FURSTENBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM

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1 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM ZIJIA WAG Abstract. We introduce the basis of ergodic theory and illustrate Furstenberg s proof of Szemerédi s theorem. Contents. Introduction 2. A brief introduction to ergodic theory Ergodicity and weak mixing Compact systems Factor and extension Conditional measures Weak mixing and compactness for extensions The structure theorem 8 3. Furstenburg s proof of Szemerédi s theorem General Strategy Szemerédi s theorem Correspondence Two fundamental systems Extension principles Conclusion 28 Acknowledgments bibliography 28 References 28. Introduction The statement of Szemerédi s theorem is very simple. Theorem. (Szemeredi). A subset of integers with positive upper Banach density has arbitrarily long arithmetic progressions. It was first proved by Szemerédi in 975 using a combinatorial and completely elementary approach. Although his method was extremely complicated, some of the important ideas such as Szemerédi s regularity lemma in graph theory came out from his proof. Two years later, a totally different approach is introduced by Furstenberg. He turned Szemerédi s theorem, a problem that looks extremely combinatorial, into an ergodic puzzle about multiple recurrence of a measure-preserving Date: AUGUST 28, 208.

2 2 ZIJIA WAG system. Later in 2002, Gowers gave a Fourier-analytic proof. The fact that the original question asked by Erdös and Turán in 936 is answered in three completely distinct ways has already made this problem highly interesting. In this paper, we discuss Furstenberg s ergodic proof of Szemerédi s theorem. Despite the elegance of Furstenberg s ergodic proof of Szemerédi s theorem, the value of this proof goes way beyond solving the problem per se. His proof sheds light on many important topics in ergodic theory, for instance, the classification of dynamical systems, conditional measures, extensions, etc. 2. A brief introduction to ergodic theory Ergodic theory studies dynamical systems. By dynamical systems, we mean certain good actions on measure spaces that exhibit interesting long-term behaviors. An Z action is just a function from the space to itself, or in other words, a dynamics. Obviously, not all functions are well-behaved. In this section, we talk about the basics of ergodic theory to set the foundation for our later discussions. Definition 2.. A measure space (, B, µ) is a space with measure µ and the σ-algebra B of measurable sets. Sometimes we ignore the σ-algebra associated to the measure space and just write (, µ) when it is not so important. However, one shall treat the σ algebra with great caution when dealing with conditional measures which will be discussed later in this paper. Remark 2.2. In this paper, we mostly assume that we are dealing with probability spaces, in which the measure of the entire space is. Definition 2.3. A map T : (, B, µ) (Y, B Y, ν) is measure-preserving if for any set A B Y, µ(t A) = ν(a). Definition 2.4. A measure-preserving map φ is an invertible measure-preserving map if the inverse of φ is measurable and well-defined almost everywhere. Definition 2.5. We call (, B, T, µ) a measure-preserving system, or equivalently a dynamical system, if T is a measure-preserving map on. Example 2.6. We define 2 Z to be the infinite product of {0, }. This space is compact by the Tychonoff s theorem. Given an element x 2 Z, we denote the k th coordinate of x by x[k]. We define a measure µ on the space 2 Z by an infinite product. Let π j be the projection onto the j th coordinate. On each copy of {0, }, we use the half-half measure ν, i.e. for each measurable set A if A = {0, }, ν[a] = 0 if A is empty, 2 otherwise. We define µ by µ(b) = n Z ν(π nb) for all the measurable rectangles in the form of j Z A j 2 and extend the measure to the entire σ algebra of measurable sets. This way of defining a measure is valid as explained in Remark 2.8. Moreover, one can define the Bernoulli shift T k on this space for any integer k. T k acts on an element Since we have a finite set, every subset is measurable. 2 Each Aj is measurable in its own copy of {0, }.

3 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 3 x 2 Z by shifting each coordinate of x to the left by k bits, i.e. x[s] = T k x[s 4] for all s Z. Example 2.7. Given a circle T = R/Z equipped with the Haar measure µ, we can define rotation R α acting by addition, i.e. R α : x x + α. This forms a measure-preserving system. In order to prove that R α is measure-preserving, it suffices to show that Rα preserves the measure of all the intervals 3. otice that for any interval (a, b) T, 4 µ(rα (a, b)) = (b α) (a α) = b a = µ((a, b)). Remark 2.8. Proving the measure-preserving property for every measurable set can be painful. However, it suffices to prove this property for a collection of sets that generates the σ-algebra. This is a standard trick that we will keep using repeatedly. Example 2.9. Instead of rotation, we can define a different dynamics on the circle T, namely the circle doubling map M 2, where M stands for multiplication. M 2 : T T is defined by M 2 (a) = 2a. For an arbitrary interval (a, b) T, 2, b+ 2 )) = ( b 2 a 2 ) + ( b+ 2 a+ 2 ) = b a = µ(a, b). Therefore, the circle doubling map M 2 is also a dynamics on T. In fact, we can show that M k, multiplication by k, is measure-preserving for every natural number k. µ(m 2 (a, b)) = µ(( a 2, b 2 ) ( a+ Remark 2.0. ow we have defined two different dynamics on the same space T (or R/Z). One natural question to ask is whether these two systems are equivalent. Although it is quite obvious that they are different given that the action a a + α is not even close to the action a 2a. However, it is hard to tell whether two dynamical systems are different or behave in some similar ways when they are in different spaces. Therefore we introduce the notion of measurable isomorphism. Definition 2.. Given a probability measure space (, B, µ) and a measurable set A B. A is null if µ(a) = 0. On the other hand, A is conull if µ(a) =. This gives us a convenient way to talk about the special sets that have zero or full measure, which we will encounter a lot in our discussion of ergodic theory. Definition 2.2. In a dynamical system (, B, T, µ) and a measurable set A B. We call A T-invariant, or invariant to T, if T A A. Moreover, if T A = A, we say that A is strictly T-invariant, or strictly invariant to T. Definition 2.3. Two systems (, B, T, µ) and (Y, B Y, T Y, ν) are measurably isomorphic if there exist conull sets B invariant to T and Y B Y invariant to T Y and an invertible measure-preserving map f : Y such that f T = T Y f for every x, i.e. that the following diagram commutes a.e. f Y T T Y otice that the above commutative diagram may only be defined on a set of full measure. Y f 3 See Remark Here we view the circle as the interval [0, ) with endpoints identified.

