Multiple recurrence and nilsequences

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1 = Multiple recurrence and nilsequences Vitaly Bergelson 1 Bernard Host 2 Bryna Kra 3 ith an Appendix by Imre Ruzsa 4 1 Department o Mathematics The Ohio State University 100 Math Toer 231 West 18th Avenue Columbus OH USA. 2 Équipe d analyse et de mathématiques appliquées Université de Marne la Vallée Marne la Vallée Cedex France. ! "$#% 3 Department o Mathematics Northestern University 2033 Sheridan Road Evanston IL USA. & % ' % ()) % 4 Alréd Rényi Institute o Mathematics %)* % ) + Hungarian Academy o Sciences POB 127 H Budapest Hungary. May Abstract. Aiming at a simultaneous extension o Khintchine s and Furstenberg s Recurrence theorems e address the question i or a measure pre- o positive measure the set serving system X -./- µ- T 0 and a set A 12. o integers n such that µ A 3 T n A 3 T 2n A T n A076 µ A ε is syndetic. The size o this set surprisingly enough depends on the length : 10 o the arithmetic progression under consideration. In an ergodic system or ; 2 and ; 3 this set is syndetic hile or < 4 it is not. The main tool is a decomposition result or the multicorrelation sequence x0 T n x0 T 2n x0+444 T n x0 dµ x0 here and n are positive integers and is a bounded measurable unction. We also derive combinatorial consequences o these results or example shoing that or a set o integers E ith upper Banach density d> is syndetic. n 1BA : d>dc E 3 E0?6 0 and or all ε 6 0 the set 4 E : n0d3 E : 2n0D3 E : 3n0EF6 d> E0 9 εg 1. Introduction 1.1. Ergodic theory results We begin by recalling to classical results o early ergodic theory. Let X -./- µ- T 0 be a measure preserving probability system ith an invertible H The irst author as partially supported by NSF grant DMS and the third author as partially supported by NSF grant DMS

2 Multiple recurrence and nilsequences 3 measure preserving transormation T. (For brevity e call X -./- µ- T 0 a system.) Let A 1I. be a set o positive measure. The Poincaré Recurrence Theorem states that µ A 3 T n A0?6 0 or ininitely many values o n. The Khintchine Recurrence Theorem [K] states that the measure µ A 3 T n A0 is large or many values o n. Beore stating the result precisely e need a deinition: Deinition 1.1. A subset E o the integers A is syndetic i A can be covered by initely many translates o E. In other ords E has bounded gaps meaning that there exists an integer L 6 0 such that every interval o length L contains at least one element o E. In [K] Khintchine strengthened the Poincaré Recurrence Theorem shoing that under the same assumptions: For every ε 6 0 J n 1KA : µ A 3 T n A0?6 µ A0 2 9 ε L is syndetic. More recently Furstenberg proved a Multiple Recurrence Theorem: Theorem (Furstenberg [F1]). Let X -./- µ- T 0 be a system let A 1M. be a set ith µ A0N6 0 and let < 1 an integer. Then lim in NO MPQ8 1 N 9 M NO 1 µ A 3 T n A 3 T 2n A 3KSSS3 T n A0T6 0 4 nr M The limin is actually a limit; see [HK]. (See also [Z2].) In particular there exist ininitely many integers n such that µ A3 T n A3 T 2n A 3USSS 3 T n A0V6 0. Furstenberg s Theorem can thus be considered as a ar reaching generalization o the Poincaré Recurrence Theorem hich corresponds to ; 1. Our interest is in the existence o a theorem that has the same relation to the Khintchine Recurrence Theorem as Furstenberg s Theorem has to the Poincaré Recurrence Theorem. More precisely under the same assumptions e as i the set J n 1BA : µ C A 3 T n A 3KSSS3 T n AEW6 µ A ε L (1.1) is syndetic or every positive integer and every ε 6 0. While it ollos rom Furstenberg s Theorem that or some constant c ; c A0X6 0 the set n 1MA : µ A 3 T n A 3BSSSY3 T n A0Z6 cg is syndetic e are asing i this can be strengthened to mae the set in (1.1) syndetic or every positive integer and c ; µ A ε or every ε 6 0. Under the hypothesis o ergodicity the anser is positive or ; 2 and ; 3 and surprisingly enough is negative or all < 4. Under the assumption o ergodicity e generalize the Khintchine Recurrence Theorem:

3 4 Vitaly Bergelson et al. Theorem 1.2. Let X -./- µ- T 0 be an ergodic system and let A 1K. be a set o positive measure. Then or every ε 6 0 the subsets J n 1BA : µ A 3 T n A 3 T 2n A0?6 µ A0 3 9 ε L (1.2) and J n 1BA : µ A 3 T n A 3 T 2n A 3 T 3n A0?6 µ A0 4 9 ε L (1.3) o A are syndetic. While ergodicity is not needed or Khintchine s Theorem it is essential in Theorem 1.2. Theorem 2.1 provides a counterexample in the general (nonergodic) case. For arithmetic progressions o length < 5 the result analogous to Theorem 1.2 does not hold. Using the result o Ruzsa contained in the Appendix in Section 2.3 e sho Theorem 1.3. There exists an ergodic system X -./- µ- T 0 such that or all integers [W< 1 there exists a set A ; A [)0?1B. ith µ A0?6 0 and or every integer n ; ` 0. µ C A 3 T n A 3 T 2n A 3 T 3n A 3 T 4n AE]\ µ A0 ^ _ 2 (1.4) In act e ind the slightly better upper bound µ A0 O clog µa Ab or some set A o positive arbitrarily small measure and some positive constant c Combinatorial results We recall a deinition: Deinition 1.4. The upper Banach density o a subset E o A is: d> E0c; lim NPQ8 sup 1 E 35g M- M : N Md e 9 1h 4 N Note that the limit exists by subadditivity and is the inimum o the sequence. Furstenberg used his Multiple Recurrence Theorem to mae the beautiul connection beteen ergodic theory and combinatorics and prove Szemerédi s Theorem: Theorem (Szemerédi [S]). A subset o integers ith positive upper Banach density contains arithmetic progressions o arbitrary inite length. Using a variation o Furstenberg s Correspondence Principle (Proposition 3.1) and Theorem 1.2 e immediately deduce:

