NIL-BOHR SETS OF INTEGERS

Transcription

1 NIL-BOHR SETS OF INTEGERS BERNARD HOST AND BRYNA KRA Abstract. W study rlations btwn substs of intgrs that ar larg, whr larg can b intrprtd in trms of siz (such as a st of positiv uppr dnsity or a st with boundd gaps) or in trms of additiv structur (such as a Bohr st). Bohr sts ar fundamntally ablian in natur and ar linkd to Fourir analysis. Rcntly it has bcom apparnt that a highr ordr, nonablian, Fourir analysis plays a rol both in additiv combinatorics and in rgodic thory. Hr w introduc a highr ordr vrsion of Bohr sts and giv various proprtis of ths objcts, gnralizing rsults of Brglson, Furstnbrg, and Wiss. 1. Introduction 1.1. Additiv combinatorics and Bohr sts. Additiv combinatorics is th study of structurd substs of intgrs, concrnd with qustions such as what on can say about sts of intgrs that ar larg in trms of siz or about sts that ar larg in trms of additiv structur. An intrsting problm is finding various rlations btwn classs of larg sts. Sts with positiv uppr Banach dnsity or syndtic sts 1 ar xampls of sts that ar larg in trms of siz. A simpl rsult rlats ths two notions: if A Z has positiv uppr Banach dnsity, thn th st of diffrncs (A) = A A = {a b: a, b A} is syndtic. An xampl of a structurd st is a Bohr st. Following a modification of th traditional dfinition introducd in [2], w say that a subst A Z is a Bohr st if thr xist m N, α T m, and an opn st Th first author was partially supportd by th Institut Univrsitair d Franc and th scond by NSF grant and by th Clay Mathmatics Institut. This work was bgun during th visit of th authors to MSRI and compltd whil th scond author was a visitor at Institut Hnri Poincaré; w thank th instituts for thir hospitality. 1 If A Z, th uppr Banach dnsity d (A) is dfind to b A [a n, b n ] limsup. b n a n b n a n A st A Z is said to b syndtic if it intrscts vry sufficintly larg intrval. 1

2 2 BERNARD HOST AND BRYNA KRA U T m such that {n Z: nα U} is containd in A (s Dfinition 2.1). It is asy to chck that th class of Bohr sts is closd undr translations. Most of th notions of a larg st that ar dfind solly in trms of siz ar also closd undr translation. Howvr, w hav important classs of structurd sts that ar not closd undr translation. On particular xampl is that of a Bohr 0 -st ([2], [4]): a subst A Z is a Bohr 0 -st if it is a Bohr st such that th st U in th prvious dfinition contains 0. A simpl application of th pigonhol principl givs that if S is an infinit st of intgrs, thn S S has nontrivial intrsction with vry Bohr 0 -st. This is anothr xampl of largnss: a st is larg if it has nontrivial intrsction with vry mmbr of som class of sts. Such notions of largnss ar gnrally rfrrd to as dual notions and ar dnotd with a star. For xampl, a -st is a st that has nontrivial intrsction with th st of diffrncs (A) from any infinit st A. Hr w study convrs rsults. If a st intrscts vry st of a givn class, thn our goal is to show that it has som sort of structur. Such thorms ar not, in gnral, xact convrss of th dirct structural statmnts. For xampl, thr xist -sts that ar not Bohr 0 -sts (s [2]). But, this statmnt is not far from bing tru. Strngthning a rsult of [2], w show (Thorm 2.8) that a -st is a picwis Bohr 0 -st, maning that it agrs with a Bohr 0 -st on a squnc of intrvals whos lngths tnd to infinity Nil-Bohr sts. Bohr sts ar fundamntally linkd to ablian groups and Fourir analysis. In th past fw yars, it has bcom apparnt in both rgodic thory and additiv combinatorics that nilpotnt groups and a highr ordr Fourir analysis play a rol (s, for xampl, [6], [8], and [7]). As such, w dfin a d-stp nil-bohr 0 -st, analogous to th dfinition of a Bohr 0 -st, but with a nilmanifold rplacing th rol of an ablian group (s Dfinition 2.3). For d = 1, th ablian cas, this is xactly th objct studid in [2]. Hr w gnraliz thir rsults for d 1. W obtain a gnralization of Thorm 2.8 on diffrnt sts, introducing th ida of a st of sums with gaps. For an intgr d 1 and an infinit squnc P = (p i : i 1) in N, th st of sums with gaps of lngth < d of P is dfind to b th st SG d (P ) of all intgrs of th form ɛ 1 p 1 + ɛ 2 p ɛ n p n,

3 NIL-BOHR SETS OF INTEGERS 3 whr n 1 is an intgr, ɛ i {0, 1} for 1 i n, th ɛ i ar not all qual to 0, and th blocks of conscutiv 0 s btwn two 1 s hav lngth < d. W rmark that P is considrd as a squnc, and not a st of intgrs. W do not assum that th p i ar distinct, nor do w assum that th squnc (p i : i 1) is incrasing. Our main rsult (Thorm 2.6) is that a st with nontrivial intrsction with any SG d -st is a picwis d-stp nilpotnt Bohr 0 -st. Acknowldgmnt. Hilll Furstnbrg introducd us to this problm and w thank him for his ncouragmnt, as wll as for hlpful commnts on a prliminary vrsion of this articl. 2. Prcis statmnts of dfinitions and rsults 2.1. Bohr sts and Nil-Bohr sts. W formally dfin th objcts dscribd in th introduction: Dfinition 2.1. A subst A Z is a Bohr st if thr xist m N, α T m, and an opn st U T m such that {n Z: nα U} is containd in A; th st A is a Bohr 0 -st if additionally 0 U. Not that ths sts can also b dfind in trms of th topology inducd on Z by mbdding th intgrs into th Bohr compactification: a subst of Z is Bohr if it contains a nonmpty opn st in th inducd topology and is Bohr 0 if it contains an opn nighborhood of 0 in th inducd topology. W can gnraliz th dfinition of a Bohr 0 -st for rturn tims in a nilsystm, rathr than just in a torus. W first giv a short dfinition of a nilsystm and rfr to Sction 3.2 for furthr proprtis. Dfinition 2.2. If G is a d-stp nilpotnt Li group and Γ G is a discrt and cocompact subgroup, th compact manifold X = G/Γ is a d-stp nilmanifold. Th Haar masur µ of X is th uniqu probability masur that is invariant undr th action x g x of G on X by lft translations. If T dnots lft translation on X by a fixd lmnt of G, thn (X, µ, T ) is a d-stp nilsystm. Using nighborhoods of a point, w dfin a gnralization of a Bohr st:

4 4 BERNARD HOST AND BRYNA KRA Dfinition 2.3. A subst A Z is a Nil d Bohr 0 -st if thr xist a d-stp nilsystm (X, µ, T ), x 0 X, and an opn st U X containing x 0 such that {n Z: T n x 0 U} is containd in A. Similar to th Bohr compactification of Z that can b usd to dfin th Bohr sts, thr is a d-stp nilpotnt compactification of Z that can b usd to dfin th Nil d Bohr 0 -sts. This compactification is a non-mtric compact spac Ẑ, ndowd with a homomorphism T and a particular point x 0 with dns orbit, and is charactrizd by th following proprtis: i) Givn any d-stp nilsystm (Z, T ) and a point x 0 Z, thr is a uniqu factor map π Z : Ẑ Z with π Z( x 0 ) = x 0. ii) Th topology of Ẑ is spannd by ths factor maps π Z. Rmark. A Bohr 0 -st can b dfind in trms of almost priodic squncs. In th sam way, a Nil d Bohr 0 -st can b dfind in trms of som particular squncs, th d-stp nilsquncs. Sinc Nil d Bohr 0 - sts ar dfind locally, it sms likly that thy can b dfind by crtain particular typs of nilsquncs, namly thos arising from gnralizd polynomials without constant trms. W do not addrss this issu hr Picwis vrsions. If F dnots a class of substs of intgrs, various authors, for xampl Furstnbrg in [5] and Brglson, Furstnbrg, and Wiss in [2], dfin a subst A of intgrs to b a picwis-f st if A contains th intrsction of a squnc of arbitrarily long intrvals and a mmbr of F. For xampl, th notions of picwis-bohr st, a picwis-bohr 0 -st, and a picwis-nil d Bohr 0 -st, can b dfind in this way. Howvr, th notion of a picwis st is rathr wak: for xampl, a picwis-bohr st dfind in this mannr is not ncssarily syndtic. Th proprtis that w can prov ar strongr than th traditional picwis statmnts, and in particular imply th traditional picwis vrsions. For this, w introduc a strongr dfinition of picwis: Dfinition 2.4. Givn a class F of substs of intgrs, th st A Z is said to b strongly picwis-f, writtn PW- F, if for vry squnc (J k : k 1) of intrvals whos lngths J k tnd to, thr xists a squnc (I j : j 1) of intrvals satisfying: i) For ach j 1, thr xists som k = k(j) such that th intrval I j is containd in J k ;

5 NIL-BOHR SETS OF INTEGERS 5 ii) Th lngths I j tnd to infinity; iii) Thr xists a st Λ F such that Λ I j A for vry j 1. Not that Λ dpnds on th squnc (J k : k 1). With this dfinition of strongly picwis, if th class F consists of syndtic sts thn vry PW- F-st is syndtic. In particular, a strongly picwis-bohr st, dnotd PW- Bohr, is syndtic. Similarly, w dnot a strongly picwis-bohr 0 - st by PW- Bohr 0 and a strongly picwis-nil d Bohr 0 - st by PW- Nil d Bohr 0 and ths sts ar also syndtic Sumsts and Diffrnc Sts. Dfinition 2.5. Lt E N b a st of intgrs. Th sumst of E is th st S(E) consisting of all nontrivial finit sums of distinct lmnts of E. A subst A of N is a S r-st if A S(E) for vry st E N with E = r. W hav: Thorm 2.6. Evry S d+1 -st is a PW- Nil d Bohr 0 -st. For clarity, w includ som xampls of ths objcts. Exampl (An S 2-st). Lt r (1/3, 1/2) b ral, α T := R/Z b irrational, and A = {n N: nα ( r, r) mod 1}. Thn w claim that A is a S 2-st. Any S 2 -st is a st of th form {m, n, m + n} for som distinct, positiv intgrs m and n. If m / A and m + n / A, thn nα mod 1 ( T \ ( r, r) ) ( T \ ( r, r) ) = [2r 1, 1 2r] ( r, r) and so n A. Exampl (A Nil 2 Bohr 0 st which is not an S 2-st). Lt r (0, 1/2) b ral, α T := R/Z b irrational, and B = {n N: n 2 α ( r, r) mod 1}. Thn B is a Nil 2 Bohr 0 -st, as can b chckd by considring th transformation on T 2 dfind by (x, y) (x + α, y + x). On th othr hand, by th Wyl Equidistribution Thorm, th st { (m 2 α, n 2 α, (m + n) 2 α): 1 m < n } is dns in T 3 and so thr xist distinct m, n N such that m 2 α / ( r, r), n 2 α / ( r, r), and (m + n) 2 α / ( r, r). Thrfor B is not an S 2-st.

6 6 BERNARD HOST AND BRYNA KRA Exampl (An S 3-st which is not an S 2-st). W claim that for r (3/7, 1/2), th st B dfind abov is an S 3-st. Indd, if th intgrs m, n, p, m+n, m+p, and n+p do not blong to B, thn (m+n) 2 α n 2 α, (n + p) 2 α p 2 α, and (m + p) 2 α m 2 α blong to [2r 1, 1 2r]. Using th idntity (m + n + p) 2 = ( (m + n) 2 n 2) + ( (n + p) 2 p 2) + ( (m + p) 2 m 2), w hav that (m + n + p) 2 α [3(2r 1), 3(1 2r)] ( r, r) and m + n + p B. W can itrat Thorm 2.6, lading to th following dfinitions from [2] and [5]: Dfinition 2.7. If S is a nonmpty subst of N, dfin th diffrnc st (S) by (S) = (S S) N = {b a: a S, b S, b > a}. If A is a subst of N, A is a r-st if A (S) for vry subst S of N with S = r; A is a -st if A (S) for vry infinit subst S of N. Thorm 2.8. Evry -st is a PW- Bohr 0 -st. Evry r st is obviously a -st and Thorm 2.8 gnralizs Thorm II of [2], whr it is shown that vry r st is a PW- Bohr 0 -st. Th class of sts of th form (S) with S = 3 coincids with th class of sts of th form S(E) with E = 2 and thus th classs 3 and S 2 ar th sam. Thorm 2.8 gnralizs th cas d = 1 of Thorm 2.6. Th convrs statmnt of Thorm 2.8 dos not hold. Howvr, it is asy to chck that vry Bohr 0 -st is a -st (s [2]). Dfinition 2.9. Lt d 0 b an intgr and lt P = (p i ) b a (finit or infinit) squnc in N. Th st of sums with gaps of lngth < d of P is th st SG d (P ) of all intgrs of th form ɛ 1 p 1 + ɛ 2 p ɛ n p n, whr n 1 is an intgr, ɛ i {0, 1} for 1 i n, th ɛ i ar not all qual to 0, and th blocks of conscutiv 0 s btwn two 1 s hav lngth < d. A subst A N is an SG d-st if A SG d (P ) for vry infinit squnc P in N. Not that in this dfinition, P is a squnc and not a subst of N. For xampl, if P = {p 1, p 2,... }, thn SG 1 (P ) is th st of all sums p m + + p n of conscutiv lmnts of P, and thus it coincids with