4 4 ZIJIA WAG Example 2.4. (T, B T, M 4, µ) is isomorphic to (T 2, B T 2, M 2 M 2, µ µ) where µ µ is the product measure and M 2 M 2 : T 2 T 2 is defined by M 2 M 2 (t, t 2 ) = (2t, 2t 2 ). It is clear that M 2 M 2 is a measure-preserving map on T 2. Here we construct a measure-preserving map φ from T to T 2 such that the diagram below commutes. φ T M 4 M 2 M 2 T φ T 2 T 2 We construct a sequence {φ n } n of maps from T to T 2 where each φ n is a measurepreserving map on a small σ-algebra. When n =, we define C B T to be the trivial σ-algebra, which only contains the entire interval [0, ) and the empty set. Similarly, we define D B T 2 to be the σ-algebra that only contains the unit square. We define φ to be some bijective map 5 from [0, )to [0, ) 2. It is clearly measure-preserving when viewed as a map from (T, C ) to (T 2, D ). When n = 2, we divide the interval into four subintervals {[0, 4 ), [ 4, 2 ), [ 2, 3 4 ), [ 3 4, )} and define C 2 B T to be the σ-algebra generated by these four subintervals. Similarly, we can divide T 2 into four squares and define the σ-algebra D 2. The function φ 2 is defined by sending the four subintervals of T into the for subsquares of T 2 in counter clockwise order starting at the top left square. Again, we can use some bijective map to ensure that the map φ 2 : (T, C 2 ) (T 2, D 2 ) is measure-preserving. By construction, the sequence {φ n } n converges to some measurable function φ and the limit φ : (T, B T ) (T 2, B T 2) is measure-preserving. We can prove that such a map is an isomorphism by viewing the points in T in digit 4 expansions and the points in T in binary expansions, which is another standard trick in the theory of dynamical systems. Remark 2.5. We think of the bijection from a different direction i.e., [0, ) 2 [0, ). There are many constructions of such a bijective map. One of the most direct ways is to interleave the terms in the continued fraction expansion of each coordinate to get a single number in [0, ). However, it is not very clear that this map is onto. Actually, we only need to know the existence of a bijective map instead of the actual form. Therefore, we can use the Cantor-Schröder-Bernstein theorem. x (x, 0) gives the injective map from [0, )to [0, ) 2. Interleaving the digits of decimal expansion on each of the coordinates, i.e (0.a a 2 a 3..., 0, b b 2 b 3 ) 0.a b a 2 b 2..., gives an injection from [0, ) 2 to [0, ). When the decimal expansion is not unique, we use the Axiom of Choice to pick a random one. As we have seen in the previous examples, the measure-preserving property is already highly nontrivial. There are already a lot to say about these systems directly from the definition of being measure-preserving. Here we prove some elementary but useful results about measure-preserving maps. Definition 2.6. Given a measure-preserving system (, B, T, µ), we define U T the associated operator, or equivalently the operator associated to T by U T f(x) = f(t x). Moreover, we define U T to be the adjoint of U T such that U T f, g = f, U T g for all f, g L2 (). 5 See Remark 2.5 for more details.

5 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 5 Remark 2.7. otice that in the Definition 2.6, we did not specify the spaces that the operator lives in. This is a rather loose definition since the actual spaces depend on the context. For example, when we are dealing with L 2 spaces like in Theorem 2.27, we assume that U T : L 2 () L 2 (). Theorem 2.8. Given a measure-preserving system (, B, T, µ), for any L function f, we have U T fdµ = fdµ. Proof. For characteristic function χ B, U T χ B dµ = µ(t B) = µ(b) = χ B since T is measure-preserving. ow for any L function f that is nonnegative, we can take a sequence of simple functions that approximate f and apply the monotone convergence theorem. Finally for a general L function f, we can define f = f + f where both parts are nonnegative and then apply the previous result. Remark 2.9. Another way to state Theorem 2.8 is to say that U T : L () L () is an isometry, i.e. U T f = f. Theorem (Poincaré Recurrence) Given a dynamical system (, B, T, µ) and A B, almost every point in A returns to A infinitely often. Proof. We need to show that there exists a set A A of full measure such that for every point a A there exists a strictly increasing integer sequence {n i } i such that T ni a A. In order to find such an A we remove from A step-by-step a countable collection of sets with measure 0. Let = {x A T n a / A for all n } = i= T i A c A A. We claim that the set has measure zero. Indeed, T n = i=n+ T i A c T n A. For m < n, we have T n T n A but T m T n A c since m < n. Therefore, the countable family {T k } contains mutually disjoint sets. We know that µ( T i ) = = i=0 µ(t i ) (they are disjoint) i=0 µ( ) (T is measure-preserving), i=0 so µ( ) = 0. We now define A = A. By construction, every point in A returns to A at least once and µ(a) = µ(a ). ow we define 2 = {x A T 2n a / A for all n } = i= T 2i A c A. ow we let A 2 = A 2. Similarly, µ(a 2 ) = µ(a ) = µ(a) since 2 has measure zero. Moreover, every point in A 2 returns to the set A at least twice. Let A = i= A i. Then A has the same measure as A and every point in A returns to A infinitely often. Remark 2.2. Theorem 2.20 does not require any high-level techniques. We can generate the result of Poincaré s theorem from the recurrence of points to recurrence of part of the set A that has positive measure. We call this property the multiple recurrence property. As we will see later, such generalization applies to arbitrary measure-preserving systems just like Poincaré recurrence.

