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It is clear that this statement immediately implies the first formulation of Sze. mere di s theorem and a compactness argument gives the converse implication. 1 2 Translation to a probability system Starting with Szemere di s the. orem one gains insight into the intersection of sufficiently many sets with positive. measure in an arbitrary probability system 2 Note that N k denotes the quantity. in Theorem 1 1, Corollary 1 2 Let 0 k N X X be a probability space and A1. AN X with Ai for i 1 N If N N k then there exist a d N. Aa Aa d Aa 2d Aa kd 6, Proof For A X let 1A x denote the characteristic function of A meaning. that 1A x is 1 for x A and is 0 otherwise Let N N k Then. Thus there exists x X such that, Then E n x An satisfies E N and so Szemere di s theorem implies. that E contains an arithmetic progression of length k By the definition of E we. have a sequence of sets with the desired property, 1 3 Measure preserving systems A probability measure preserving system. is a quadruple X X T where X X is a probability space and T X X. is a bijective measurable measure preserving transformation This means that for. all A X T 1 A X and, In general we refer to a probability measure preserving system as a system.
Without loss of generality we can place several simplifying assumptions on. our systems We assume that X is countably generated thus for 1 p. Lp is separable We implicitly assume that all sets and functions are measurable. with respect to the appropriate algebra even when this is not explicitly stated. Equality between sets or functions is always meant up to sets of measure 0. 2A algebra is a collection X of subsets of X satisfying i X X ii for any A X we. also have X A X iii for any countable,W V collection An X we also have n 1 An X A. algebra is endowed with operations and c which correspond to union intersection and. taking complements By a probability system we mean a triple X X where X is a measure. space X is a algebra of measurable subsets of X and is a probability measure In general. we use the convention of denoting the algebra X by the associated calligraphic version of the. measure space X,ERGODIC METHODS IN ADDITIVE COMBINATORICS 3. 1 4 Furstenberg multiple recurrence In a system one can use Szeme. re di s theorem to derive a bit more information about intersections of sets If. X X T is a system and A X with A 0 then,A T 1 A T 2 A T n A. are all sets of the same measure and so all have measure Applying Corol. lary 1 2 to this sequence of sets we have the existence of a d N with. T a A T a d A T a 2d T a kd A 6, Furthermore the measure of this intersection must be positive If not we could. remove from A a subset of measure zero containing all the intersections and obtain. a subset of measure at least without this property In this way starting with. Szemere di s Theorem we have derived Furstenberg s multiple recurrence theorem. Theorem 1 3 Furstenberg 16 Let X X T be a system and let A X. with A 0 Then for any k 1 there exists n N such that. 1 1 A T n A T 2n A T kn A 0,2 Ergodic theory to combinatorics.
2 1 Strong form of multiple recurrence We have seen that Fursten. berg multiple recurrence can be easily derived from Szemere di s theorem More. interesting is the converse implication showing that one can use ergodic theory. to prove regularity properties of subsets of the integers and in particular derive. Szemere di s theorem This is what Furstenberg did in his landmark paper 16. and the techniques introduced in this paper have been used subsequently to deduce. other patterns in subsets of integers with positive upper density See Section 9. Moreover Furstenberg s proof lead to new questions within ergodic theory about. the structure of measure preserving systems In turn this finer analysis of measure. preserving systems has had implications in additive combinatorics We return to. these questions in Section 3, Furstenberg s approach to Szemere di s theorem has two major components. The first is proving a certain recurrence statement in ergodic theory like that of. Theorem 1 3 The second is showing that this statement implies a corresponding. statement about subsets of the integers We now make this more precise. To use ergodic theory to show that some intersection of sets has positive mea. sure it is natural to average the expression under consideration This leads us to. the strong form of Furstenberg s multiple recurrence. Theorem 2 1 Furstenberg 16 Let X X T be a system and let A X. with A 0 Then for any k 1,2 1 lim inf A T n A T 2n A T kn A. is positive, In particular this implies the existence of infinitely many n N such that the. intersection in 1 1 is positive and Theorem 1 3 follows In Section 3 we discuss. how to prove Theorem 2 1, 2 2 The correspondence principle The second major component in. Furstenberg s proof is using this multiple recurrence statement to derive a statement. about integers such as Szemere di s theorem This is the content of Furstenberg s. correspondence principle, Theorem 2 2 Furstenberg 16 17 Let E Z have positive upper density.