7 NIL-BOHR SETS OF INTEGERS 7 th st (S) whr S = {s, s + p 1, s + p 1 + p 2,... }. Thrfor SG 1-sts ar th sam as -sts. For a squnc P, SG 2 (P ) consists of all sums of th form m 1 i=m 0 p i + m 2 i=m 1 +2 p i + + whr k N and m 0, m 1,..., m k+1 m i+1 m i + 2 for i = 0,..., k. m k i=m k 1 +2 p i + m k+1 i=m k +2 p i, ar positiv intgrs satisfying Thorm Evry SG d-st is a PW- Nil d Bohr 0 -st. Sinc SG 1-sts ar th sam as -sts, Thorm 2.10 gnralizs Thorm 2.8. If P = d + 1, thn SG d (P ) = S(P ) and thus Thorm 2.10 gnralizs Thorm 2.6. In gnral, a Nil d Bohr 0 -st is not a -st. To construct an xampl, tak an irrational α and lt B b th st of n Z such that n 2 α is clos to 0 mod 1. Thn B is a Nil 2 Bohr 0 -st. On th othr hand, by induction w can build an incrasing squnc n j of intgrs such that n 2 jα is clos to 1/3 mod 1, whil n i n j α mod 1 is clos to 0 for i < j. Taking S to b th st of such n j, w hav that (S) dos not intrsct B. This lads to th following qustion: Qustion Is vry Nil d Bohr 0 -st an SG d-st? As alrady notd, th answr to this qustion is positiv for d = 1 and w conjctur that it is positiv in gnral. As our charactrizations of th sts SG d and th class SG d ar complicatd, w ask th following: Qustion Find an altrnat dscription of th sts SG d and of th class SG d Th mthod. Th first ingrdint in th proof is a modification and xtnsion of th Furstnbrg Corrspondnc Principl. Th classical Corrspondnc Principl givs a rlation btwn sts of intgrs and masur prsrving systms, rlating th siz of th sts of intgrs to th masur of som sts of th systm. It dos not giv rlations btwn structurs in th st of intgrs undr considration and rgodic proprtis of th corrsponding systm. Som information of this typ is providd by our modification (originally introducd in [9]). W thn ar lft with studying crtain proprtis of th systms that aris from this corrspondnc. As in svral rlatd problms, th proprtis of th systm that w nd ar linkd to crtain factors

8 8 BERNARD HOST AND BRYNA KRA of th systm, which ar nilsystms. This mthod and ths factors wr introducd in th study of convrgnc of som multipl rgodic avrags in [8]. Working within ths factor systms, w conclud by making us of tchniqus for th analysis of nilsystms that hav bn dvlopd ovr th last fw yars. In th ablian stting, a fundamntal tool is th Fourir transform, but no analog xists for highr ordr nilsystms 2. Anothr classical tool availabl in th ablian cas is th convolution product, but this too is not dfind for gnral nilsystms. Instad, in Sction 5 w build som spacs and masurs that tak on th rol of th convolution. As an xampl, if G is a compact ablian group w can considr th subgroup {(g 1, g 2, g 3, g 4 ) G 4 : g 1 + g 2 = g 3 + g 4 } of G 4, and w tak intgrals with rspct to its Haar masur. This rplacs th rol of th convolution product. Ths constructions ar thn usd to prov th ky convrgnc rsult (Proposition 6.4). By studying th limit, Thorm 2.6 is dducd in Sction 7. By furthr itrations, Thorms 2.8 and 2.10 ar provd in Sction 8. Th stratgy usd in th proofs of our main rsults (Thorm 2.6 and 2.10) rquirs th us of substantial tchnical machinry. Although Thorm 2.8 is a particular cas of Thorm 2.10, to hlp th radr to undrstand th main idas of th papr w includ a short proof of this rsult in Sction 4. As wll, w us this to point out th diffrncs btwn this simplr stting and th gnral contxt. This proof is almost slf-containd, as it only rlis on th modifid Corrspondnc Principl of Sction Prliminaris 3.1. Notation. W introduc notation that w us throughout th rmaindr of th articl. If X is a st and d 1 is an intgr, w writ X [d] = X 2d and w indx th 2 d copis of X by {0, 1} d. Elmnts of X [d] ar writtn as x = (x ɛ : ɛ {0, 1} d ). W writ lmnts of {0, 1} d without commas or parnthss. W also oftn idntify {0, 1} d with th family P([d]) of substs of [d] = {1, 2,..., d}. In this idntification, ɛ i = 1 is th sam as i ɛ and = Th thory of rprsntations dos not hlp us, as th intrsting rprsntations of a nilpotnt Li group ar infinit dimnsional.

9 NIL-BOHR SETS OF INTEGERS 9 For ɛ {0, 1} d and n Z d, w writ ɛ = ɛ ɛ d and ɛ n = ɛ 1 n ɛ d n d. If p: X Y is a map, thn w writ p [d] : X [d] Y [d] for th map (p, p,..., p) takn 2 d tims. In particular, if T is a transformation on th spac X, w dfin T [d] : X [d] X [d] as T T... T takn 2 d tims and w call T [d] th diagonal transformation. W dfin th fac transformations T [d] i for 1 i d by: (T [d] i x) ɛ = { T (x ɛ ) if ɛ i = 1 x ɛ othrwis. Thus for d = 2, th diagonal transformation is T T T T and th fac transformations ar Id T Id T and Id Id T T. In a slight abus of notation, w dnot all transformations, vn in diffrnt systms, by th lttr T (unlss th systm is naturally a Cartsian product). For convninc, w assum that all functions ar ral valud Rviw of nilsystms. Dfinition 3.1. If G is a d-stp nilpotnt Li group and Γ G is a discrt and cocompact subgroup, th compact manifold X = G/Γ is a d-stp nilmanifold. Th Haar masur µ of X is th uniqu probability masur that is invariant undr th action x g x of G on X by lft translations. If T dnots lft translation on X by a fixd lmnt of G, thn (X, µ, T ) is a d-stp nilsystm. (W gnrally omit th σ-algbra from th notation, writing (X, µ, T ) for a masur prsrving systm rathr than (X, B, µ, T ), whr B dnots th Borl σ-algbra.) A d-stp nilsystm is an xampl of a topological distal dynamical systm. For a d-stp nilsystm, th following proprtis ar quivalnt: transitivity, minimality, uniqu rgodicity, and rgodicity. (Not that th first thr of ths proprtis rfr to th topological systm, whil th last rfrs to th masur prsrving systm.) Also, th closd orbit of a point in a d-stp nilsystm, ndowd with th rstriction of th original transformation, is a d-stp nilsystm, and it follows that this closd orbit is minimal and uniquly rgodic. S [1] for proofs and gnral rfrncs on nilsystms. W also spak of a nilsystm (X = G/Γ, T 1,..., T d ), whr T 1,..., T d ar translations by commuting lmnts of G. All th abov proprtis hold for such systms. In particular, vry closd orbit is uniquly rgodic undr th inducd transformations.

10 10 BERNARD HOST AND BRYNA KRA W also mak us of invrs limits of systms, both in th topological and masur thortic snss. All invrs limits ar implicitly assumd to b takn along squncs. Invrs limits for a squnc of rgodic nilsystms ar th sam in both th topological and masur thortic snss: this follows bcaus a masur thortic factor map btwn two rgodic nilsystms is ncssarily continuous (s for xampl Appndix A of [10]). Many proprtis of th nilsystms also pass to th invrs limit. In particular, in a topological invrs limit of d-stp nilsystms, vry closd orbit is minimal and uniquly rgodic Structur Thorm. Assum now that (X, µ, T ) is an rgodic systm. W rcall a construction and dfinitions from [8], but for consistncy w mak som small changs in th notation. For an intgr d 0, a masur µ [d] on X [d] was built in [8]. Hr w dnot this masur by µ (d). Th masur µ (d) is invariant undr T [d] and undr all th fac transformations T [d] i, 1 i d. Each of th projctions of th masur µ (d) on X is qual to th masur µ. If f is a boundd masurabl function on X, thn f(x ɛ ) dµ (d) (x) 0 ɛ [d] and w dfin f d to b this xprssion raisd to th powr 1/2 d. Thn d is a sminorm on L (µ). A main rsult from [8] is that this is a norm if and only if th systm is an invrs limit of (d 1)-stp nilsystms. Mor prcisly, a summary of th Structur Thorm of [8] is: Thorm 3.2. Assum that (X, µ, T ) is an rgodic systm. Thn for ach d 2, thr xist a systm (Z d, µ d, T ) and a factor map π d : X Z d satisfying: i) (Z d, µ d, T ) is th invrs limit of a squnc of (d 1)-stp nilsystms. ii) For ach f L (µ), f E(f Z d ) π d d = 0. For ach d 1, w call (Z d, µ d, T ) th HK-factor of ordr d of (X, µ, T ). Th factor map π d : X Z d is masurabl, and a priori has no rason to b continuous. For l d, Z l is a factor of Z d, with a continuous factor map.