6 6 ZIJIA WAG Definition A measure-preserving system (, B, T, µ) has multiple recurrence of order k if for each A B that has positive measure, there exist n such that ( k ) µ T in A > 0. i=0 Despite the similarity between Theorem 2.20 and Definition 2.22, the latter is actually much more sophisticated and involves some of the deeper results in ergodic theory. In fact, the whole point of this paper is to prove the multiple recurrence property for a general measure-preserving system. We will show that this is equivalent to proving Szemerédi s theorem in Theorem Ergodicity and weak mixing. Definition A measure-preserving system (, B, T, µ) is ergodic if for any strictly invariant set A B such that T A = A either µ(a) = or µ(a) = 0. In the above definition of ergodicity, we can actually replace sets with functions and give a different characterization of ergodicity. The proof is similar to that of Theorem 2.8, where we prove the proposition for simple functions first and pass through limits. Theorem A measure-preserving system (, B, T, µ) is ergodic if and only if any measurable function f such that f(t x) = f(x) for almost every x is constant almost everywhere. Example Consider the dynamical system (T, R α ) where T is the circle and R α : x x + α. For any R α invariant function f L 2 (T), let n= c ne 2πint be the Fourier expansion of f(t). f(r α t) = c n e 2πin(t+α) = n= n= c n e 2πinα c n e 2πint. By the uniqueness of Fourier coefficients, we know that c n = 0 for all n except n = 0, where c n = c n e 2πinα holds trivially since e 2πinα =. Therefore, f is constant and irrational rotation on the circle is ergodic. Example ow consider the circle doubling system (T, M 2 ) where M 2 acts on T by M 2 (x) = 2x. We use the same technique as the previous example. Suppose we have a L 2 function f(t) that has Fourier expansion n= c ne 2πint. ow we have f(2t) = n= c ne 2πin(2t) = n= c ne 2πi(2n)t. Therefore, c k = c 2k = c 4k... for every integer k. otice that the function f(t) is in L 2, so the sequence {c n } n Z should be square summable. This is not possible if there is some k 0 such that c k 0. As a result, f must be constant and the system (T, M 2 ) is ergodic. Theorem (Mean ergodic theorem) Given a measure-preserving system (, B, T, µ), we define P the orthogonal projection onto the closed subspace V = {h L 2 () U T h = h} L 2 ().

7 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 7 Then we have in L 2 for all f L 2 (). UT n f P f Proof. Observe that L 2 () = V W where W = {U T g g g L 2 ()} L 2 (). To see this, it suffices to show that W = V. If h V, then h, U T g g = h, U T g h, g = U T h, U T g h, g = h, g h, g (by Theorem 2.8) =0. If h W, we need to show that h = U T h. Indeed, we know that 0 = h, U T g g = h, U T g h, g = U T h, g h, g for all g L2 (). This means that (2.28) h = U T h where U T is the adjoint of U T defined in Definition 2.6. Therefore U T h h 2 2 = U T h h, U T h h = U T h, U T h + h, h 2 U T h, h =2 h, h 2 h, U T h =2 h, h 2 h, h (by 2.28) =0. ow we have proved our observation and we are ready to prove the mean ergodic theorem. Given any f L 2 (), there exists a sequence of L 2 functions {h i } i W such that f = P f + h where h i h. 6 otice that UT n f = It suffices to show that we have = =P f + U n T P f + P f + U n T h. U n T h U n T h (P f V ) U n T h 0. Let h i = U T g i g i for each h i. Then UT n h i = 2 = 2 g i 2 UT n (U T g i g i ) U T g i g i Whenever we talk about convergence in this proof, we assume it is convergence in L 2 norm.

8 8 ZIJIA WAG which goes to 0 as. For any ɛ > 0, we choose i large enough such that h h i 2 < ɛ 2 and then choose large enough such that U T nh i < ɛ 2. 2 Finally, UT n h 2 < = <ɛ. U T (h h i ) + UT n h i 2 2 U T (h h i ) 2 + ɛ 2 (h h i ) 2 + ɛ 2 (by Theorem 2.8) Remark Even though Theorem 2.27 bears the name of mean ergodic, it actually applies to general measure-preserving systems that are not ergodic. otice that when using mean ergodic theorem on ergodic systems, we have a stronger U T n result, namely that L2 fdµ fdµ for any L2 function f by Theorem Indeed, this is a direct consequence of the fact that the only strictly invariant sets in an ergodic system either has zero or full measure. Combining with the property that L 2 convergence implies convergence under the weak topology in the Hilbert space L 2, here we provide a weaker version of the mean ergodic theorem. Corollary Given an ergodic system (, B, T, µ) and f, g L 2, we have lim fut n gdµ = fdµ gdµ Proof. By the mean ergodic theorem, we know that UT n gdµ L2 gdµ. Therefore, UT n gdµ L2 gdµ Actually, we only need that UT n gdµ L2 gdµ

9 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 9 which is true since L 2 convergence implies weak convergence. ow we have lim f, UT n g = f, g = f, gdµ = fdµ gdµ. Corollary 2.3. A measure-preserving system (, B, T, µ) is ergodic if and only if lim µ(a T n B) = µ(a)µ(b) for all A, B B. Proof. If (, B, T, µ) is ergodic, we apply Corollary 2.30 by taking f, g to be the characteristic functions of the sets A, B. It is clear that lim µ(a T n B) = µ(a)µ(b). On the other hand, suppose lim µ(a T n B) = µ(a)µ(b). Recall that in order to show that (, B, T, µ) is ergodic, we need to show that any strictly invariant set V B has either zero or full measure. We take A = V c and B = V. otice that µ(a T n B) = µ(a B) = µ(b c B) = 0 for any integer n. Therefore, µ(a)µ(b) µ(a T n B) = 0, meaning that either A = B c or B has zero measure, i.e. T is ergodic. While the mean ergodic theorem only applies to L 2 functions, the theorem below is more general since it is true for all L functions. The trade-off here is that the convergence is only guaranteed to be pointwise. Theorem (Birkhoff). Given a measure-preserving ergodic system (, B, T, µ) and any L function f, we have lim UT n f = fdµ n almost everywhere and pointwise in L. Definition A measure-preserving system (, B, T, µ) is mixing, or strong mixing if lim n µ(a T n B) = µ(a)µ(b) for all A, B B. Definition A measure-preserving system (, B, T, µ) is weak mixing if µ(a T n B) µ(a)µ(b) = 0 for all A, B B. lim Remark Recall that it is a fact in analysis that if a sequence of real numbers {a n } R converges to some real number a, then the Cesaro sum has the same limit, i.e. a n a. Therefore, strong mixing implies weak mixing. Theorem If a measure-preserving system (, B, T, µ) is weak mixing, then it is ergodic. Proof. Take a strictly invariant set A B. By Definition 2.34, we have lim µ(a T n A) µ(a)µ(a) = 0. Since A = T n A for any integer n, we know that µ(a) = µ(a) 2. Therefore, µ(a) is either 0 or. This implies the ergodicity of (, B, T, µ).