There exist a system X X T and a set A X with A d E such that. T m1 A T mk A d E m1 E mk,for all k N and all m1 mk Z. Proof Let X 0 1 Z be endowed with the product topology and the shift. map T given by T x n x n 1 for all n Z A point of X is thus a sequence. x x n n Z and the distance between two points x x n n Z y y n n Z. is defined to be 0 if x y and to be 2 k if x 6 y and k min n x n 6 y n. Define a a n n Z 0 1 Z by,0 otherwise, and let A x X x 0 1 Thus A is a clopen closed and open set. The set A X plays the same role as the set E Z for all n Z. T n a A if and only if n E, By definition of d E there exist sequences Mi and Ni of integers with. Ni such that,lim E Mi Mi Ni 1 d E,It follows that,lim 1A T n a lim 1E n d E. Let C be the countable algebra generated by cylinder sets meaning sets that. are defined by specifying finitely many coordinates of each element and leaving the. others free We can define an additive measure on C by. B lim 1B T n a, where we pass if necessary to subsequences Ni Mi such that this limit exists.
for all B C Note that C is countable and so by diagonalization we can arrange. it such that this limit exists for all elements of C. We can extend the additive measure to a additive measure on all Borel. sets X in X which is exactly the algebra generated by C Then is an invariant. measure meaning that for all B C,T B lim 1B T n 1 a B. ERGODIC METHODS IN ADDITIVE COMBINATORICS 5,Furthermore. A lim 1A T n a d E, If m1 mk Z then the set T A T mk A is a clopen set its indicator. function is continuous and,T m1 A T mk A lim 1T m1 A T mk A T n a. lim 1 E m1 E mk n,d E m1 E mk, We use this to deduce Szemere di s theorem from Theorem 1 3 As in the proof.
of the correspondence principle define a 0 1 Z by,0 otherwise. and set A x 0 1 Z x 0 1 Thus T n a A if and only if n E. By Theorem 1 3 there exists n N such that,A T n A T 2n A T kn A 0. Therefore for some m N T m a enters this multiple intersection and so. a m a m n a m 2n a m kn 1,But this means that,m m n m 2n m kn E. and so we have found an arithmetic progression of length k 1 in E. 3 Convergence of multiple ergodic averages, 3 1 Convergence along arithmetic progressions Furstenberg s multiple. recurrence theorem left open the question of the existence of the limit in 2 1 More. generally one can ask if given a system X X T and f1 f2 fk L. 3 1 lim f1 T n x f2 T 2n x fk T kn x, exist Moreover we can ask in what sense in L2 or pointwise does this limit.
exist and if it does exist what can be said about the limit Setting each function. fi to be the indicator function of a measurable set A we are back in the context of. Furstenberg s theorem, For k 1 existence of the limit in L2 is the mean ergodic theorem of von. Neumann In Section 4 2 we give a proof of this statement For k 2 existence of. the limit in L2 was proven by Furstenberg 16 as part of his proof of Szemere di s. theorem Furthermore in the same paper he showed the existence of the limit in. L2 in a weak mixing system for arbitrary k we define weak mixing in Section 5 5. and outline the proof for this case, For k 3 the proof of existence of the limit in 3 1 requires a more subtle. understanding of measure preserving systems and we begin discussing this case in. Section 5 8 Under some technical hypotheses the existence of the limit in L2. for k 3 was first proven by Conze and Lesigne see 8 9 then by Furstenberg. and Weiss 22 and in the general case by Host and Kra 32 More generally we. showed the existence of the limit for all k N, Theorem 3 1 Host and Kra 34 Let X X T be a system let k N. and let f1 f2 fk L Then the averages,f1 T n x f2 T 2n x fk T kn x. converge in L2 as N, Such a convergence result for a finite system is trivial For example if X.