11 NIL-BOHR SETS OF INTEGERS Th cas of an invrs limit of nilsystms. If (X, µ, T ) is an invrs limit of (d 1)-stp rgodic nilsystms, w dfin 3 X (d) to b th closd orbit in X [d] of a point x 0 = (x 0,..., x 0 ) (for som arbitrary x 0 X) undr th transformations T [d] and T [d] i for 1 i d. Whn (X, µ, T ) is an rgodic (d 1)-stp nilsystm, thn (X (d), T [d], T [d] 1,..., T [d] d ) is an rgodic (d 1) nilsystm and th masur µ(d) dscribd abov is its Haar masur ([8], Sction 11). Whn (X, µ, T ) is an invrs limit of (d 1)-stp rgodic nilsystms, thn systm (X (d), T [d], T [d] 1,..., T [d] d ) is an invrs limit of rgodic nilsystms. It is minimal, uniquly rgodic and its uniqu invariant masur is th masur µ (d) Furstnbrg corrspondnc principl rvisitd. By l (Z), w man th algbra of boundd ral valud squncs indxd by Z. Lt A b a subalgbra of l (Z), containing th constants, invariant undr th shift, closd and sparabl with rspct to th norm of uniform convrgnc. W rfr to this simply as an algbra. In applications, finitly many substs of Z ar givn and A is th shift invariant algbra spannd by indicator functions of ths substs. Givn an algbra, w associat various objcts to it: a dynamical systm, an rgodic masur on this systm, a squnc of intrvals, tc. W giv a summary of ths objcts without proof, rfrring to [9] for furthr dtails A systm associatd to A. By Glfand s rprsntation, thr xist a topological dynamical systm (X, T ) and a point x 0 X such that th map φ C(X) ( φ(t n x 0 ): n Z ) l (Z) is an isomtric isomorphism of algbras from C(X) onto A. (W us C(X) to dnot th collction of continuous functions on X.) In particular, if S is a subst of Z with 1 S A, thn thr xists a subst S of X that is opn and closd in X such that (1) for vry n Z, T n x 0 S if and only if n S Som avrags and som masurs associatd to A. Thr also xist a squnc I = (I j : j 1) of intrvals of Z, whos lngths tnd 3 W ar forcd to us diffrnt notation from that in [8], as othrwis th prolifration of indics would b uncontrollabl.

12 12 BERNARD HOST AND BRYNA KRA to infinity, and an invariant rgodic probability masur µ on X such that 1 (2) for vry φ C(X), φ(t n x 0 ) φ dµ as j +. I j n I j Givn a subst S of Z, w can chos th intrvals I j such that S I j I j d (S) as j +, whr d (S) dnots th uppr Banach dnsity of S. In particular, w can assum that th intrvals I j ar containd in N Notation. In th squl, whn a = (a n : n Z) is a boundd squnc, w writ 1 lim Av n,i a n = lim a n j + I j n I j if this limit xists, and st limsup Avn,I a n = limsup j + 1 I j n I j a n W omit th subscripts n and/or I if thy ar clar from th contxt Avrags and factors of ordr k. Rcall that Z k dnots th HKfactor of ordr k of (X, µ, T ) and that π k : X Z k dnots th factor map. Th squnc of intrvals I = (I j : j 1) can b chosn such that: Proposition 3.3. For vry k 1, thr xists a point k Z k such that π l,k ( k ) = l for l < k and such that for vry φ C(X) and vry f C(Z k ), lim Av I φ(t n x 0 )f(t n k ) = φ f π k dµ = E(φ Z k ) f dµ k. This formula xtnds (2). Th nxt corollary is an xampl of th rlation btwn intgrals on th factors Z k and PW- Nil Bohr-sts. Mor prcis rsults ar provd and usd in th squl. Corollary 3.4. Lt S b a subst of Z such that 1 S blongs to th algbra A and lt S b th corrsponding subst of X. Lt f b a.

13 NIL-BOHR SETS OF INTEGERS 13 nonngativ continuous function on Z k with f( k ) > 0, whr k is as in Proposition 3.3. If 1 S (x) f π k (x) dµ(x) = 0 thn Z \ S is a PW- Nil k Bohr 0 -st. Proof. Lt Λ = {n Z: f(t n k ) > f( k )/2}. Thn Λ is a Nil k Bohr 0 - st. Indd, th function f can b approximatd uniformly by a function of th form f p, whr f is a continuous function on a k-stp nilsystm Z and p: Z k Z is a factor map. By Proposition 3.3 and dfinition (1) of S, th avrags on I j of 1 S (n)f(t n k ) convrg to zro. Thus I j S Λ lim = 0. j + I j Thrfor, th subst E = j I j \ (S Λ) contains arbitrarily long intrvals J l, l 1. For vry l, J l (Z \ S) J l Λ It is asy to chck that givn a squnc of intrvals (J k : k 1) whos lngths tnd to infinity, w can choos th intrvals (I j : j 1) satisfying all of th abov proprtis, and such that ach intrval I j is a subintrval of som J k. To s this, w first rduc to th cas that th intrvals J k ar disjoint and sparatd by sufficintly larg gaps. W st S to b th union of ths intrvals. W hav d (S) = 1 and w can choos intrvals I j with S I j / I j 1. For vry j N, thr xists k j such that I j J kj / I j 1 as j +. W st I j = I j J kj and th squnc (I j : j 1) satisfis all th rqustd proprtis Dfinition of th uniformity sminorms. W rcall dfinitions and rsults of [9] adaptd to th prsnt contxt. W kp notation as in th prvious sctions; in particular, Z k and k ar as in Proposition 3.3. Lt I b as in Sction 3.4 and lt B b th algbra spannd by A and squncs of th form (f(t n k ): n Z), whr f is a continuous function on Z k for som k. By Proposition 3.3, for vry squnc a = (a n : n Z) blonging to th algbra B, th limit lim Av I,n a n xists. Givn a squnc a B, for h = (h 1,..., h d ) Z d, lt c h = lim Av I,n a n+ɛ h. ɛ [d]