10 0 ZIJIA WAG Remark We can finally conclude that: Mixing Weak mixing Ergodic. ow we provide some characterizations of weak mixing systems via product spaces. The following lemma, which characterizes bounded real sequences that has zero Cesaro sum, is a fundamental and elementary result in real analysis. Lemma If a n R is a nonnegative bounded sequence that has zero Cesaro sum, i.e. lim a n = 0, then there exists an index set J with density 7 zero such that lim a n = 0. n,n / J This lemma yields a direct corollary on weak mixing systems when we take a n to be µ(a T n B) µ(a)µ(b). Corollary Given a weak mixing system (, B, T, µ) and A, B B, there exists an index set J with density zero such that lim µ(a T n B) µ(a)µ(b) = 0. n,n/ J Remark Corollary 2.39 shows that weak mixing systems are very similar to strong mixing systems when we ignore an index set with zero density. Theorem 2.4. Given a measure-preserving system (, B, T, µ) following are equivalent:. (, B, T, µ) is weak mixing. 2. Given any ergodic system (Y, B Y, T Y, ν), ( Y, B B Y, T T Y, µ ν) is ergodic. 3. (, B B, T T, µ µ) is weak mixing. Proof. (3) (): Take A, B B, we consider sets A, B B B. otice that for any measurable set C B, µ µ(c ) = µ(c). We can conclude that is weak mixing by the weak mixing of. () (3): It suffices to check the definition of weak mixing for rectangle sets A B, C D B B. By Corollary 2.39, there exists index sets K, L with density zero such that lim µ(a T n C) µ(a)µ(c) = 0, n,n/ K lim µ(b T n D) µ(b)µ(d) = 0. n,n/ L otice that a finite union of zero density sets has zero density. We take J = K L and have lim µ(a T n C) µ(a)µ(c) = 0, n,n/ J lim µ(b T n D) µ(b)µ(d) = 0. n,n/ J 7 We have defined density in Remark 3.2.

11 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM Therefore, lim µ µ((a B) T n (C D)) µ µ(a B)µ µ(c D) n,n/ J n,n/ J µ µ((a T n C) (B T n D)) µ(a)µ(b)µ(c)µ(d) n,n/ J µ(a T n C)µ(B T n D) µ(a)µ(b)µ(c)µ(d) =0. This proves that T T is weak mixing since J has zero density. () (2): It suffices to prove Corollary 2.3 for rectangular sets in Y. For any pair of rectangular sets A B, C D B B Y, we need to show that lim Actually, lim µ ν((a B) T n (C D)) = µ ν(a B)µ ν(c D). + lim µ ν((a B) T n (C D)) µ(a T n C)ν(B T n D) µ(a)µ(c)ν(b T n D) (µ(a T n C) µ(a)µ(c))ν(b T n D) =µ(a)µ(c)ν(b)ν(d) + (µ(a T n C) µ(a)µ(c))ν(b T n D) (by the ergodicity of (Y, B Y, T Y, ν)) µ(a)µ(c)ν(b)ν(d) + (µ(a T n C) µ(a)µ(c)) (since ν(b T n D) ) =µ(a)µ(c)ν(b)ν(d) (since (, B, T, µ) is weak mixing) =µ ν(a B)µ ν(c D). This shows that ( Y, B B Y, T T Y, µ ν) is ergodic. (2) () Assume that for any ergodic system (Y, B Y, T Y, ν), ( Y, B B Y, T T Y, µ ν) is ergodic. We first consider the ergodic system (Y, B Y, id Y, ν) where Y only contains a single element e and id Y is the identity element on Y. There is a canonical isomorphism φ : Y that sends (x, e) Y to x. Therefore (, T ) is ergodic since it is isomorphic to ( Y, T id Y ) and ( Y, T id Y ) is ergodic by (2). We can further deduce that (, T T ) is ergodic by applying (2) again knowing that (, T ) is ergodic. Recall that in order to show that (, B, T, µ) is weak mixing, we need to prove that n µ(a T B) µ(a)µ(b) = 0 for all A, B B. Actually, lim

12 2 ZIJIA WAG it suffices to show that lim general result in real analysis. Indeed, n µ(a T B) µ(a)µ(b) 2 = 0 by a (2.42) (2.43) (2.44) (2.45) (2.46) lim 2 lim 2µ(A)µ(B) lim (µ(a T n B) µ(a)µ(b))2 µ(a T n B)2 + lim µ(a T n B)µ(A)µ(B) µ(a T n B)2 + µ(a) 2 µ(b) 2 µ(a T n B). µ(a) 2 µ(b) 2 In order to compute the two terms lim µ(a T n B)2 and lim µ(a T n B), we apply the ergodicity of (, T T ) and deduce that lim µ(a T n B) µ µ((a ) (T T ) n (B )) =(µ µ)(a )(µ µ)(b ) =µ(a)µ(b), lim µ(a T n B)2 (µ µ)((a A) (T T ) n (B B)) =(µ µ)(a A)(µ µ)(b B) =µ(a) 2 µ(b) 2.

