ergodic theory studies long-time behaviour of e.g. elementsT i f , where (T i f )(x) = f (T ix), and the properties of sums,products, averages, etc. of such elements.




Two Ergodic Theorems

Let (X ,B, µ,T ) be a measure preserving system and define the

average Anf := 1n

n−1∑i=0

T i f .

Mean Ergodic Theorem For any f ∈ L2(X ,B, µ) the averagesAnf converge in the L2-norm.

Pointwise Ergodic Theorem For any f ∈ L1(X ,B, µ) theaverages Anf converge pointwise almost everywhere.

Moreover, if the space is ergodic – i.e. X and ∅ are the onlyT -invariant sets – the averages converge to the integral of f .




Ergodic Theory - Applications to Combinatorics

Furthermore, there is a (no longer) surprising connection betweenergodic theory and finite combinatorics:

Furstenberg Recurrence Theorem If (X ,B, µ) is a measurespace and T1, . . . ,Tl are commuting measure preservingtransformations, then for any set A ∈ B with µ(A) > 0 there existsan integer n ≥ 1 with

µ(A ∩ T−n1 A ∩ T−n

2 A ∩ . . . ∩ T−nl A) > 0.




Ergodic Theory - Applications to Combinatorics

This recurrence theorem allows one to proof Szemeredi’s theorem:

Szemeredi’s Theorem For any δ > 0 and any k ∈ IN thereexists an N = N(δ, k) such that for any interval [a, b] ⊂ ZZ of

length ≥ N and A ⊆ [a, b] of density ≥ δ, i.e. |A|b−a ≥ δ, contains

an arithmetic progression of length k.

The challenge is: How to translate the abstract concepts andtechniques of ergodic theory – limits, projections, etc. – intoconcrete combinatorial results?




Mean Ergodic Theorem

Let us state the Mean Ergodic theorem in a Hilbert space setting:

Mean Ergodic Theorem Let (X , 〈·, ·〉) be a Hilbert space, letT : X → X be an isometry and let f ∈ X be given. Then

∀ε > 0∃n ∈ IN∀m > n(‖Amf − Anf ‖ ≤ ε).

This also holds if T is nonexpansive, i.e. ‖Tf ‖ ≤ ‖f ‖ for all f ∈ X .

Can we compute (a bound on) n in the parameters, i.e. f ,T , ε andpossibly depending on the Hilbert space?




Mean Ergodic Theorem - Noncomputability Results

Just as with PCM, we may construct a Hilbert space and anisometry T : X → X such that there can be no computable rate ofconvergence for the averages.

The same can be done for measure spaces – so this is not a featureof the more general setting of Hilbert spaces.

The general idea is to code the Halting problem into a measurespace and a measure preserving transformation, such that acomputable rate of convergence would solve the halting problem.




Mean Ergodic Theorem - Computability Results

Instead we consider, as with PCM, the Dialectica-transform:

Mean Ergodic Theorem Let (X , 〈·, ·〉) be a Hilbert space, letT : X → X be an isometry and let f ∈ X be given. Then

∀ε > 0∀M : IN → IN∃n ∈ IN(‖AM(n)f − Anf ‖ ≤ ε),

where we assume that M is monotone increasing; again, we couldadd notation to express local stability.

This now has the suitable logical form for logical metatheorems toguarantee that a bound on n can be extracted. The bound will notdepend on the particular space, nor on the transformation T , butonly on ε, M and a bound ‖f ‖ ≤ b.





The sketch of the proof for the Mean Ergodic Theorem is asfollows:

I We can decompose the space X into componentsU = {u − Tu|u ∈ X} and V = {v ∈ X |v = Tv}.

I For elements u − Tu, we have ‖An(u − Tu)‖ ≤ 2‖u‖/n.

I For elements v ∈ V , we have Anv = v .

So for f = u + v , the averages Anf converge to v , where the ratecan be given in terms of ‖u‖, i.e. the projection of f onto U.

It is this projection onto U that makes the proof non-constructive!





Since the averages Anf exist in the subspace of X spanned by{T i f }, it suffices to consider the projection of f onto the subspaceUf := {T i f − T i+1f |i = 0, 1, . . .}.

We can explicitly describe a sequence gi = ui − Tui converging tothe projection of f onto Uf :

g0 = 〈f ,f−Tf 〉‖f−Tf ‖2 (f − Tf ),

gi+1 = gi + 〈f−gi ,Ti f−T i+1f 〉

‖T i f−T i+1f ‖2 (T i f − T i+1f )




Mean Ergodic Theorem - Computability ResultsThe computation then roughly goes as follows:

I ‖AM(n)f − Anf ‖ ≤‖AM(n)(f − gi )− An(f − gi )‖+ ‖AM(n)gi‖+ ‖Angi‖.

I Let ai = ‖gi‖. If |ai − aj | is small, then ‖gi − gj‖ is small.

I If ‖gi − gj‖ is suitably small for a suitable j , then‖AM(n)(f − gi )− An(f − gi )‖ is small.

I If n,M(n) are large relative to ‖ui‖, ‖AM(n)gi‖, ‖Angi‖ aresmall.

Thus sufficient local stability for the bounded monotone sequenceai together with upper bounds on ‖ui‖ allows us to derive localstability for Anf .





But: to compute upper bounds on ‖ui‖ we need lower bounds on‖Ti f − T i+1f ‖ - which in general could be = 0.

The solution is the following observation:

I If e.g. ‖f − Tf ‖ is very small, we may get local stability for‖AM(0)f − f ‖ via the triangle inequality.

I Otherwise, we have a lower bound on ‖f − Tf ‖.

In other words, the final step is giving an appropriate computationalinterpretation to r = 0 ∨ r 6= 0 for a particular real number r .