14 14 BERNARD HOST AND BRYNA KRA Thn lim H 1 H d H 1 h 1,...,h d =0 xists and is nonngativ. W dfin a I,d to b this limit raisd to th powr 1/2 d. Proposition 3.5. Lt (Z, T ) b an invrs limit of k-stp nilsystms and f b a continuous function on Z. Thn for vry δ > 0 thr xists C = C(δ) > 0 such that for vry squnc a = (a n : n Z) blonging to th algbra B and for vry z Z, limsup Av I a n f(t n z) δ a + C a I,k+1. Proof. By dnsity, w can rduc to th cas that (Z, T ) is a k-stp nilsystm and that th function f is smooth. In this cas, th rsult is containd in [9] undr th hypothsis that th systm is rgodic. Indd, by Proposition 5.6 of this papr, f is a dual function on X. By th Modifid Dirct Thorm of Sction 5.4 in [9], thr xists a constant f k 0 with c h limsup AvI a n f(t n z) f k a I,k+1. This givs th announcd inquality with C = f k In th proofs of [9] w can chck that th hypothsis of rgodicity is not usd. Th nxt proposition was provd in Sction 3 of [9] and follows from th Structur Thorm (Thorm 3.2). Proposition 3.6. Lt φ b a continuous function on X with φ 1, k 1 an intgr, and f a continuous function on Z k with f 1. Thn ( φ(t n x 0 ) f(t n k ): n Z ) 2 I,k+1 f E(φ Zk ) 1/2k+1. L 1 (µ k ) 4. Th cas d = Th contxt. To hlp th radr undrstand th main idas, and as a warm up, w start with a proof of Thorm 2.8. Throughout, w includ commnts on th diffrncs btwn th cas d = 1 and th gnral sttings of Thorms 2.6 and Th proof dos not mak us of th machinry dvlopd in th rst of th papr othr than th modifid corrspondnc principl of Sction 3.4. Lt A b a subst of Z and assum that A is not a PW- Bohr 0 -st. W show that A is not a -st. Sinc th familis of -sts and of SG 1-sts ar th sam, it suffics to show that A is not a SG 1-st.

15 NIL-BOHR SETS OF INTEGERS A systm associatd to th st A. W rcall th construction of Sction 3.4. Lt A b an algbra, in th sns of Sction 3.4, containing 1 A. Lt (X, T ), x 0 X, I, and µ b associatd to this algbra as in Sction 3.4 and assum that th intrvals I j ar includd in N. Lt f b th continuous function on X associatd to th squnc 1 1 A : f(t n x 0 ) = 1 if n / A ; f(t n x 0 ) = 0 if n A. Sinc A dos not contain arbitrarily long intrvals, th dnsity of Z \ A in th intrvals I j dos not tnd to zro. Thus f dµ > 0. Lt (Z, ν, T ) b th Kronckr factor of (X, µ, T ) and lt π : X Z b th factor map. W rcall that Z is a compact ablian group and that ν is its Haar masur. W us additiv notation for Z and th transformation T : Z Z is givn by T z = z + α for som α Z. For vry boundd masurabl function φ on X, writ φ = E(φ Z). Th lmnt 1 Z 1 = Z in Proposition 3.3 can b chosn to b th unit lmnt 0 of Z, and this proposition stats that: For vry continuous function φ on X and vry continuous function h on Z, (3) lim Av I φ(t n x 0 )h(nα) = φ h π dµ = φ h dν Two positivity rsults. W prov a positivity rsult (this is a rformulation of Lmma 7.1 in th prsnt contxt): Claim 4.1. Lt h b a boundd, nonngativ masurabl function on th Kronckr (Z, ν) with h dν > 0. Thn (4) f(s)h(t)h(s + t) dν(s) dν(t) > 0. Proof of Claim 4.1. For s Z, dfin H(s) = h(s + t)h(t) dν(t). Thn H(0) > 0 and H is a continuous function on Z. Th subst Λ of Z dfind by is a Bohr 0 -st. Λ = {n Z: H(nα) > H(0)/2}

16 16 BERNARD HOST AND BRYNA KRA By dfinition of th functions f and H, w hav that f(s)h(t)h(s + t) dν(s) dν(t) = f(s)h(s) dν(s) = lim Av I f(t n x 0 )H(nα) H(0) 2 limsup j + (Z \ A) Λ I j whr th middl quality follows from (3). This limsup is not qual to zro, as othrwis th st A (Z \ Λ) would contain arbitrarily long intrvals J i, maning that A would contain Λ J i and would b a PW- Bohr-st. In th gnral cas, th functions f and h ar dfind on an invrs limit of nilsystms. Sinc convolution products ar not dfind in this contxt, th corrsponding rsult is mor difficult to stat. Th intgral in (4) is rplacd by an intgral with rspct to th Haar masur of som submanifold of a Cartsian powr of Z k dfind in Sction 5.2. Th intgral dfining H(s) is rplacd by th intgral with rspct to th Haar masur of som othr submanifold, dpnding on s, dfind in Sction 5.3. In th gnral cas, th positivity of H(0) is shown in Proposition 5.3 and th continuity of th function H in Proposition 5.2. Claim 4.2. Lt h b a continuous nonngativ function on X with h dµ > 0. Thr xists an intgr n, blonging to som intrval Ii, I j, with h(t n x 0 ) > 0 and T n h f dµ > 0. Proof of Claim 4.2. Sinc h dν = h dµ > 0, by Claim 4.1, ( ) (5) h(t) f(s) h(s + t) dν(s) dν(t) > 0. Th function dfind by th innr intgral in this formula is continuous on Z and thus using (3) as abov, w hav that lim Av n,i h(t n x 0 ) f(s) h(s + nα) dν(s) > 0. Sinc f and h ar th conditional xpctations of th functions f and h, rspctivly, on th Kronckr factor Z of X, w hav that (6) lim Av n,i f(s) h(s + nα) dν(s) f(x)h(t n x) dµ(x) = 0

17 NIL-BOHR SETS OF INTEGERS 17 Thus lim Av n,i h(t n x 0 ) f(x)h(t n x) dµ(x) > 0 and th xistnc of th intgr n with th announcd proprtis follows. Th convrgnc rsult (3) usd in this cas dos not suffic for th gnral cas and is rplacd by th dpr Proposition 6.4. Th proof of this proposition occupis most of Sction 6 and uss th uniformity sminorms introducd in [9] and whos proprtis ar rcalld in Sction 3.5. Proposition 6.1 gnralizs (6) End of th proof. By induction, using Claim 4.2 at ach stp, w dfin a squnc of positiv intgrs (n j : j 1), such that th functions h (j) on X, dfind inductivly by h (0) = f and h (j) = T n j h (j 1) f for j 1 satisfy h j 1 (T n j x 0 ) > 0 and T n j h (j 1) f dµ > 0 for vry j 1. By dscnding induction on i with j fixd, for 1 i j w hav that h (i 1) (T n i+n i+1 + +n j x 0 ) > 0. Thus f(t n i+n i+1 + +n j x 0 ) > 0 and so n i + n i n j / A. Stting E = (n j : j 1) w hav that A SG 1 (E) = and A is not an SG 1-st. Th proof of Thorm 2.10 uss a similar, but mor intricat, induction. 5. Som masurs associatd to invrs limits of nilsystms 5.1. Standing assumptions. W assum that vry topological systm (Z, T ) is implicitly ndowd with a particular point, calld th bas point. Evry topological factor map is implicitly assumd to map bas point to bas point. For vry k 1, w tak th bas point of Z k to b th point k introducd in Sction If (Z, T ) is a nilsystm with Z = G/Γ, thn by changing th group Γ if ndd, w can assum that th bas point of Z is th imag in Z of th unit lmnt of G.