4 Multiple recurrence and nilsequences 5 Corollary 1.5. Let E be a set o integers ith positive upper Banach density. Then or every ε 6 0 the sets and J n 1MA : d>ic E 3 J n 1BA : d> C E 3 are syndetic. E : n0d3 E : n0d3 E : 2n0D3 E : 2n0 E 6 d> E : 3n0EF6 d> E0 3 9 ε L E0 4 9 ε L Roughly speaing this means that given a set E ith positive upper Banach density or many values o n E contains many arithmetic progressions o length 3 (or o length 4) ith dierence n. (The dierence o the arithmetic progression a- a : n-444- a : ng is the integer n 6 0.) The olloing result ollos rom the proo o Theorem 1.3 and shos that the analogous result does not hold or longer progressions: Corollary 1.6. For every positive integer [ there exists a set o integers E ; E [ 0 ith positive upper Banach density such that d> C E 3 E : n0d3 E : 2n0D3 E : 3n0D3 E : 4n0 E \ d> or every nonzero integer n. E0 ^ _ Nilsequences We no explain the main ideas behind the ergodic theoretic results o Section 1.1. Fix an integer < 1. Given an ergodic system X -./- µ- T 0 and a set A 1U. o positive measure the ey ingredient or our ergodic results is the analysis o the sequence µ C A 3 T n A 3 T 2n A 3KSSSj3 T n AEF4 More generally or a real valued unction 1 L µ0 e consider the multicorrelation sequence I - n0 :;l x0ms T n x0ms 444S T n x0 dµ x0x4 (1.5) I When ; 1 Herglotz s Theorem states that the correlation sequence 1- n0 is the Fourier transorm o some positive inite measure σ ; σ on the torus n : I 1- n0 :;l S o T n dµ ;qpσ n0 :;r!s e 2πint dσ t0v4 By decomposing the measure σ into its continuous part σ c and its discrete part σ d e can rite the sequence I 1- n0 as the sum o to sequences I 1- n0z;ltσ c n0d: tσ d n0u4 The sequence tσ c n0g tends to 0 in uniorm density:

5 6 Vitaly Bergelson et al. Deinition 1.7. Let a n : n 1vA/G be a bounded sequence. We say that a n tends to zero in uniorm density and e rite UD-Lima n ; 0 i lim NPQ8 sup 1 M8 NO 1 Md)e M a n nr M ; 0 4 Equivalently UD-Lima n ; 0 i and only i or any ε 6 0 the set n 1 A : a n 6 ε G has upper Banach density zero (c. Bergelson [Ber] deinition 3.5). The sequence tσ d n0g is almost periodic and hence there exists a compact abelian group G a continuous real valued unction φ on G and a 1 G such that tσ d n0x; φ a n 0 or all n. The compact abelian group G is an inverse limit o compact abelian Lie groups. Thus any almost periodic sequence can be uniormly approximated by an almost periodic sequence arising rom a compact abelian Lie group. We ind a similar decomposition or the multicorrelation sequences I or < 2. The notion o an almost periodic sequence is replaced by that o a nilsequence hich e no deine: Deinition 1.8. Let < 1 be an integer and let X ; G_ Λ be a -step nilmaniold. Let φ be a continuous real (or complex) valued unction on G and let a 1 G and e 1 X. The sequence φ a n S e0g is called a basic -step nilsequence. A -step nilsequence is a uniorm limit o basic -step nilsequences. (For the precise deinition o a nilmaniold see section 4.1.) Note that a 1-step nilsequence is the same as an almost periodic sequence. Examples o 2-step nilsequences are given in Section 4.3. While an inverse limit o compact abelian Lie groups is a compact group an inverse limit o -step nilmaniolds is not in general the homogeneous space o some locally compact group (see Rudolph [R]). This explains hy the deinition o a nilsequence is not a direct generalization o the deinition o an almost periodic sequence. The general decomposition result is: Theorem 1.9. Let X -./- µ- T 0 be an ergodic system let 1 L µ0 and let < 1 be an integer. The sequence I - n0g is the sum o a sequence tending to zero in uniorm density and a -step nilsequence. We explain ho Theorems 1.9 and 1.2 are related. Deinition Let a n : n 1KA/G be a bounded sequence o real numbers. The syndetic supremum o this sequence is synd-supa n :; supy c 1Bz : n 1MA : a n 6 cg is syndetic {54 - n0

6 ~ ~ Multiple recurrence and nilsequences 7 In Section 4.3 e sho that the syndetic supremum o a nilsequence is equal to its supremum. We use the olloing simple lemma several times: Lemma Let a n G and b n G be to bounded sequences o real numbers. I UD-Lim a n 9 b n 0x; 0 then synd-supa n ; synd-supb n. Thereore the syndetic supremums o the sequences µ A3 T n A3 T 2n A0G and µ A3 T n A3 T 2n A3 T 3n A0G are equal to the supremums o the associated nilsequences and e are reduced to shoing that they are greater than or equal to µ A0 3 and µ A0 4 respectively. This is carried out in Section 8 completing the proo o Theorem Conventions and notation Given a system X -./- µ- T 0 in general e omit the σ-algebra rom our notation and rite X - µ- T 0. For a system X - µ- T 0 a actor is used ith to meanings: it is a T - invariant sub-σ-algebra or a system Y- ν- S0 and a measurable map π : X } Y such that πµ ; ν and S o π ; π o T. These to deinitions coincide under the identiication o the σ-algebra o Y ith the invariant sub-σalgebra π O 1 Z0 o.. In a slight abuse o vocabulary e say that Y is a actor o X. I is an integrable unction on X e denote the conditional expectation o on the Y 0 or the unction on Y deined by actor by ~ Z0c;~ Z0. We rite ~ Y 0Do π. This expectation is characterized by or all g 1 L ν0 - X S g o π dµ ; Y Y 0mS gdν 4 Throughout the article e implicitly assume that the term bounded unction means bounded and measurable Outline o the paper In Section 2 e construct to examples the irst shoing that ergodicity is a necessary assumption or Theorem 1.2 and the second a counterexample or progressions o length < 5 (Theorem 1.3). In Section 3 e use a variant o Furstenberg s Correspondence Principle to derive combinatorial consequences o the ergodic theoretic statements. The bul o the paper is devoted to describing the decomposition o multicorrelation sequences and proving Theorem 1.2. We start by revieing nilsystems and construction o certain actors in Section 4 and then in Section 5 explicitly describe the limit o an average along arithmetic progressions in a nilsystem. In Section 6 e introduce technical notions needed or the decomposition and in

7 8 Vitaly Bergelson et al. Section 7 e complete the proo o the decomposition. Section 8 combines these results and proves Theorem 1.2. Acnoledgments: We than I. Ruzsa or the combinatorial construction (contained in the Appendix) hich is the starting point or the construction o the counterexample o Theorem 1.3. We also than E. Lesigne or pointing us to the version o the Correspondence Principle e use and A. Leibman or useul explanations about nilsystems. 2. Combinatorial and ergodic counterexamples In this section c denotes a universal constant ith the understanding that its value may change rom one use to the next. Let m s denote the Haar measure on the torus nt; zx_a A counterexample or a nonergodic system In order to sho that ergodicity is necessary in Theorem 1.2 e use the olloing result o Behrend: Theorem (Behrend [Beh]). For every integer L 6 0 there exists a subset Λ o L 9 1G having more than Lexp 9 c logl0 elements that does not contain any nontrivial arithmetic progression o length 3. Theorem 2.1. There exists a (nonergodic) system X - µ- T 0 and or every integer [W< 1 there exists a subset A o X o positive measure so that or every nonzero integer n. µ C A 3 T n A 3 T 2n AE \ 1 2 µ A0 ^ 4 (2.1) We actually construct a set A o arbitrarily small positive measure ith µ C A3 T n A3 T 2n AE \ µ A0 O cloga µa Ab b or every integer n ; ` 0 and a positive universal constant c. Proo. Let X ;rn2ƒmn endoed ith its Haar measure µ ; m s ƒ m s and ith the transormation T given by T x- y0v; x- y : x0. Let Λ be a subset o L 9 1G not containing any nontrivial arithmetic progression o length 3. Deine j B ; jd Λ 2L - j 2L : 1 4L E - (2.2) hich e consider as a subset o the torus and set A ; nvƒ B.