13 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 3 ow we can apply (2.46) and have that lim 2µ(A)µ(B) lim (µ(a T n B) µ(a)µ(b))2 µ(a T n B)2 + µ(a) 2 µ(b) 2 µ(a T n B) =µ(a) 2 µ(b) 2 + µ(a) 2 µ(b) 2 2µ(A)µ(B)µ(A)µ(B) =0. Corollary If (, B, T, µ) and (Y, B Y, T Y, ν) are both weak mixing, then ( Y, B B Y, T T Y, µ ν) is weak mixing. Proof. Take an arbitrary ergodic system (Z, B Z, T Z, κ). Since (Y, B Y, T Y, ν) is weak mixing, by the equivalence that we established in Theorem 2.4, we have that the system (Y Z, B Y B Z, T Y T Z, ν κ is ergodic. We apply Theorem 2.4 again and we know that ( Y Z, B B Y B Z, T T Y T Z, µ ν κ) is ergodic because (, B, T, µ) is weak mixing. Since (Z, B Z, T Z, κ) is chosen arbitrarily, we can conclude that ( Y, B B Y, T T Y, µ ν) is weak mixing Compact systems. We have seen some compact systems already but here we provide the formal definition about what we mean by compact systems. Definition A system (, B, T, µ) is compact if the orbit of every L 2 functions on is precompact. Remark We sometimes call functions that have precompact orbits almost periodic functions. Another way of saying a dynamical system is compact is that every L 2 functions on it is almost periodic. We now give a different characterization of dynamical compactness and we will be using this to prove that compact systems are SZ. Theorem A system (, B, T, µ) is compact if and only if for any f L 2 () and ɛ > 0 the set {k f U k T f 2 < ɛ} has bounded gaps. Remark 2.5. Integer sets that have bounded gaps are called syndetic sets. Below is a formal definition for syndetic sets but it is much easier to just think of them as sets that has bounded gaps. Definition A set A is syndetic if there exists a finite set of integers {a i } i k such that k i= A a i. Proof of Theorem Given an arbitrary L 2 function f, let A ɛ be {k f U k T f 2 < ɛ}. It suffices to show that Orb(f) is totally bounded. Indeed, for any ɛ > 0 there exists a finite set of integers {ɛ i } i k such that k i= A ɛ ɛ i. This means that Orb(f) k ɛi i= B(UT, ɛ) 8. This proves totally boundedness since ɛ is arbitrary. 8 B(x, r) means the open ball around the point x with radius r.

14 4 ZIJIA WAG Example It is helpful to think of rotations when we are dealing with compact systems. Recall the rotation system on the circle {T, R α, µ} where α / Q. Given a L 2 function f, the orbit of f is Orb(f) = {f(x + nα)} n. It might not be obvious at first glance that the set Orb(f) is precompact. However, the conclusion is quite obvious if we use the second characterization of compactness given in Theorem This follows from the fact that the orbit of any point on a circle is equidistributed under irrational rotations. Remark The rotations does not have to be on the circle. In fact, we can define rotation on any compact abelian group. Definition A Kronecker system is defined by (G, R α ) where G is a compact abelian group and R α acts on G by translation by α, i.e. R α (g) = gα for some α G. We conclude the introduction to basic dynamical systems by stating the two useful results about compact systems and weak mixing systems []. Theorem If a compact system is ergodic, then it is isomorphic to a kronecker system. Remark ow we have seen the two fundamental structures of dynamical systems, namely the weak mixing systems and the compact systems. Interestingly enough, the two fundamental systems are opposite to each other in the following sense. Theorem A dynamical system is weak mixing if and only if it has no nontrivial compact factors. Remark In other words, a measure-preserving system is either weak mixing or has at least one nontrivial compact factor Factor and extension. We have already encountered a special kind of extension in Example 2.4. Here, we generalize the idea of an isomorphism and introduce the notion of an extension. Definition Given two measure-preserving systems = (, B, µ, T ) and Y = (Y, B Y, ν, T Y ), we say that Y is a factor of if there is some measurepreserving factor map φ defined almost everywhere such that the diagram below commutes. T Y We say that is an extension of Y. φ T Y Remark 2.6. A factor map is weaker than an isomorphism in the sense that we do not require the factor map to be invertible. Example Every measure-preserving system has a trivial factor, namely the factor that consists of a single element. Factors may also be created by taking sub σ algebras. Given a measure-preserving system (, B, µ, T ) and B B a proper sub σ algebra, we can view the system (, B, µ, T ) as a factor of (, B, µ, T ) where the factor map is given by the identity map id. Despite the fact that id Y φ

15 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 5 is clearly defined almost everywhere, id is not an isomorphism. ote that B is strictly smaller than B. We will see a generalization of this example later in Example 2.7 where we talk about condition measures Conditional measures. The concept of conditional measure is useful when we are working with more than one dynamical systems at the same time, e.g. relatively weak mixing extensions. It allows us to construct a measure with a given sub σ algebra. In our case of scenario, the smaller σ algebra is usually generated by either the fibers of the measure-preserving map that we are working with or the pullback of the σ algebra of the factor as we will see in Example 2.7. Before going directly into conditional measures, we first take a look at conditional expectations. Definition Given a probability space (, B, µ) and an integrable function f, the expectation of f is defined by E(f) = fdµ9. If C B is a sub σ algebra, the conditional expectation of f on C is E(f C), where E(f C) is the unique element in L (, C, µ) such that the following is true: () The function E(f C) is a measurable function on (, C, µ). (2) For each C C, C E(f C)dµ = C fdµ. Remark The existence and uniqueness of conditional expectation is a consequence of the Radon-ikodym theorem. We will only use the notions discussed in this section as black boxes since they are not the focus of this paper. Example (, B, µ) is a probability space and {P i } i n a partition of 0. We consider C, a finite sub σ algebra generated by the given partition {P i } i n. The conditional expectation of an integrable function f is given by E(f C)(x) = if x P i. This is clearly measurable under C. Moreover, for any measurable P i fdµ µ(p i) set C C, we have that C E(f C)dµ = C fdµ. This is trivial for each C {P i } i n but every set in C is a finite union of the elements in the partition. ote that the conditional expectation is defined almost everywhere, therefore it does not matter if an element in the partition has measure zero. Theorem Given a probability space (, B, µ) and sub σ algebras D C B and f L (, C, µ), g L (, B, µ),the following are true: () E(fg C) = fe(g C), (2) E(E(g C) D) = E(g D). Remark The first statement in Theorem 2.66 implies that an integrable function that is measurable in the smaller sub σ algebra can be considered as a constant thus can be pulled out. One can also think of σ algebras as the collection of information and the conditional expectation with respect to a given sub σ algebra is just the expectation of an event given the information that we have at hand. If we have no information at hand, i.e. the sub σ algebra is trivial, then the best guess we can give is the usual expectation. Example When D is the trivial σ algebra, we have that E(E(f C)) = E(f), which is exactly what we required in the Definition We use E and E to distinguish the usual expectation and the conditional expectation 0 A collection of subsets of a given space is called a partition if the elements in the collection are mutually disjoint and the union of all the elements is the entire space.