18 18 BERNARD HOST AND BRYNA KRA 5.2. Th masurs µ (m). Proposition 5.1. Lt (X, µ, T ) b an rgodic invrs limit of rgodic k-stp nilsystms, ndowd with th bas point X, and lt m 1 b an intgr. a) Th closd orbit of th point [m] = (,,..., ) of X (m) undr th transformations T [m] i, 1 i m, is =ɛ [d] X (m) = {x X (m) : x = }. b) Lt µ (m) b th uniqu masur on this st invariant undr ths transformations. Thn th imag of µ (m) undr ach of th natural projctions x x ɛ : X [m] X, ɛ [d], is qual to µ. c) Lt (Y, ν, T ) b an invrs limit of k-stp nilsystms and lt p: X Y b a factor map. Thn ν (m) is th imag of µ (m) undr p [m] : X [m] Y [m]. d) Lt (Y, ν, T ) b th HK-factor of ordr (m 1) of X and p: X Y b th factor map. Thn th masur µ (m) is rlativly indpndnt with rspct to ν (m), maning that whn f ɛ, ɛ [d], ar 2 m 1 boundd masurabl functions on X, f ɛ (x ɛ ) dµ (m) (x) = E(f ɛ Y )(y ɛ ) dν (m) (y). =ɛ [d] (Th xistnc of ths intgrals follows from b).) Th uniqunss of th masur µ (m) in b) follows from th fact that X (m) is a closd orbit in th systm (X (m), T [m] 1,..., T m [m] ), which is an invrs limit of nilsystms (Sction 3.3.1). Following our convntion, w assum in c) and d) that Y is ndowd with a bas point and that p maps th bas point to th bas point. Proof. W first prov a) and d) assuming that X is a nilsystm. (Whil th proof is containd in [9], w sktch it hr in ordr to introduc som objcts and som notation.) Th nilmanifold and cubs. Writ X = G/Γ and lt τ b th lmnt of G dfining th transformation T of X. W can assum that th bas point of X is th imag in X of th unit lmnt of G. Sinc (X, µ, T ) is rgodic, w can also assum that G is spannd by th connctd componnt G c of th idntity and τ (s for xampl Sction 4 of [3]). This implis that th commutator subgroups G j, j 2, ar also connctd. As xplaind in Sction 3.3.1, X (m) is a nilmanifold: X (m) = G (m) /Γ (m), whr G (m) is a subgroup of G [m] and Γ (m) = Γ [m] G (m). W rcall a

19 NIL-BOHR SETS OF INTEGERS 19 convnint prsntation of G (m) (s Appndix B of [9] or Appndix E of [7]). For g G and F P([m]), w writ g F for th lmnt of G [m] givn by { ( for vry ɛ [m], g F g if ɛ F ; ) ɛ = 1 othrwis. Lt α 1,..., α 2 m b an numration of all substs of [d] such that α i is nondcrasing. In particular, α 1 =. For 1 i 2 m, lt F i = {ɛ: α i ɛ [m]}. For vry i, F i is an uppr fac of th cub P([m]), maning a fac containing th vrtx [d]; its codimnsion is α i. Thn F 1,..., F 2 m is an numration of all th uppr facs, in nonincrasing ordr with rspct to codimnsion. In particular, F 1 is th whol cub P([m]). W also assum that α i = {i 1} for 2 i m + 1. Thn th group G (m) is th subst of G [m] consisting of lmnts h that can b writtn as (7) h = g F 1 1 g F g F 2 d, whr g 2 d i G αi for vry i (whr, by convntion, G 0 = G) and ach lmnt of G (m) has a uniqu xprssion of this form. Th diagonal transformation T [m] of X (m) is th translation by th lmnt τ F 1 = τ [m] of G (m) and, for 1 i m, th i-th fac transformation is th translation by th lmnt τ [m] i := τ F i+1. Rcall that for ɛ [m], (τ [m] i ) ɛ = Proof of a). W dfin { τ if i ɛ ; 1 othrwis. G (m) = { g G (m) : g = 1 }. This group is closd and normal in G (m) and vry lmnt of G (m) can b writtn in a uniqu way as h [m] g with h G and g G (m). Morovr, G (m) is th st of lmnts of G (m) that ar writtn as in (7) with g 1 = 1. From this, it is asy to dduc that th commutator subgroup of this group is qual to G (m) (G 2 ) [m]. Clarly, th subst X (m) of X (m) is invariant undr G (m) and it follows from th prcding dscription that th action of this group on this st is transitiv. Thrfor, th subgroup of G (m) Γ (m) is cocompact in G (m) := Γ (m) G (m) and w can idntify X (m) = G (m) /Γ (m).

20 20 BERNARD HOST AND BRYNA KRA Sinc th groups G j, j 2, ar connctd, th connctd componnt of th idntity of G (m) contains all lmnts of th form (7) whr g 1 = 1 and g i lis in th connctd componnt of th idntity of G for 2 i m + 1. Sinc G is spannd by th connctd componnt of its idntity and τ, G (m) is spannd by th connctd componnt of its idntity and th lmnts τ F i = τ [m] i for 1 i m. Morovr, by using th abov dscription of G (m), it is not difficult to chck that th action inducd by T [m] i, 1 i m on th compact ablian group G (m) /(G (m) ) 2 Γ (m) is rgodic. By a classical critria [11], th action of th transformations T [m] i on X (m) is rgodic and thus minimal. In particular, X (m) is th closd orbit of th point [m] undr ths transformations. This provs a) Proof of d). Th HK-factor of ordr (m 1) (Y, ν, T ) of (X = G/Γ, µ, T ) is Y = G/ΓG m ndowd with its Haar masur. For vry ɛ [m] with ɛ and vry w G m, w hav w {ɛ} G (m) of X (m) is invariant undr translation and thus th Haar masur µ (m) by this lmnt. Th rsult follows Proof of th proposition in th gnral cas. For a), th gnralization to invrs limits is immdiat. b) Lt ɛ [d] with ɛ. Lt i ɛ. Thn for vry x X (m) w hav T x ɛ = (T [m] i x) ɛ. Sinc th masur µ (m) its imag undr th projction x x ɛ is invariant undr T and thus is qual to µ. Proprty c) is immdiat. d) Lt th functions f ɛ b as in th statmnt; without loss w can assum that f ɛ 1 for vry ɛ. Lt (X i, µ i, T i ), i 1, b an incrasing squnc of k-stp nilsystms with invrs limit (X, µ, T ) and lt π i : X X i, i 1, b th (pointd) factor maps. For vry ɛ, ɛ [d], w hav that and thus (8) =ɛ [d] f ɛ E(f ɛ X i ) π L i 0 as i + 1 (µ) E(f ɛ X i ) π i (x ɛ ) dµ (m) (x) is invariant undr T (m) i, =ɛ [d] f ɛ (x ɛ ) dµ (m) (x) as i +. For vry i, lt (W i, σ i, T ) b th HK-factor of ordr (m 1) of X i, q i : X i W i th factor map and r i = q i π i.