8 \ 4 Multiple recurrence and nilsequences 9 For every integer n ; ` 0 e have T n x- y0ˆ; x- y : nx0 and µ C A 3 T n A 3 T 2n AE ; + sx Šs 1 B y0 1 B y : nx0 1 B y : 2nx0 dm s y0 dm s x0 ;I+ sx Šs 1 B y0 1 B y : x0 1 B y : 2x0 dm s y0 dm s x0u4 We no bound this last integral. Let x- y 1 n be such that the expression in this integral is nonzero. The three points y- y : x and y : 2x belong to B and e can rite y ; i 2L : a ; y : x ; j 2L : b ; y : 2x ; or integers i- j- belonging to Λ and a- b- c 1Œg 0-1_ 4L0 i 9 2 j : 2L ; 9 a : 2b 9 c L - 1 2L 0 2L : c mod 10. Then and thus i 9 2 j : ; 0. The integers i- j- orm an arithmetic progression in Λ and so the only possibility is that they are all equal giving that the three points y- y : x- y : 2x belong to the same subinterval o B. Thereore x 1 9 1_ 8L0-1_ 8L00 mod 10 and or every n ; ` 0 µ C A 3 T n A 3 T 2n AE ; + sc s 1 B y0 1 B y : x0 1 B y : 2x0 dm s y0 dm s x0 m s B0 4L We have m s B0ˆ; Λ _ 4L0. By Behrend s Theorem e can choose Λ o cardinality on the order o Lexp 9 c logl0. By taing L suiciently large an easy computation gives the bound (2.1). Ž 2.2. Quadratic conigurations Out next goal is to sho that the results o Theorem 1.2 do not hold or arithmetic progression o length < 5. We start ith a deinition designed to describe certain patterns that do not occur. Deinition 2.2. An integer polynomial is a polynomial taing integer values on the integers. When P is a nonconstant integer polynomial o degree \ 2 the subset J P 00 - P 10 - P 20 - P 30 - P 40 L o A is called a quadratic coniguration o 5 terms ritten QC5 or short.

9 10 Vitaly Bergelson et al. Note that any QC5 contains at least 3 distinct elements. An arithmetic progression o length 5 is a QC5 corresponding to a polynomial o degree 1. We mimic the construction in Theorem 2.1 or this setup: Lemma 2.3. Let Λ be a subset o L 9 1G not containing any QC5. For j 1 Λ let I j n be the interval I j ;qg j 4L - j 4L : 1 16L 0V4 and let B be the union o the intervals I j or j 1 Λ. Let x- y- z 15n be such that the ive points belong to B. Then 2y belongs a i ; x : iy : i 2 z mod 10 - i ; mod 10 to the interval C 4L - 1 4L E. Proo. We consider a a 4 as real numbers belonging to the interval g 0-10 ; by the deinition o B they actually all belong to g 0-1_ 40. For i ; let j i 1 E be the integer such that a i 1 I ji. The ive elements a a 4 o n satisy the relations a 3 ; a 0 9 3a 1 : 3a 2 mod 10 and a 4 ; a 1 9 3a 2 : 3a 3 mod 10 4 The real number a 0 9 3a 1 : 3a 2 belongs to the interval J ; j j 1 : 3 j 2 4L L - j j 1 : 3 j 2 : 4L 1 4L and this interval is contained in 9 3_ As a 3 ; a 0 9 3a 1 : 3a 2 mod 10 and a 3 1 g 0-1_ 40 the equality a 3 ; a 0 9 3a 1 : 3a 2 holds in z and thus a 3 1 J. Moreover a 3 1 I j3 and or every j ; ` j j 1 : 3 j 2 the interval I j has empty intersection ith the interval J. We get that j 3 ; j j 1 : 3 j 2. By the same computation e have that j 4 ; j j 2 : 3 j 3. From these to relations it ollos that there exists an integer polynomial Q ith j i ; Q i0 or i ; This polynomial must be constant or otherise j j 4 G ould be a QC5 in Λ. Thereore the ive points a i i ; belong to the same subinterval I j o B. Since 2y ; 9 3a 0 : 4a 1 9 a 2 mod 10 e have that 2y 1 O L 4L. Ž The next counterexample relies on a combinatorial construction communicated to us by Imre Ruzsa; his construction is reproduced in the Appendix. Theorem 2.4 (I. Ruzsa). For every integer L 6 0 there exists a subset Λ o L 9 1G having more than Lexp 9 c logl0 elements that does not contain any QC5.

10 Multiple recurrence and nilsequences Counterexample or longer progressions in ergodic theory. Proo (Proo o Theorem 1.3). We irst deine a particular system. Recall that n denotes the torus and m s its Haar measure. Deine X ; n ƒbn and µ ; m s ƒ m s. Let α 1 n be an irrational and let X be endoed ith the transormation T given by T x- y0z; x : α- y : 2x : α0n4 It is classical that this system is ergodic. This also ollos rom the discussion in Section 4.2. Let Λ be a subset o L 9 1G not containing any QC5 B the subset o n deined in Lemma 2.3 and A ;lntƒ B. For every integer n and every x- y0v1 X e have T n x- y0ˆ; x : nα- y : 2nx : n 2 α0v4 Thus or n ; ` 0 e have µ A 3 T n A 3 T 2n A 3 T 3n A 3 T 4n A0 ; + sx Šs 1 B y0 1 B y : 2nx : n 2 α0 1 B y : 4nx : 4n 2 α0 1 B y : 6nx : 9n 2 α0 1 B y : 8nx : 16n 2 α0 dm s x0 dm s y0 4 Let x- y 1Kn and n ; ` 0 be such that the expression in the last integral is not zero. By Lemma 2.3 4nx belongs mod 10 to the interval O L 4L. Since y 1 B e have µ A 3 T n A 3 T 2n A 3 T 3n A 3 T 4n m s B0 Λ A0 \ ; 2L 32L 4 2 By Ruzsa s Theorem e can choose Λ o cardinality on the order o Lexp 9 c logl0. By choosing L suiciently large an easy computation gives the bound (1.4). Ž 2.4. Counterexample or longer progressions or sets o integers ith positive density. Proo (Proo o Corollary 1.6). Let X - µ- T 0 and A be the system and the subset o X deined in Section 2.3. Fix some x 1 X and deine E ; m 1 A : T m x 1 AG. It is classical that the topological dynamical system X - T0 is uniquely ergodic; this also ollos rom the discussion in Section 4.2 and Theorem 4.1. Since the boundary o A has zero measure e have d> E0 ; µ A0. By the same argument d> C E 3 E : n0i3 E : 2n0i3 E : 3n0+3 E : 4n0 E ; µ A 3 T n A 3 T 2n A 3 T 3n A 3 T 4n A0. The same argument as in the proo o Theorem 1.3 gives the announced result. Ž