Z N Z then X consists of all partitions of X and is the uniform probability. measure meaning that the measure of a set is proportional to the cardinality of. the set The transformation T is given by T x x 1 mod N It is then trivial. to check the convergence of the average in 3 1 However although the ergodic. theory is trivial in this case there are common themes to be explored Throughout. these notes an effort is made to highlight the connection with recent advances in. additive combinatorics see 39 for more on this connection Of particular interest. is the role played by nilpotent groups and homogeneous spaces of nilpotent groups. in the proof of the ergodic statement, Much of the present notes is devoted to understanding the ingredients in the. proof of Theorem 3 1 and the role of nilpotent groups in this proof Other exposi. tory accounts of this proof can be found in 31 40 In this context 2 step nilpotent. groups first appeared in the work of Conze Lesigne in their proof of convergence. for k 3 and a k 1 step nilpotent group plays a similar role in convergence. for the average in 3 1 Nilpotent groups also play some role in the combinatorial. setup and this has been recently verified by Green and Tao see 26 28 for pro. gressions of length 4 which corresponds to the case k 3 in 3 1 For more on. this connection see the lecture notes of Ben Green in this volume. 3 2 Other results Using ergodic theory other patterns have been shown to. exist in sets of positive upper density and we discuss these results in Section 9 We. briefly summarize these results A striking example is the theorem of Bergelson and. Leibman 6 showing the existence of polynomial patterns in such sets Analogous. to the linear average corresponding to arithmetic progressions existence of the. associated polynomial averages was shown in 35 45 One can also average along. cubes existence of these averages and a corresponding combinatorial statement. was shown in 34 For commuting transformations little is known and these partial. results are summarized in Section 9 1 An explicit formula for the limit in 3 1 was. given by Ziegler 56 who also has recently given a second proof 57 of Theorem 3 1. 4 Single convergence the case k 1, 4 1 Poincare recurrence The case k 1 in Furstenberg s multiple recur. rence Theorem 1 3 is Poincare recurrence,ERGODIC METHODS IN ADDITIVE COMBINATORICS 7. Theorem 4 1 Poincare 49 If X X T is a system and A X with. A 0 then there exist infinitely many n N such that A T n A 0. Proof Let F x A T n x A for all n 1 Thus F T n F for. all n 1 and so for all integers n 6 m,T m A T n A, In particular F T 1 F T 2 F are all pairwise disjoint sets and each set in this. sequence has measure equal to F If F 0 then,a contradiction of being a probability measure.
Therefore F 0 and the statement is proven, In fact the same proof shows a bit more by a simple modification of the. definition of F we have that almost every x A returns to A infinitely often. 4 2 The von Neumann ergodic theorem Although the proof of Poincare. recurrence is simple unfortunately there seems to be no way to generalize it to. show multiple recurrence In order to find a method that generalizes for multiple. recurrence we prove a stronger statement than Poincare recurrence taking the. average of the expression under consideration and showing that the lim inf of this. average is positive It is not any harder for k 1 only to show that the limit of. this average exists and is positive This is the content of the von Neumann mean. ergodic theorem We first give the statement in a general Hilbert space. Theorem 4 2 von Neumann 55 If U is an isometry of a Hilbert space H. and P is orthogonal projection onto the U invariant subspace I f H U f. f then for all f H,4 1 lim U f P f, Thus the case k 1 in Theorem 3 1 is an immediate corollary of the von. Neumann ergodic theorem,Proof If f I then, for all N N and so obviously the average converges to f On the other hand if. f g U g for some g H then,U nf g U N g, and so the average converges to 0 as N Set J g U g g H If fk J. and fk f J then,1 1 n 1 NX 1,U f fk U fk,n 0 n 0 n 0.
U kf fk k 1 U n, Thus for f J the average 1 N n 0 U n f also converges to 0 as N. We now show that an arbitrary f H can be written as a combination of. functions which exhibit these behaviors meaning that any f H can be written. as f f1 f2 for some f1 I and f2 J If h J then for all g H. 0 hh g U gi hh gi hh U gi hh gi hU h gi hh U h gi, and so h U h and h U h Conversely reversing the steps we have that if h I. Centre de Recherches Math ematiques CRM Proceedings and Lecture Notes Volume 2007 Ergodic Methods in Additive Combinatorics Bryna Kra Abstract Shortly after Szemer edi s