21 NIL-BOHR SETS OF INTEGERS 21 W hav shown that for vry i, th masur (µ i ) (m) is rlativly indpndnt with rspct to (σ i ) (m). Using c) twic, w hav that th scond intgral in (8) is qual to E(f ɛ W i ) r i (x ɛ ) dµ (m) (x). =ɛ [d] As th systms X i form an incrasing squnc, th systms W i also form an incrasing squnc. Lt (W, σ, T ) b th invrs limit of this squnc. This systm is a factor of X, and writing r : X W for th factor map, w hav that E(f ɛ W i ) r i E(f ɛ W ) r in L 1 (µ) for vry ɛ. W gt (9) =ɛ [d] f ɛ (x ɛ ) dµ (m) (x) = This mans that th masur µ (m) to σ (m). =ɛ [d] E(f ɛ W ) r(x ɛ ) dµ (m) (x). is rlativly indpndnt with rspct Sinc W is an invrs limit of (m 1)-stp nilsystms and is a factor of X, it is a factor of th HK-factor Y of ordr (m 1) of X. If for som ɛ w hav E(f ɛ Y ) = 0, thn w hav E(f ɛ W ) = 0 and th scond intgral in (9) is qual to zro. Th rsult follows. Passing to invrs limits adds tchnical issus to ach proof. Ths issus ar not difficult and th passag to invrs limits uss only routin tchniqus, as in th prcding proof. Howvr, it dos gratly incras th lngth of th argumnts, and so in gnral w omit this portion of th argumnt in th squl Th masurs µ (m),x. In this sction, again (X, µ, T ) is an rgodic invrs limit of k-stp nilsystms, with bas point X. For x X w writ X (m),x = {x X (m) : x = and x {m} = x}. Th st X, (m) is th imag of th st X (m,1) prmutation of coordinats. introducd blow by a Proposition 5.2. For ach x X, thr xists a masur µ (m),x, concntratd on X,x (m), such that i) Th imag of µ (m),x undr ach projction x x ɛ : X [m] X, ɛ, ɛ {m}, is qual to µ.

22 22 BERNARD HOST AND BRYNA KRA ii) If f ɛ, ɛ [m], ɛ [1], ar 2 m 2 boundd masurabl functions on X, thn th function F on X givn by F (x) = f ɛ (x ɛ ) dµ (m),x (x) ɛ [d] ɛ, ɛ {m} is continuous. iii) Morovr, for vry boundd masurabl function f on X, f(x)f (x) dµ(x) = f(x {m} ) f ɛ (x ɛ ) dµ (m) (x). ɛ [d] ɛ,[m] Proof. It suffics to prov this Proposition in th cas that (X, µ, T ) is k-stp nilsystm, as th gnral cas follows by standard mthods. W writ X = G/Γ as usual. W can assum that is th imag in X of th unit lmnt 1 of G. W dfin G (m), = { g G (m) : g = g {m} = 1 }. This group is closd and normal in G. It is th st of lmnts of G (m) that can b writtn as in (7) with g = 1 and g i = 1 for th valu of i such that α i = {m}. Rcall that [m] = (,,..., ). It is asy to chck that G (m), [m] = X, (m). It follows that Γ (m), := Γ [m] G (m), is cocompact in G, (m) and that X, (m) can b idntifid with th nilmanifold G (m), /Γ (m),. W writ µ (m), for th Haar masur of this nilmanifold. Lt F = {ɛ [m]: m ɛ}. W rcall that for g G, g F G (m) is dfind by { (g F g if m ɛ ) ɛ = 1 othrwis. By dfinition of th sts X,x (m), th imag of X, (m) undr translation by g m [m] is qual to X,g. (m) Sinc G (m), is normal in G (m), th imag of th masur µ (m), undr g F is invariant undr G (m),. Morovr, if g, h G satisfy g = h, thn w hav that g = hγ for som γ Γ. Sinc γ F [m] = [m] and by normality of G (m), again, th masur µ (m), is invariant undr γ F and thus th imags of µ (m), undr g F and h F ar th sam. Thrfor, for vry x X w can dfin a masur µ (m),x on X,x (m) by (10) µ (m),x = g F µ (m), for vry g G such that g = x.

23 NIL-BOHR SETS OF INTEGERS 23 In particular, for vry h G and vry x X, (11) µ (m),h x = hf µ (m),x. If T is th translation by τ G, thn T m [m] and so for vry intgr n, (12) µ (m),t n x = T m [m] n µ (m) For 1 i < m, τ [m] i undr T [m] i is th translation by τ F and,x. G (m), and thus, for vry x X, µ (m),x is invariant. As abov, it follows that this masur satisfis th first proprty of th proposition. To prov th othr proprtis, th first statmnt of th proposition implis that w can rduc to th cas that th functions f ɛ ar continuous. By (10), th map x µ (m),x is wakly continuous and th function F is continuous. W ar lft with showing that µ (m) = µ (m),x dµ(x). T [m] i For 1 i < m, sinc for vry x th masur µ (m),x is invariant undr, th masur dfind by this intgral is invariant undr this transformation. By (11), µ (m),t x = T m [m] µ (m),x for vry x and it follows that th masur dfind by th abov intgral is invariant undr T m [m], it is qual to th Haar masur µ (m) it is concntratd on X (m) nilmanifold (rcall that (X (m). Sinc of this, T [m] 1,..., T m [m] ) is uniquly rgodic) A positivity rsult. In this sction, again (X, µ, T ) is an rgodic invrs limit of k-stp nilsystms, with bas point X. In th nxt proposition, th notation ɛ = ɛ 1... ɛ m {0, 1} m is mor convnint that ɛ [m]. W rcall that {0, 1} m corrsponds to [m] and that {0, 1} m corrsponds to {m} [m]. For ɛ {0, 1} m+1, ɛ 1... ɛ m corrsponds to ɛ [m]. Proposition 5.3. Lt f ɛ, ɛ {0, 1} m, b 2 m 1 boundd masurabl ral functions on X. Thn f ɛ1...ɛ m (x ɛ ) dµ (m+1), (x) ɛ {0,1} m+1 ɛ ɛ ( ɛ {0,1} m ɛ f ɛ (x ɛ ) dµ (m) (x)) 2.

24 24 BERNARD HOST AND BRYNA KRA Proof. W first rduc th gnral cas to that of an rgodic k-stp nilsystm. If (X, µ, T ) is an invrs limit of an incrasing squnc of k-stp rgodic nilsystms, thn th spacs X (m) and X, (m+1), as wll as th masurs µ (m) and µ (m+1),, ar th invrs limits of th corrsponding objcts associatd to ach of th nilsystms in th squnc of nilsystms convrging to X. Thus it suffics to prov th proposition whn (X, T, µ) is an rgodic k-stp nilsystm. W writ X = G/Γ as usual. Th groups G (m), Γ (m), G (m+1), and Γ (m+1), hav bn dfind and studid abov. W rcall that X (m) = G (m) /Γ (m) and that µ (m) is th Haar masur of this nilmanifold. Also, X, (m+1) = G (m+1), /Γ (m+1), and µ (m+1), is th Haar masur of this nilmanifold. It is convnint to idntify X [m+1] with X [m] X [m], writing a point x X [m+1] as x = (x, x ), whr x = (x ɛ1...ɛ m0 : ɛ {0, 1} m ) and x = (x ɛ1...ɛ m1 : ɛ {0, 1} m ). Th diagonal map (m) X (x, x), that is, for x X [m] and ɛ {0, 1} m+1, : X[m] X [m+1] is dfind by (m) X (x) = ( (m) X (x)) ɛ = x ɛ 1...ɛ m. W rmark that (m) X (X(m) ) X, (m+1). W us similar notation for lmnts of G [m+1] and dfin th diagonal map (m) G : G[m] G [m+1]. W hav that (m) (G (m) ) G (m+1), and, for vry g = (g, g ) G, (m+1) ; in othr words, G (m+1) G (m) G (m) and w hav that G (m) It follows that G (m+1), X (m+1),, For vry x X (m), st G (m) w hav that g and g blong to G (m). W dfin = {g G (m) : (1 [m], g) G (m+1), } is a closd normal subgroup of G (m) and that = { } (g, hg): g G (m), h G (m). = { (x, h x): x X (m) }, h G (m+1). V x = {y X (m) : (x, y) X, (m+1) } = {h x: h G (m+1) }. Thn V x is a nilmanifold, quotint of th nilpotnt Li group G (m+1) by th stabilizr of x. Lt ν x b th Haar masur of this nilmanifold.