11 12 Vitaly Bergelson et al. 3. Translation beteen combinatorics and ergodic theory 3.1. Correspondence principle In order to obtain combinatorial corollaries o our ergodic theoretic results e need the olloing version o Furstenberg s Correspondence Principle: Proposition 3.1. Let E be a set o integers ith positive upper Banach den- sity. There exist an ergodic system X -./- µ- T 0 and a set A 1. ith µ A0; d> E0 such that µ T m 1 A 3BSSS3 T m A0 \ d> C E : m 1 0i3BSSS3 E : m 0 E or all integers < 1 and all m m 1BA. The observation that it suices to prove the ergodic theoretic results or ergodic systems as transmitted to us by Lesigne (personal communication); the proo e give is almost entirely contained in Furstenberg [F2]. Proo. We proceed as in the proo o Lemma 3.7 o [F2]. Let 0-1G e be endoed ith the product topology and the shit map T given by T x0 n ; x n8 1 or all n 1IA. Deine e 1 0-1G e by setting e n ; 1 i n 1 E and e n ; 0 otherise. Let X be the closure o the orbit o e under T meaning the closure o J T m e: m 1KA L. Set A ; x 1 X : x 0 ; 1G. It is a clopen (closed and open) subset o X. It ollos rom the deinition that or every integer n e have T n e 1 A i and only i n 1 E. By deinition o d> E0 there exist to sequences M i G and N i G o integers ith N i }š: such that 1 lim N i ip E 35g M i - M i : N i 9 1h } d> In the proo o Lemma 3.7 in [F2] it is shon that there exists an invariant probability measure ν on X such that ν A0 is equal to the above limit and thus is equal to d> E0. By using the ergodic decomposition o ν under T e have that there exists an ergodic invariant probability measure µ on X ith µ A0V< d> E0. Let m m be integers. The set T m 1 A 3MSSS3 T m A is a clopen set. Its indicator unction is continuous and by Proposition 3.9 o [F2] there exist to sequences K i G and L i G o integers ith L i } : such that µ T m 1 A 3BSSS3 T m K 1 i 8 L i O 1 A0c; lim i L i nr K i ; lim i 1 L i E : m 1 0D3BSSS3 E : m 0D3 E0V4 1 T m 1AœŠ œ T m A T n e0 K i K i : L i 9 1G \ d>dc E : m 1 0D3KSSS3 E : m 0E/4 By using this bound ith ; 1 and m 1 ; 0 e have that µ A0 \ d> E0 and thus µ A0x; d> E0. Ž Using the modiied correspondence principle and Theorem 1.2 e immediately deduce Corollary 1.5.

12 Multiple recurrence and nilsequences Combinatorial consequence or progressions o length 3 and 4 Szemerédi s Theorem can be ormulated in a inite version: Theorem (Szemerédi [S]). For every integer < 1 and every δ 6 0 there exists N - δ 0 such that or all N < N - δ 0 any subset E o NG ith at least δn elements contains an arithmetic progression o length. Similar to the inite version o Szemerédi s Theorem e can derive a inite version albeit a somehat eaer one o Corollary 1.5. We begin ith a remar. Write ž xÿ or the integer part o the real number x. From the inite version o Szemerédi s Theorem it is not diicult to deduce that every subset E o NG ith at least δn elements contains at least ž c - δ 0 N 2 Ÿ arithmetic progressions o length here c - δ 0 is some constant. Thereore the set E contains at least ž c - δ 0 NŸ progressions o length ith the same dierence. In vie o Corollary 1.5 it is natural to as the olloing question: Question. Is it true that or every δ 6 0 and every ε 6 0 there exists N ε - δ 0 such that or every N 6 N ε - δ 0 every subset E o NG ith E < δn elements contains at least 1 9 ε0 δ 3 N arithmetic progressions o length 3 ith the same dierence and at least 1 9 ε0 δ 4 N arithmetic progressions o length 4 ith the same dierence? 1 We are not able anser this question but can prove a eaer result ith a relatively intricate ormulation: Corollary 3.2. For all real numbers δ 6 0 and ε 6 0 and every integer K 6 0 there exists an integer M δ - ε- K0 6 0 such that or all N 6 M δ - ε - K NG ith E < δn there exist: and every subset E a subinterval J o NG ith length K and an integer s 6 0 such that E 3 E 9 s0d3 E 9 2s0D3 J < 1 9 ε0 δ 3 K 4 (3.1) a subinterval J o NG ith length K and an integer s "6 0 such that E 3 E 9 s 0D3 E 9 2s 0D3 E 9 3s 0D3 J < 1 9 ε0 δ 4 K 4 (3.2) Statement (3.1) means that E 3 J contains at least 1 9 ε0 δ 3 K starting points o arithmetic progressions o length 3 included in E all ith the same dierence. Statement (3.2) has the analogous meaning or progressions o length 4. 1 Recently Ben Green gave a positive anser to this question or progressions o length 3 (preprint available at math.co/ ).

13 14 Vitaly Bergelson et al. Proo. We only prove the result or progressions o length 3 as the proo or progressions o length 4 is identical. Assume that the result does not hold. Then there exist δ 6 0 ε 6 0 K 6 0 a sequence N i G tending to : and or every i a subset E i o N i G ith E i < δn i such that or every subinterval J o length K o N i G relation (3.1) (ith E i substituted or E) is alse. We can assume that N i 6 K or every i. By induction e build a se- quence M i G o positive integers ith M i M i : 2N i and M i8 1 6 M i : N i : N i8 1 or every i. Deine the set E to be the union o the sets M i : E i. We have d> E0c; limsup i E i _ N i < δ. Fix an integer s 6 0. By construction or every integer M there exist i and an interval J o length K included in g M i : 1- M i : N i h such that E 35g M- M : K 9 1h E 3 J. We deduce that sup M E 3 E 9 s0d3 E 9 2s0D35g M- M : K 9 1h sup E 3 ; sup i J M i 8 1 M i 8 N i J R K E 9 s0+3 E 9 2s0+3 J 4 (3.3) By construction i or some integer m 1 E e have m : s 1 E m : 2s 1 E and m 1Œg M i : 1- M i : N i h then the integers m : s and m : 2s also belong to the same interval. Thereore or J g M i : 1- M i : N i h E 3 E 9 s0d3 E 9 2s0+3 J ; C E i 3 E i 9 s0d3 E i 9 2s0+3 J 9 M i 0 E : M i 4 Putting this into Equation (3.3) e have that sup M E 3 E 9 s0d3 ; sup i by deinition o the sets E i. E 9 2s0D35g M- M : K 9 1h sup E i 3 E i 9 s0d3 E i 9 2s0D3 I \ I 1 ª N i«i R K We deduce that or every s 6 0 e have d> C E 3 E 9 s ε0 δ 3 and Corollary 1.5 provides a contradiction. Ž 1 9 ε0 δ 3 K E 9 2s0 E \ The anser to the similar question or longer arithmetic progressions is negative: there exist signiicant subsets o NG that contain very e arithmetic progressions o length < 5 ith the same dierence. Proposition 3.3. For all integers [ 6 0 there exists δ 6 0 such that or ininitely many values o N there exists a subset E o NG ith E < δn that contains no more than 1 2δ^ N arithmetic progressions o length 5 ith the same dierence.