16 6 ZIJIA WAG Definition Let (, B, µ) be a probability space. Suppose that we have a sub σ algebra C B. Then for almost every x, we can define a family of of probability measures (on ) {µ x } x. We call them the conditional measures. The conditional measures satisfy the following properties: () E(f C)(x) = f(t)dµ x(t), (2) The map x µ x is measurable with respect to C, 2 (3) When C is countably generated, µ x = µ y if and only if they are in the same atom of C. 3 Remark otice that some regularity conditions on the sub σ algebra is required to make sense of statement (3) in Definition Although countability of the sub σ algebra is a fair assumption to make, e.g. in Example 2.7, conditional measures can still be defined without countability. However, (3) would not work if the sub σ algebra is not countably generated. This is just because the intersection of uncountably many measurable sets is not necessarily measurable. Example 2.7. Just as we have mentioned at the start of the introduction to conditional measure, we are going to use this tool when we need to work with extensions. Suppose (, B, µ, T x ) φ (Y, B Y, ν, T y ) is an extension between two dynamical systems. C = φ B Y gives a sub σ algebra of B. Using C we can define a family of conditional measures {µ x } x for almost every x and a C measurable map x δx µ x. Actually, we are working with the φ B Y, so µ x = µ x are the same if x and x are in the same fiber of φ. Moreover, we can think of the conditional measures as {µ y } y Y. A canonical measurable map y δy µ y can be given by the diagram below δ x {µ x } x φ φ δ y Y {µ y } y Y. Since the spaces L 2 (, C, µ, T x ) and L 2 (Y, B Y, ν, T y ) are isomorphic, we can introduce the following notation used by Furstenberg []. Given a function f L 2 (, B, µ), we define the conditional expectation E(f Y)(y) = fdµ y. Recall that in order for φ to be an isomorphism, we need the following diagram to commute a.e. φ T x T y Y Y. Therefore, we have the following identities deduced from Theorem Given a C measurable function g, E(gf Y) = ge(f Y), U Ty E(f Y) = E(U Tx f Y). φ ote that the family of measures depends on the choice of C. 2 We can think of the space of the measures on as the linear functionals on L (). 3 Given a measure space (, B, µ) where B is countably generated, the atom containing the point x is defined to be the intersection of all the measurable sets containing x.

17 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM Weak mixing and compactness for extensions. We can generalize the compactness and weak mixing properties of dynamical systems to relative compactness and relative weak mixing properties with the help of extensions and conditional measures. All the definitions below assumes that = (, B, µ, T x ) is an extension of Y = (Y, B Y, ν, T y ). Definition A function f L 2 (, µ) is almost periodic with respect to Y if for every ɛ > 0, there exists a finite collection of functions {g i } i k L 2 (, µ) such that min i r U n T f g i L 2 µy < ɛ almost everywhere for all n. Remark ote that the definition of relative almost periodicity involves the conditional measure µ y constructed in Example 2.7. They are probability measures for instead of Y but there is a measurable map y µ y. Definition The extension Y is compact if the set of functions almost periodic with respect to Y is dense in L 2 (, µ). In order to generalize weak mixing, we need to first introduce the notion of relatively independent joining. There are several different constructions of relatively independent joining, here we introduce the formulation used by Einsiedler and Ward [2]. Definition Given two measure-preserving systems (, B, µ, T ) and (Y, B Y, ν, T Y ), a joining is a T T Y invariant measure δ defined on the space ( Y, B B Y ) such that the projections of δ onto and Y coordinates are µ and ν respectively, i.e. () δ(a Y ) = µ(a ) for all A B, (2) δ(a Y ) = ν(a Y ) for all A Y B Y. Example The product measure µ ν is always a joining by construction. Definition Given two invertible measure-preserving systems (, B, µ, T ) and (Y, B Y, ν, T Y ) that shares a common non-trivial factor (Z, B Z, δ, T Z ) via factor maps φ and φ Y, we define the relatively independent joining µ δ ν(, or Z Y ) in the following way. Let B be φ B Z and B Y be φ Y B Z. We can define conditional measures on and Y using the sub σ algebras B and B Y. Let µ φ (x) be µ x and ν φy (y)] be ν y and µ δ ν = µ z ν zdδ(z). Z Definition The extension Y is weak mixing relative to Y if the system (, µ Y µ, T T ) is ergodic. Remark Recall that in Theorem 2.4, we have shown that a dynamical system (Z, C, δ, S) is weak mixing if and only if the product system (Z Z, C C, δ δ, S S) is ergodic. ow back to our definition of relatively weak mixing. ote that the relatively independent joining µ Y µ is exactly µ µ if Y is a trivial factor of. In other words, is relatively weak mixing with respect to its trivial factor if and only if the system (, B B, µ µ, T T ) is ergodic, i.e. is weak mixing in the usual sense.