With this analysis, the following bounds were obtained byAvigad-Towsner-G.:

Define:

i0 = 0, nk = d bε2

ik∑l=0

M(2lbε )e

ik + 1 = ik + d215M(nk )4b4

ε4 e

Let d = 512b2

ε2 , then for some n ≤ N(b, ε,M) = 2ndbε , we have that

‖AM(n)f − Anf ‖ < ε.




Mean Ergodic Theorem - Observations

I Instead of full convergence of the averages, one can obtainonly local stability.

I To obtain local stability of the averages, it suffices to obtainlocal stability (not convergence) of the projections onto Uf .

I Observations like, “‖f − Tf ‖ = 0 implies convergence of theaverages” are weakened to “‖f − Tf ‖ < δ implies localstability of the averages”, allowing the crucial upper boundson the norms ‖ui‖.




Mean Ergodic Theorem - Further Comments

I If the measure preserving system is ergodic, we can compute afull rate of convergence for Anf - this is because here Anfconverges to the integral of f , and one can compute a fullrate of convergence from ‖f ∗‖, where f ∗ = limn→∞ Anf .

I Kohlenbach and Leustean analysed a proof by Birkhoff of thegeneralization of Mean Ergodic Proof to uniformly convexBanach spaces. This analysis yields much improved bounds.




Szemeredi’s Theorem - Sketch of Proof

We call a measure preserving system weak mixing if

limn→∞

1

n

∑i<n

|µ(T−iA ∩ B)− µ(A)µ(B)| = 0,

for any measurable sets A,B.

We call a measure preserving system compact, if the orbit

{A,T−1A,T−2A,T−3A, . . .}

is compact for any measurable set A.

Informally, weak mixing corresponds to randomness andcompactness to regularity.




Szemeredi’s Theorem - Sketch of proof

Both conditions easily imply Szemeredi’s Theorem - but not everymeasure preserving system is either weak mixing or compact.

However, if a measure preserving system (X ,B, µ,T ) is not weakmixing, it has a non-trivial T -invariant compact factor.

A factor can be thought of as a coarsening of the σ-algebra B to aT -invariant sub-σ-algebra B′.





Furthermore, if a measure preserving system (X ,B, µ, T ) is notweak mixing relative to a factor B′, it has an intermediatenon-trivial T -invariant compact factor B′′ such that (X ,B′′, µ,T )is compact relative to B′.

Iterating this construction and taking unions at limit stages, thisprocess will – assuming the system is separable – terminate atsome countable ordinal.





A factor B is Szemeredi (short: SZ), if it satisfies

lim infn→∞

1

n

n∑i=0

(µ(A ∩ T−i1 A ∩ T−i

2 A ∩ . . . ∩ T−il A)) > 0,

for every set A ∈ B.

I The trivial factor B0 is SZ.

I SZ is preserved under compact extensions.

I SZ is preserved under limits.

I SZ is preserved under weak mixing extensions.




Szemeredi’s Proof - Comments on the Proof

I The construction of the measure space from the “dense”subset of the natural numbers corresponds to an applicationof Weak (binary) Koenig’s Lemma.

I Once weak mixing (relative to an SZ factor) is established,obtaining recurrence is straightforward.

I The transfinite iteration is constructively “acceptable”.

I The source of non-constructivity is the construction of thecompact extension using projections and limits akin to theMean Ergodic Theorem.




Szemeredi’s Proof - Comments on the Proof

I The transfinite sequence of compact extensions exhausts thecountable ordinals, i.e. for every countable ordinal there is ameasure space where the induction stops only at that ordinal.

I Avigad and Towsner have shown that the proof can beformalized in ID1 and have given a functional interpretation ofID1 thus yielding a computational interpretation of this proof.

I A more refined analysis by Avigad and Towsner yieldsimproved ordinal bounds for the application of Furstenberg’sstructure theorem to Szemeredi’s Theorem.




Szemeredi’s Theorem - Improved Ordinal BoundsIn Furstenberg’s proof of Szmeredi’s Theorem one establishes weakmixing relative to a factor. Weak mixing, in general, was:

limn→∞

1

n

∑i<n

|µ(T−iA ∩ B)− µ(A)µ(B)| = 0.

To eventually apply this property to obtain recurrence it suffices tohave approximate weak mixing

∀m ≥ n1

m

∑i<m

|µ(T−iA ∩ B)− µ(A)µ(B)| < ε,

for a suitable ε and suitably many factors in the sequence ofcompact extensions. Avigad and Towsner show approximate weakmixing can always be obtained below ωωω

.




Final Remarks

I Non-constructivity arises from use of limits and projections,not “choice” or “compactness arguments” or “transfiniteconstructions”.

I In general, use of limits and projections to obtaincombinatoric results is weakened to use of approximate limitsand projections.

I Plenty of avenues for future work - extracting “exact” boundsfrom Furstenberg’s proof, analysing methods by Gowers,Tao-Green, etc.




Final Remarks

References:

I Kohlenbach, “Applied Proof Theory: Proof Interpretationsand their Use in Mathematics”, Springer Monographs inMathematics, 2008.

I Avigad, G., Towsner, “Local Stability of Ergodic Averages”,TAMS 362, pp. 261-280 (2010).

I Avigad, “The Metamathematics of Ergodic Theory”, APAL157, pp. 64-79 (2009).

I Avigad, Towsner, “Metastability in the Furstenberg-ZimmerTower”, draft.


Documents

Proof mining in ergodic theory - a survey · Ergodic Theory Analysis of Mean Ergodic Theorem Analysis of Szemeredi’s Theorem Proof Mining - Metatheorems Based on G¨odel’s (’Dialectica’)