14 Multiple recurrence and nilsequences 15 We give only the main ideas o the proo as it lies a bit ar rom the main ocus o this paper. Let L and B be as in the proo o Theorem 1.3 (Sections 2.2 and 2.3). Let α be an irrational hich is ell approximated by a sequence p j _ q j G o rationals ith q j prime and deine: F ; n 1M : n 2 α 1 B mod 10G 4 Let N be one o the primes q j and E ; F NG. Then E _ N is close to m B0. Let s be a positive integer. Let the integer n be such that the arithmetic progression n- n: s n: 4sG is included in E. By Lemma 2.3 2nsα belongs modulo 1 to the interval O 1-1 4L 4L 0. Note that n and s are smaller than N ; q j. By approximating α by p j _ q j and using the primality o q j e see that the number o possible values o n or a given s is bounded by cn_ L or some positive constant c. Thereore E contains eer than cn_ L progressions o length 5 ith the same dierence. For L suiciently large e get the announced bound. Once again the bound e actually ind is δ O cloga δb N or some constant c 6 0. Ž 4. Preliminaries 4.1. Nilsystems We revie some deinitions and properties needed in the sequel. The notation introduced here is used throughout the rest o this paper. Let G be a group. For g- h 1 G e rite g g- hh"; g O 1 h O 1 gh. When A and B are to subsets o G e rite g A- Bh or the subgroup o G spanned by g a- bh : a 1 A- b 1 BG. The loer central series G ; G 1 G 2 SSS G j G j o G is deined by G 1 ; G and G j8 1 ; g G- G j h or j < 1 4 Let < 1 be an integer. We say that G is -step nilpotent i G 8 1 ; Let G be a -step nilpotent Lie group and let Λ be a discrete cocompact subgroup. The compact maniold X ; G_ Λ is called a -step nilmaniold. The group G acts naturally on X be let translation and e rite g- x0v±} g S x or this action. There exists a unique Borel probability measure µ on X invariant under this action called the Haar measure o X. The undamental properties o nilmaniolds ere established by Malcev [M]. We mae use o the olloing property several times hich appears in [M] or connected groups and is proved in Leibman [Lei2] in a similar ay or the general case: For every integer j < 1 the subgroups G j and ΛG j are closed in G. It ollos that the group Λ j ; Λ 3 G j is cocompact in G j. 1G.

15 16 Vitaly Bergelson et al. Let t be a ixed element o G and let T : X } X be the transormation x ±} t S x. Then X - T0 is called a -step topological nilsystem and X - µ- T 0 is called a -step nilsystem. Fundamental properties o nilsystems ere established by Auslander Green and Hahn [AGH] and by Parry [Pa1]. Further ergodic properties ere proven by Parry [Pa2] and Lesigne [Les] hen the group G is connected and generalized by Leibman [Lei2]. We summarize various properties o nilsystems that e need: Theorem 4.1. Let X ; G_ Λ - µ- T 0 be a -step nilsystem ith T the translation by the element t 1 G. Then: 1. X - T0 is uniquely ergodic i and only i X - µ- T 0 is ergodic i and only i X - T 0 is minimal i and only i X - T 0 is transitive (that is i there exists a point x hose orbit T n x: n 1BA/G is dense). 2. Let Y be the closed orbit o some point x 1 X. Then Y can be given the structure o a nilmaniold Y ; H_ Γ here H is a closed subgroup o G containing t and Γ is a closed cocompact subgroup o H. 3. For any continuous unction on X and any sequences o integers M i G and N i G ith N i } : the averages converge or all x 1 X. Assume urthermore that 1 N i M i 8 N i O 1 nr M i T n x0 (H) G is spanned by the connected component o the identity and the element t. Then: 4. The groups G j j < 2 are connected. 5. The nilsystem X - µ- T 0 is ergodic i and only i the rotation induced by t on the compact abelian group G_ G 2 Λ is ergodic. 6. I the nilsystem X - µ- T 0 is ergodic then its Kronecer actor is Z ; G_ G 2 Λ ith the rotation induced by t and ith the natural actor map X ; G_ Λ } G_ G 2 Λ ; Z. For connected groups parts 1 2 and 3 o this theorem can be deduced rom [AGH] [F3] and [Pa1] hile parts 4 and 6 are proved in [Pa1]. When G is connected and simply connected and more generally hen G can be imbedded as a closed subgroup o a connected simply connected -step nilpotent Lie group all parts o this theorem ere proved in [Les]. In the case that the group is simply connected the result ollos rom Lesigne s proo. The general case or parts and 6 ollo rom [Lei2]. The proo o part 4 as transmitted to us by Leibman (personal communication) and e outline it here.

16 Multiple recurrence and nilsequences 17 Proo. (Part 4) Assume that property (H) holds and let Ga 0b be the connected component o the identity 1 o G. The second commutator G 2 is spanned by the commutators o the generators o G. Since g t- thm; 1 thereore e have G 2 ;²g Ga 0b - Gh. For h 1 G the map g ±}³g g- hh is continuous rom Ga 0b to G 2 and maps 1 to 1. Thus its range is included in the connected component o 1 in G 2. We get that G 2 is connected. We proceed by induction or the commutator subgroups o higher order. Assume that the n-th commutator subgroup G n is connected. Proceeding as above using the connectedness o G n e have that or g 1 G n and h 1 G g g- hh belongs to the connected component o 1 in G n8 1. Thus G n8 1 is connected. Ž We also use (in Section 5.2) a generalization o part 5 o Theorem 4.1 or to commuting translations on a nilmaniold but it is just as simple to state it or [ commutating translations: Theorem 4.2 (Leibman [Lei2] Theorem 2.17). Let X ; G_ Λ be a -step nilmaniold endoed ith its Haar measure µ. Let t t^ be commuting elements o G and let T T^ be the associated translations on X. Assume that (H ) G is spanned by the connected component o the identity and the elements t t^. Then the action o A ^ on X spanned by T T^ the induced action on G_ G 2 Λ is ergodic. is ergodic i and only i We return to the case o a single transormation. Let X - µ- T 0 be an ergodic -step nilsystem. There are several ays to represent X as a nilmaniold G_ Λ. For our purposes e reduce to a particular choice o the representation. Assume that X ; G_ Λ and let t 1 G be the element deining T. The connected component Ga 0b o the identity in G projects to an open subset o X. By ergodicity the subgroup Ga 0b - tµ o G spanned by Ga 0b and t projects onto X. Substituting this group or G and Λ 3 Ga 0b - tµ or Λ e have reduced to the case that hypothesis (H) is satisied. Let Λ be the largest normal subgroup o G included in Λ. Substituting G_ Λ or G and Λ_ Λ or Λ hypothesis (H) remains valid and e have reduced to the case that (L) Λ does not contain any nontrivial normal subgroup o G To examples We start by revieing the simplest examples o 2-step nilsystems.