18 8 ZIJIA WAG 2.6. The structure theorem. The theorem that we are going to introduce is usually referred to as the Furtsenberg-Zimmer structure theorem, which is a general result in ergodic theory and has nothing to do with Szemerédi s theorem at first glance. Instead of proving the structure theorem, we will only give a simplified version of it used by Furstenberg in his proof of Szemerédi s theorem []. Theorem (, B, µ, T ) is a measure-preserving system. Suppose it has a proper factor (Y, B Y, µ Y, T Y ), then one of the following is true: () The extension Y is relatively weak mixing. (2) There exists some intermediate factor (Z, B Z, µ Z, T Z ) such that the proper extension Z Y is compact (i.e. Y and Z are not isomorphic). Remark 2.8. The structure theorem can be viewed as a way of decomposing an arbitrary dynamical system where the decomposition results in a tower of extensions and each extension is either relatively weak mixing or compact as the picture below shows. φ... φ2 φ φ Each i represents a factor and each extension φ i is either weak mixing or compact. The structure theorem can be very powerful with the help of transfinite induction. Imagine that we need to prove some property about a general dynamical system, we only need to check that such property lifts through both weak mixing and compact extensions and that there exists a maximal factor that satisfies such property. Actually, this is exactly how we are going to prove Szemerédi s theorem. 3. Furstenburg s proof of Szemerédi s theorem In this section, we illustrate Furstenberg s proof of Szemerédi s theorem. The proof consists of three parts: showing that multiple recurrence of any measurepreserving system implies Szemerédi s theorem, proving the multiple recurrence property for two basic dynamical systems and finally using the structure theorem from Section 2.6 and the extension principles that we are going to establish to prove Szemerédi s theorem. 3.. General Strategy. In order to prove Szemerédi s theorem using ergodic theory, the first step is to establish some sort of correspondence between a sequence of numbers and a measure-preserving system. This is done by proving the correspondence principle in Theorem 3.9, which reduces our problem to proving that every measure-preserving system has some property SZ to be defined in Definition 3.2. As we have seen in Theorem 2.80, an arbitrary dynamical system can be decomposed into a tower of factors, where each extension is either relatively weak mixing or compact. φ... φ2 φ φ ow we show that the SZ property can be passed through both relatively weak mixing extensions and compact extensions by the extension principles proved in Theorem 3.27 and Theorem Finally, using Zorn s lemma and a lemma by Furstenberg, we will be able to prove Szemerédi s theorem.

19 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM Szemerédi s theorem. Before starting to prove the theorem, we shall introduce some conventions and backgrounds. Definition 3.. Given a set of integers A Z, the upper Banach density of A µ(a [,]) is defined to be lim sup 2+, where µ is the usual counting measure for the space of integers. Remark 3.2. This notion of upper Banach density might seem weird at first glance. There s actually a definition of natural density which replaces the lim sup with lim. Obviously there exist sets which doesn t possess natural density, but the notion of upper Banach density applies to all sets. One can define the upper Banach density µ(a [,M]) in a different but more conventional way, e.g. lim sup M M. It is not hard to show that these two definitions are actually equivalent in the sense that the property of having positive upper density is preserved. Indeed, we do not care about the exact numerical value of the upper density. Throughout this paper, we are going to use the first definition. Example 3.3. The set of odd numbers has density 2 density. and has the same upper Definition 3.4. Given a set of integers A Z, we say that it contains k-term arithmetic progression if there exists integers n, d Z such that n + id Z for i [0, k ]. Alternatively, we say that the set A contains k-ap. Theorem 3.5. (Szemerédi). Any set of integers A Z that has positive upper density contains k-ap for all k Correspondence. In this part, we introduce the notion of multiple recurrence. In fact, multiple recurrence of any order for any measure-preserving system is equivalent to Szemerédi s theorem. However for the sake of our argument, it suffices to show that multiple recurrence of all orders implies Szemerédi property. Once we have established the correspondence between Szemerédi s theorem and multiple recurrence, we can use the ergodic theory gadgets that we introduced in the second section to prove Szemerédi s theorem. To get started, we first recall the definition for multiple recurrence. Definition 3.6. A measure-preserving system (, B, T, µ) has multiple recurrence of order k if for each A B that has positive measure, there exist n such that ( k ) µ T in A > 0. i=0 In order to prove the correspondence principle, we need to introduce a classical result from functional analysis. Theorem 3.7. (Banach-Alaoglu) Given a separable normed linear space, the closed unit ball in the dual space is sequentially compact under the weak* topology.

20 20 ZIJIA WAG Remark 3.8. It is obvious that in Euclidean spaces, unit balls are closed and bounded therefore compact. However, a theorem of Riesz states that closed unit balls in infinite dimensional spaces are not compact in the usual norm topology. However, compactness of unit balls is regained in infinite dimensional spaces under the weak-* topology by the Banach-Alaoglu theorem. This theorem is extremely useful here since we are always working with infinite dimensional linear spaces in the form of C(Y ) where Y is compact. C(Y ) is therefore separable. Theorem 3.9. Multiple recurrence of any order for all measure-preserving systems implies Szemerédi s theorem. Proof. Given an integer sequence {a n } Z with positive upper Banach density and a fixed natural number k we show that {a n } contains k-ap. Recall the dynamical system (2 Z, T ) where T is the left Bernoulli shift. The sequence {a n } naturally corresponds to a point x in our space 2 Z where 4 (3.0) { x[j] = if j {an } x[j] = 0 otherwise Let A = {T n x} n Z be the closure of the two-sided 5 orbit of x. The subspace A is still compact since it is closed and 2 Z is compact. Let B A be the subset of elements such that the 0 th coordinate is. Observe that under the usual measure defined in section 2, B has measure 0 since B is an at most countable union of singletons, which has zero measure under the measure that we defined before. Therefore, we need to find a proper measure µ under which µ(b) > 0 in order to use the multiple recurrence property. The construction we give here is a standard one. Moreover, we need to relate the measure with the upper Banach density of the corresponding sequence. For any point y A, we define δ y to be the delta measure supported at the point y. Observe that 2 + n= δ T n x(b) = {a j} [, ]. 2 + We call the measure defined by the average of the delta measures ν = 2 + n= δ T n x. 4 We have defined x[j] to be the j th coordinate of x. 5 Usually when we refer to orbits, we assume that the number of times that we iterate is always positive, i.e. we consider sets in the form of {T n x} n.