17 18 Vitaly Bergelson et al Let G ; A ƒ nvƒ n ith multiplication given by - x- y0 - x - y 0c; : - x : x - y : y : 2x 0 4 Then G is a Lie group. Its commutator subgroup is 0GWƒ 0GWƒ n and G is 2-step nilpotent. The subgroup Λ ;la ƒ 0GQƒ 0G is discrete and cocompact. Let X denote the nilmaniold G_ Λ and e maintain the notation o the preceding Section ith one small modiication. Here Z ;n m ; m s is the Haar measure o n and the actor map π : X } Z is given by - x- y0z±} x; it is thus more natural to use additive notation or Z. Let α be an irrational point in n a ; 1- α- α0 and T : X } X the translation by a. Then X - µ- T 0 is a 2-step nilsystem. Note that hypotheses (H) and (L) are satisied. Since α is irrational the rotation Z- m- T 0 is ergodic and X - µ- T 0 is ergodic by part 6 o Theorem 4.1. We give an alternate description o this system. The map - x- y0 ±} x- y0 rom G to n 2 induces a homeomorphism o X onto n 2. Identiying X ith n 2 via this homeomorphism the measure µ becomes equal to m s ƒ m s and the transormation T o X is given or x- y0v1mn 2 ; X by T x- y0z; x : α- y : 2x : α0n4 This is exactly the system used in the construction o the counterexample in Section (2.3) We also revie another standard example o a 2-step ergodic nilsystem. Let G be the Heisenberg group z ƒuz ƒuz ith multiplication given by x- y- z0" x - y - z 0c; x : x - y : y - z : z : xy 0 4 (4.1) Then G is a 2-step nilpotent Lie group. The subgroup Λ ;²A ƒia ƒia is discrete and cocompact. Let X ; t ; G_ Λ and let T be the translation by and t 3 1Uz. We have that t 1 - t 2 - t 3 0u1 G ith t 1 - t 2 independent over ¹ G_ Λ - T 0 is a nilsystem. Hypothesis (H) is obviously satisied since G is connected. Here the compact abelian group G_ G 2 Λ is isomorphic to n 2 and the rotation on n 2 by t 1 - t 2 0 is ergodic. Thereore the system G_ Λ - T 0 is uniquely ergodic. Note that hypothesis (L) is not satisied by G and Λ. The reduction explained above consists here in taing the quotient o G and Λ by the subgroup Λ ; 0Gºƒ 0G/ƒBA. We get that X is the quotient o G_ Λ by Λ_ Λ here G_ Λ ;qz ƒuz»ƒ5n ith multiplication given by (4.1) and Λ_ Λ i; A ƒ A ƒ 0G Nilsequences For clarity e repeat some o the deinitions given in the introduction. Let X ; G_ Λ be a -step nilmaniold φ be a continuous unction on X e 1 X

18 Multiple recurrence and nilsequences 19 and t 1 G. The sequence a n G given by a n ; φ t n S e0 is called a -step basic nilsequence. We say that a bounded sequence is a -step nilsequence i it is a uniorm limit o -step basic nilsequences. Let X e t and φ be as above and let Y be the closed orbit o e. By part 2 o Theorem 4.1 Y- T 0 can be given the structure o a nilsystem. Since this system is transitive it is minimal by part 1 o the same theorem. Let S ; sup nd e a n and ε 6 0. The set U ; x 1 Y : φ x0x6 S 9 εg is a nonempty open subset o Y. By minimality o Y- T 0 the set n 1MA : t n S e 1 U G is syndetic and thus synd-supa n < S 9 ε. Thereore or every basic nilsequence a n G e have synd-supa n ; supa n 4 This property passes to uniorm limits and is thereore valid or every nilsequence. The Cartesian product o to -step nilsystems is again a -step nilsystem and so the amily o basic -step nilsequences is a subalgebra o [. Thereore the amily o -step nilsequences is a closed subalgebra o [. This algebra is clearly invariant under translation and invariant under complex conjugation. We give to examples o 2-step nilsequences arising rom the to examples o 2-step nilsystems given above Let X - T 0 be the nilsystem deined in Section We identiy X ith niƒ¼n. Let e ; For every integer n e have T n e ; nα- n 2 α0. Let and [ be to integers and let φ be the unction on X given by φ x- y0½; exp 2πi x :Œ[ y00. The sequence is a 2-step nilsequence. nd e J exp 2πi n :Œ[ n 2 0 α0 L Let X - T 0 denote the system deined in Section We use the irst representation o this system. We irst deine a continuous unction on X. Let be a continuous unction on z tending suiciently ast to 0 at ininity. For x- y- z0u1 z 3 deine ψ x- y- z0 :; exp 2πiz0 d e exp 2πix0 y : 0X4 Then ψ is a continuous unction on z 3 and an immediate computation gives that or all x- y- z0u1mz 3 and or all p- q- r0 1BA 3 ψ C x- y- z0" p- q- r0ef; ψ x- y- z0x4 Thereore the unction ψ on G ; z 3 induces a continuous unction φ on the quotient X o G by Λ ;»A 3. Let e be the image o in X. For

19 ~ 20 Vitaly Bergelson et al. every integer n e have φ t n S e0; ψ t n 0 and t n ; Thereore the sequence a n G given by a n ; expc 2πint 3 0 exp 2πi n n is a 2-step nilsequence. t 1 t 2 0 d e nt 1 - nt 2 - nt 3 : na no 1b 2 t 1 t 2 0. exp 2πint 1 0 nt 2 : Construction o certain actors In this Section X - µ- T 0 is an ergodic system. We revie the construction o some actors in Host and Kra [HK]. These are the actors that control the limiting behavior o the multiple ergodic averages associated to the expressions I - n0. We begin recalling some ell non acts about the Kronecer actor The Kronecer actor and the ergodic decomposition o µ ƒ µ Let Z X0 - m- T 0 denote the Kronecer actor o X - µ- T 0 and let π : X } Z X0 be the actor map. When there is no ambiguity e rite Z instead o Z X0. We recall that Z is a compact abelian group endoed ith a Borel σ-algebra ¾ and Haar measure m. The transormation T : Z } Z has the orm z ±} αz or some ixed element α o Z. For s 1 Z e deine a measure µ s on X ƒ X by x0 x 0 dµ s x- x 0c; Z0 z0vs ~ Z0 sz0 dµ z0u4 (4.2) X X ~ Z For every s 1 Z the measure µ s is invariant under T ƒ T and is ergodic or m-almost every s. The ergodic decomposition o µ ƒ µ under T ƒ T is µ ƒ µ ; µ s dm s0 4 Z The actors Z. We recall some constructions o Sections 3 and 4 in [HK]. For an integer < 0 e rite X ; X 2 and T : X } X or the map T ƒ T ƒ5444ƒ T taen 2 times. We deine a probability measure µ on X invariant under T by induction. Set µ 0 ; µ. For < 0 let be the σ-algebra o T -invariant subsets o X. Then µ 8 1 is the relatively independent square o µ over. This means that i F - F are bounded unctions on X XÀ Á 1Â F x 0 F x 0 dµ 8 1 x - x 0 :; F XÀ Â 0 ~ F 0 dµ - (4.3)