21 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM 2 Since the sequence {a n } has positive upper Banach density, we can assume that there exists a sequence of natural numbers { i } such that lim ν j (B) j j 2 j + j n= j δ T n x(b) {a n } [ j, j ] j 2 j + sup {a n } [, ] 2 + > 0. By construction {ν j } is a subset of the unit sphere in the space of measures. Therefore, there is a subsequence { jl } k such that the sequence {ν jl } k converges to some measure µ under the weak* topology. Moreover, µ(b) > 0 by weak* convergence since B is closed. We also need to verify that µ makes the system measure-preserving. Indeed, ν jl U T ν jl jl = δ T 2 jl + n x n= jl ( = δt jl 2 jl + δ ) x T jl + x, jl + n= jl + δ T n x which goes to 0 as l approaches infinity. Therefore, the limit µ is invariant under T. ow we are ready to use the multiple recurrence hypothesis on the dynamical system (A, T, ν) and the set B. By multiple recurrence of order k, there exists some positive integer n such that ν( k i=0 T in B) > 0. Therefore, there is at least a point y {T n x} n k i=0 T in B since B is open in A. Suppose y = T m x, then the set {T m+in x} 0 i k is contained in B. In other words, {a n } contains the k-term arithmetic progression {m + in} 0 i k. Since k is arbitrary, we have Szemerédi s theorem. Remark 3.. Recall that we have shown the Poincaré recurrence (Theorem 2.20). It is a very elementary result yet similar to the complicated Szemerédi s theorem. otice that for weak mixing systems, we automatically have multiple recurrence of order 2 by definition. Weak mixing systems seem easier to deal with. Therefore, we prove a stronger result which looks quite similar to the weak mixing property, which also involve Cesaro limits. We call this property SZ for simplicity, which is short for Szemerédi. Definition 3.2. We call a measure-preserving system (, B, T, µ) SZ if for each A B that has positive measure and any positive integer k, ( k ) lim inf µ T in A > 0 n= Remark 3.3. otice that in the proof of Szemerédi s theorem, we provide different criteria that characterize the Szemerédi property, for instance, Definion 3.2. i=0

22 22 ZIJIA WAG Some of these criteria are equivalent but the others are not. The following theorem shows that being SZ is stronger than having multiple recurrence of any order. Theorem 3.4. A measure-preserving system (, B, T, µ) has multiple recurrence of any order if it is SZ. Proof. For any positive integer k, is SZ gives us ( k ) lim µ T in A > 0. n= i=0 ( k ) Then there must exist some integer n such that µ i=0 T in A > 0. If not, ( lim n= µ k ) i=0 T in A would be 0 for all. The limit as would also be 0. ow we give an alternative characterization of the SZ property using functions. We will be using this to prove that weak mixing systems are SZ. As we will see later, this formulation will also allow us to exploit certain special properties of compactness and relative compactness since they are characterized by almost periodic functions. Theorem 3.5. A measure-preserving system (, B, T, µ) is SZ if and only if for any function f L such that f is nonnegative almost everywhere and f > 0, we have (3.6) lim inf where U T is the associated operator of T. k i=0 U in T fdµ > 0, Proof. Suppose Theorem 3.6 is true, let f = χ A be the characteristic function of a set A with positive measure. Then we have the SZ property. On the other hand the SZ property guarantees Theorem 3.6 for all characteristic functions of sets that have positive measures. We know that Theorem 3.6 is also true for all nonnegative L functions f with positive integral if we take a sequence of simple functions approaching f. ow we apply the monotone convergence theorem 6. Example 3.7. Consider the dynamical system (T, R α ) where T is the circle and R α : x x + α is an irrational rotation acting on the circle. We use the function formulation of SZ in Theorem 3.5. For a fixed, the Furstenberg ergodic average looks like k i=0 f(t nα)dµ(t). We have proved in Example 2.25 that irrational rotations are ergodic, which means that {nα} n is equidistributed on the circle. Therefore, when the ergodic average is equivalent to k f(x it)dµ(x)dµ(t). T T i=0 We can see that the double integral is positive by considering t sufficiently small. ote that the expectation of f is strictly positive by assumption. When t is sufficiently small, the double integral is close to CE(f) k, where C is some positive constant depending on the restriction that we put on t. 6 This is the reason that we require f to be nonnegative a.e.

23 FURSTEBERG S ERGODIC THEORY PROOF OF SZEMERÉDI S THEOREM Two fundamental systems. We prove that the two dynamical systems that are very well-structured has the SZ property, namely the compact systems and weak mixing systems Weak mixing systems. In order to prove that weak mixing systems are SZ, we need to introduce the following lemma, which is a very classical result in real analysis. It is sometimes known as the van der Corput trick. Lemma 3.8. (van der Corput). Suppose we have a bounded sequence {v n } n in some Hilbert space. We define a sequence of real numbers {a n } n 0 R such that a i sup n= v n+i, v n. If lim a n = 0, then we have lim n= v n = 0 Theorem 3.9. Given a weak mixing system (, B, T, µ) and a finite set of k functions {f i } i k L (), we have k k lim UT in f i = f i dµ in L 2. i= Remark It is clear that Theorem 3.9 implies the function formulation of SZ stated in Theorem 3.6 where we take a single function instead of k different functions. It is actually not quite surprising why in the case of weak mixing systems we not only know that the limit is larger than zero when all the functions are nonnegative and have positive expectation, but also have information on exactly what the limit converges to. Indeed we know that if (, B, T, µ) is weak mixing, then µ(a T n B) µ(a)µ(b) goes to 0 for any measurable set A, B B. Proof. The proof goes by induction. When k =, by mean ergodic theorem and the fact that weak mixing implies ergodicity, we know that lim U T nf exists and equals to the expectation of f. For simplicity, we will only show how the k = case implies the k = 2 case instead of doing a general inductive step. Therefore, we need to show that lim U n T f U 2n T f 2 = otice that if f = c for some constant c, then lim =c lim =c = i= f dµ U n T f U 2n T f 2 U 2n T f 2 f 2 (By the k = case) f f 2. f 2 dµ.