20 Ê Multiple recurrence and nilsequences 21 here x ; x - x 0 is an element o X 8 1 considered under the natural identiication o X 8 1 ith X ƒ X. For a bounded unction on X e can deine :;Äà XÀ  2 O 1 jr 0 x j 0 dµ x0jå 1Æ 2 (4.4) because this last integral is nonnegative. It is shon in [HK] that or every integer < 1 S is a seminorm on L µ0. The seminorms deine actors o.. Namely the sub-σ-algebra ¾ O 1 X0 o. is characterized by or 1 L µ0 -~ ¾ O 1 X00c; 0 i and only i ; 0 4 (4.5) X0. This gives that The actor Z X0 is the actor o X associated to ¾ X0 is the trivial actor Z 1 X0 is the Kronecer actor. When there is no ambiguity e rite Z and ¾ instead o Z X0 and ¾ X0. Z The actors associated to the measures µ s For each s 1 Z such that X ƒ X - µ s - T ƒ T 0 is ergodic or each integer < 1 a measure µ s 0 on X ƒ X0 can be deined in the ay that µ as deined rom µ. Furthermore a seminorm ÇxS!Ç s on L µ s 0 can be associated to this measure in the same ay that the seminorm LjS Ç is associated to µ. In Section 3 o [HK] it is shon that under the natural identiication o X ƒ X0 ith X 8 1 e have µ 8 1 ; Z µ s 0 dm s0 4 It ollos rom deinition (4.4) that or every 1 L µ0 2 8 Á 1 1 ; From this e immediately deduce: Z È 2 s dm s0u4 Proposition 4.3. Let < 2 be an integer and let be a bounded unction on X. I has zero conditional expectation on ¾ then or m-almost every s 1 Z the unction È considered as a unction on X ƒ X - µ s 0 has zero conditional expectation on ¾ O 1 X ƒ X - µ s - T ƒ T Inverse limits o nilsystems. We say that the system X - T 0 is an inverse limit o a sequence o actors + X j - T 0G i. jg jd)é is an increasing sequence o sub-σ-algebras invariant under the transormation T such that jd)é. j ;Ë. up to null sets. I each system X j - T 0 is isomorphic to a -step nilsystem then e say that X - T 0 is an inverse limit o -step nilsystems. Theorem 10.1 o [HK] states that or every ergodic system X - µ- T 0 and every integer < 1 the system Z X0 is an inverse limit o -step nilsystems.

21 22 Vitaly Bergelson et al Arithmetic progressions We continue assuming that X - µ- T 0 is an ergodic system. From Theorem 12.1 o [HK] and the characterization (4.5) o the actor Z O 1 e have: Theorem 4.4. Let < 2 be an integer and bounded unctions on X. I at least one o these unctions has zero conditional expectation on ¾ O 1 then or all sequences M i G and N i G o integers ith N i } : lim ip 1 N i M i 8 N i O 1 0 nr M i x0 1 T n x0 2 T 2n x0+444 T n x0 dµ x0z; 0 4 Corollary 4.5. Let < 2 be an integer and let be bounded unctions on X. I at least one o these unctions has zero conditional expectation on ¾ then 0 x0 1 T n x0 2 T 2n x0+444 T n x0 dµ x0 converges to zero in uniorm density. Proo. Let M i G and N i G be to sequences o integers ith N i }Ì:. For m-almost every s 1 Z one o the unctions 0 È 0... È has zero conditional expectation on ¾ X ƒ X - µ s - T ƒ T 0 by Proposition 4.3 and thus the averages on M i M i : N i 9 1G o 0 x0 0 x 0 1 T n x0 1 T n x T n x0 T n x 0 dµ s x- x 0 converge to zero by Theorem 4.4 applied to the system X ƒ X - µ s - T ƒ T 0 and to the unctions 0 È È. Integrating ith respect to s e get 1 N i M i 8 N i O 1 0 nr M i and the result ollos. Ž x0 1 T n x0+444 T n x0 dµ x0 2 } 0 Recall that or a bounded measurable unction on X and an integer < 1 e deined I - n0x; x0 T n x0+444 T n x0 dµ x0x4 Even more generally one can consider the same expression ith : 1 distinct bounded unctions Hoever this gives no added inormation or the problems e are studying and so e restrict to the above case. Corollary 4.6. Let < 1 be an integer let be a bounded unction on X and let g ; ~ - n0 converges to zero in uniorm density. ¾ 0. Then I - n0 9 I g

22 E Multiple recurrence and nilsequences 23 Let and g be as in this Corollary. We consider g as a unction deined on Z. Note that the unctions and g have the same integral. Since the system Z is an inverse limit o a sequence o -step nilsystems the unction g can be approximated arbitrarily ell in L 1 -norm by its conditional expectation on one o these nilsystems. We use this remar in the proo o Theorem 1.2 in Section The limit o the averages In this section < 1 is an integer X ; G_ Λ - µ- T 0 is an ergodic -step nilsystem and the transormation T is translation by the element t 1 G. We eep the notation o Section 4.1 and assume that hypotheses (H) and (L) are satisied. Recall that e let G j denote the j-th commutator o G and that Λ j ; Λ 3 G j. We have that G ; G 1 but sometimes it is convenient to use both notations in the same ormula. For 1 L µ0 e irst study the averages o the sequence I - n0. This establishes a short proo o a recent result by Ziegler [Z1]. We use some algebraic constructions based on ideas o Petresco [Pe] and Leibman [Lei1]. We explain the idea behind this construction. It is natural to deine an arithmetic progression o length : 1 in G as an element o G 8 1 o the orm g- hg- h 2 g h g0 or some g- h 1 G. Unortunately these elements do not orm a subgroup o G 8 1. Hoever such elements do span the subgroup G (deined in the next section) hich could thus be called the group o arithmetic progressions o length : 1 in G. Similarly one is tempted to deine an arithmetic progression o length : 1 in X as a point in X 8 1 o the orm x- h S x- h 2 S x h S x0 or some x 1 X and h 1 G. Once again it is more ruitul to tae a broader deinition calling an arithmetic progression in X a point rom the set X (again deined belo) hich is the image o the group G under the natural projection on X Some algebraic constructions Deine the map j : G ƒ G 1 ƒ G 2 ƒ5sss!ƒ G } j g- g 1 - g g 0x; and let G denote the range o the map j: G ; G 8 1 by C g- gg 1 - gg 2 1 g gg 10 1 g j G ƒ G 1 ƒ G 2 ƒusssƒ G 0V SSS g 0 Similarly e deine a map j> : G 1 ƒ G 2 ƒusssƒ G } G by j> g 1 - g g 0c; C g 1 - g 2 1 g g 10 1 g 20 2 SSS g 0 E 4

23 24 Vitaly Bergelson et al. Finally e deine G> ; j> ;ÍJ G 1 ƒ G 2 ƒ5sssƒ G 0 h 1 - h h 0N1 G : 1- h 1 - h h 0?1 GL 4 The olloing results are ound in Leibman [Lei1]: Theorem G is a subgroup o G The commutator group G0 2 o G is It ollos rom part 1 that G0 2 ; G 3 G ; j G 2 ƒ G 2 ƒ G 2 ƒ G 3 ƒ5sss!ƒ G G> is a subgroup o G. Moreover or g 1 G h 1 - h h 0?1 G 1 ƒ G 2 ƒ5sss!ƒ G and g 1 - g g 0x; j> e have g O 1 g 1 g- g O 1 g 2 g-444- g O 1 g g0x; j> It ollos that h 1 - h h 0 - g O 1 h 1 g- g O 1 h 2 g g O 1 h g0 (5.1) 4. For g 1 - g g 0?1 G> and g 1 G e have g O 1 g 1 g- g O 1 g 2 g-444- g O 1 g g0n1 G> 4 The maps j and j> are injective continuous and proper (the inverse image o a compact set is compact). It ollos that G and G> are closed subgroups o G 8 1 and G respectively and thus are Lie groups. We also deine the to elements t ; o G and the element 1- t- t t 0 and t ; t> ; t- t t 0 t- t t0 o G>. Write T and T or the translations by t and t on X 8 1 respectively and T> or the translation by t> on X.

24 Multiple recurrence and nilsequences The nilmaniold X. Deine Λ ; G 3 Λ 8 1. Then Λ is a discrete subgroup o G and it is easy to chec that Λ ; j Λ ƒ Λ 1 ƒ Λ 2 ƒ5sss!ƒ Λ 0. Thereore Λ is cocompact in G. We rite X ; G_ Λ and let µ denote the Haar measure o this nilmaniold. Note that X is imbedded in X 8 1 in a natural ay. Since t and t belong to G this nilmaniold is invariant under the transormations T and T. Lemma 5.2. The nilmaniold X is ergodic (and thus uniquely ergodic) or the action spanned by T and T. Proo. Since the groups G j j 6 1 are connected and G satisies condition (H) it ollos that the group G is spanned by t t and the connected component o the identity. By Theorem 4.2 it suices to sho that the action induced by T and T on G_ G0 2 Λ is ergodic. By part 2 o Theorem 5.1 the map g 1 - g g 0c±} g 1 mod G 2 - g 2 mod G 2 0 induces an isomorphism rom G_ G0 2 onto G_ G 2 ƒ G_ G 2. Thus the compact abelian group G_ G0 2 Λ can be identiied ith G_ G 2 Λ ƒ G_ G 2 Λ and the transormations induced by T and T are Id ƒ T and T ƒ T respectively. The action spanned by these transormations is obviously ergodic. Ž 5.3. The nilmaniolds X x. For x 1 X e deine X x ;ÍJ x 1 - x x 0N1 X : x- x 1 - x x 0N1 X L 4 Clearly or every x 1 X the compact set X x is invariant under translations by elements o G>. We give to each o these sets the structure o a nilmaniold quotient o this group. Fix x 1 X and let a be a lit o x in G. The point x- x-444- x0 ( times) clearly belongs to X x. Let x 1 - x x 0X1 X x. The point x- x 1 - x x 0 belongs to X and e can lit it to an element o G that e can rite g- gg 1 - gg gg 0 ith g 1 G and g 1 - g g 0½1 G>. Writing λ ; a O 1 g and h i ; aλg i λ O 1 a O 1 or 1 \ i \ e have λ 1 Λ 1- h 1 - h h 0 belongs to G by Remar 4 above and g- gg 1 - gg gg 0c; 1- h 1 - h h 0"S a- a- a a0ms λ - λ - λ λ 0 4 This gives that x 1 - x x 0 is the image o x- x-444 x0 under translation by h 1 - h h 0 hich belongs to G>.

25 26 Vitaly Bergelson et al. Thereore the action o G> on X x is transitive. The stabilizer o x- x x0 or this action is the group Λ x :;ÍJ aλ 1 a O 1 - aλ 2 a O aλ a O 1 0 : λ 1 - λ λ 0?1 Λ 3 G> L 4 Λ 3 G> is a discrete subgroup o G> and it is easy to chec that it is equal to j> Λ 1 ƒ Λ 2 ƒ5sssƒ Λ 0 and thus is cocompact in G>. It ollos that Λ x is a discrete and cocompact subgroup o G>. We can thus identiy X x ith the nilmaniold G> _ Λ x. Let µ x denote the Haar measure o X x. Lemma 5.3. µ ; δ x È µ x dµ x0. X Proo. Let µ be the measure deined by this integral. This measure is concentrated on X. By Lemma 5.2 it suices to sho that it is invariant under T and T. Recall that T > is the translation by t> ; t- t t 0 hich belongs to G> and thus this transormation preserves X x and µ x or every x. Thereore or every x 1 X the measure δ x È µ x is invariant under T ; Id ƒ T> and so µ is invariant under this transormation. Let x 1 X. Consider the image o µ x under T ƒisss ƒ T ( times). This measure is concentrated on X Tx and by remar 4 in Section 5.1 it is easy to chec that it is invariant under G>. Thus it is equal to the Haar measure µ T x. Thereore the image o δ x È µ x under T is δ T x È µ Tx. It ollos that µ is invariant under T. Ž 5.4. The limit o the averages. Given this bacground e give a short proo o Ziegler s result [Z1]: Theorem 5.4 (Ziegler [Z1]). Let be continuous unctions on X and let M i G and N i G be to sequences o integers such that N i }Î:. For µ-almost every x 1 X 1 N i as i } M i 8 N i O 1 1 T n x0 2 T 2n x0+444 T n x0 nr M i } 1 x x x 0 d µ x x 1 - x x 0 (5.2) Proo. For x 1 X the point x- x x0 belongs to the nilmaniold X x. By part 3 o Theorem 4.1 applied to the nilsystem X x - T> 0 and this point the averages in Equation (5.2) converge everyhere to some unction φ. Thereore e are let ith computing this unction.

26 Ï Multiple recurrence and nilsequences 27 Let be a continuous unction on X. We have x0 φ x0 dµ x0 ; lim ip 1 N i ; lim ip 1 N 2 i M i 8 N i O 1 nr M i M i 8 N i O 1 mr M i x0 M i 8 N i O 1 nr M i j T jn x0 dµ x0 jr 1 T m x0 j T jn8 m x0 dµ x074 jr 1 The point x- x-444- x0 belongs to X and or all n and m its image under T n T 0 m is the point T m x- T n8 m x-444- T n8 m x0. By Lemma 5.2 X is uniquely ergodic or the action spanned by T and T and the average in the last integral converges everyhere to Using Lemma 5.3 e have X x x x 0 d µ x 0 - x x 0 4 X x0 φ x0 dµ x0n; ; and the result ollos. Ž X x x x 0 d µ x 0 - x x 0 X x0i x x 0 d µ x x x 0 X x 1 dµ x0 Corollary 5.5. For µ-almost every x 1 X the nilsystem X x - µ x - T > 0 is ergodic. Proo. Recall that T > preserves µ x or every x. Let Ï o continuous unctions on X that is dense in Ð be a countable amily X0 in the uniorm norm. By Theorem 5.4 there exists a subset X 0 o X ith µ X 0 0c; 1 such that 1 N NO 1 nr 0 jr 1 j T jn x0z}ñ j x j 0 d µ x x 1 - x x 0 jr 1 as N }š: or every x 1 X 0 and or all unctions Ï. Since is dense the same result holds or arbitrary continuous unctions. It ollos that or x 1 X 0 the orbit o x- x-444- x0 under T > is dense in the support o the measure µ x. Since the support o this measure is X x e have that the action o T > on X x is transitive. By Theorem 4.1 X x - µ x - T > 0 is ergodic. Ž