25 NIL-BOHR SETS OF INTEGERS 25 For x X (m) and g G (m), w hav that (13) th imag of ν x undr translation by g is qual to ν g x. Indd, this imag is supportd on V g x and is invariant undr G (m) sinc G (m) is normal in G (m). W claim that (14) µ (m+1), = δ x ν x dµ (m) (x). Th masur on X, (m+1) dfind by this intgral is invariant undr translation by lmnts of th form (1 [m], h) with h G (m) (not that ach δ x ν x is invariant undr such translations). By (13), th masur dfind by this intgral is also invariant undr translation by (g, g) for g G (m). Thrfor this masur is invariant undr G (m+1),. Sinc it is supportd on X, (m+1), it is qual to th Haar masur µ (m+1), of this nilmanifold. Th claim is provn. By (13) again, ν h x = ν x for h G (m). Lt F dnot th σ-algbra of G (m) -invariant Borl sts. For vry boundd Borl function F on X (m), (15) F dν x = E(F F)(x) µ (m) -a.. To s this, w not that th function dfind by this intgral is invariant undr translation by G (m) and thus is F-masurabl. Convrsly, if F is F-masurabl, thn for µ (m) almost vry x, it coincids ν x - almost vrywhr with a constant and so th intgral is qual almost vrywhr to F (x). Thus for a boundd Borl function F on X (m), using (14) and (15), w hav that ( ) F (x )F (x ) dµ, (m+1) (x) = F (x ) F (x ) dν x (x ) dµ (m) (x ) = F E(F F) dµ (m) = E(F F) 2 dµ (m) ( ) 2 F dµ (m) Th masurs µ (m,r). In this sction again, (X, µ, T ) is an rgodic invrs limit of k-stp nilsystms, with bas point X. Lt m and r b intgrs with 0 r < m.,

26 26 BERNARD HOST AND BRYNA KRA Lt m,r : X [m r] X [m] b th map givn by for x X [m r] and ɛ = ɛ 1... ɛ m {0, 1} m, W dfin: (16) (17) µ (m,r) X (m,r) ( ) = m,r X (m r) and is th imag of µ (m r) undr m,r. ( m,r x) ɛ = x ɛr+1...ɛ m. Rcall that X (m r) is th closd orbit of [m r] undr th transformations T [m r] i for 1 i m r and that µ (m r) is th uniqu probability masur of this st invariant undr ths transformations. W hav m,r [m r] = [m], and, for 1 i m r, m,r T [m r] i Thrfor: X (m,r) = T [m] r+i m,r. is th closd orbit of th point [m] X (m) undr th transformations T [m] i for r + 1 i m and µ (m,r) is th uniqu probability masur on this st invariant undr ths transformations. For xampl, X (m,0) = X (m) X (m) and µ (m,0) = µ (m). X (r+1,r) = { [r] } [r] X (r+1), whr [r] dnots th diagonal of X [r]. µ (r+1,r) is th product of th Dirac mass at [r] by th diagonal masur of X [r]. Sinc th imag of µ (m r) undr th projctions x x ɛ with ɛ, ar qual to µ, w hav that: Th imags of µ (m,r) undr th projctions x x ɛ for ɛ [m], ɛ [r], ar qual to µ. Thrfor, if h ɛ, ɛ [m], ɛ [r], ar 2 m 2 r masurabl functions on X with h ɛ 1, w hav that (18) ɛ [m] ɛ [r] h ɛ (x ɛ ) dµ (m,r) (x) min h ɛ L 1 (µ). ɛ [m] ɛ [r] 6. A convrgnc rsult In this sction, w prov th ky convrgnc rsult (Proposition 6.4) Contxt. W rcall our contxt, as introducd in Sctions 3.4 and 3.5. Th systm (X, T ) is associatd to th subalgbra A of l (Z), µ is an rgodic invariant probability masur on X, associatd to th avrags on th squnc I = (I j : j 1) of intrvals.

27 NIL-BOHR SETS OF INTEGERS 27 For vry k 1, lt (Z k, µ k, T ) b th factor of ordr k of (X, µ, T ). W rcall that this systm is an invrs limit of (k 1)-stp nilsystms, both in th topological and th rgodic thortical snss. Th systm (Z k, T ) is distal, minimal and uniquly rgodic, and Z k is givn with a bas point k. In a futil attmpt to kp th notation only mildly disagrabl, whn th bas point k is usd as a subindx, w omit th subscript k. W writ π k : X Z k for th factor map. W rcall that this map is masurabl, and has no rason for bing continuous. For l k, Z l is a factor of Z k, with a factor map π l,k : Z k Z l which is continuous and π l,k ( k ) = l. W us various diffrnt mthods of taking limits of avrags of squncs indxd by Z r. For xampl, in Proposition 6.1, w avrag ovr any Følnr squnc in Z r. In th squl, w us itratd limits: if ( a n : n = (n 1,..., n r ) Z r) is a boundd squnc, w dfin th itratd limsup of a as Itr limsup Av I,n1,...,n r a n1,...,n r = limsup... limsup j 1 j r 1 I j1... I jr n 1 I j1... n r I jr a n1,...,n r. W dfin th Itr lim Av a n analogously, assuming that all of th limits xist An uppr bound. Th nxt proposition is provd in Sction 13 of [8]: Proposition 6.1. Lt (X, µ, T ) b an rgodic systm and (Z d, T, ν) b its factor of ordr d. Lt f ɛ, ɛ [d], b 2 d boundd masurabl functions on X. For n = (n 1,..., n d ) Z d, lt a n = f ɛ (T n ɛ x) dµ(x) and b n = E(f ɛ Z d )(T n ɛ z) dµ d (z). ɛ [d] ɛ [d] Thn a n b n convrgs to zro in dnsity, maning that th avrags of [a n b n on any Følnr squnc in Z d convrg to zro. Lmma 6.2. Lt k 1, 0 r d and h ɛ, ɛ [d + 1], ɛ [r], b 2 d+1 2 r continuous functions on Z k. Thn for vry δ > 0, thr xists C = C(δ) > 0 with th following proprty: Lt ψ ɛ, ɛ [r], b 2 r squncs blonging to B (as dfind in Sction 3.5) with absolut valu 1. Thn th itratd limsup in n 1,..., n r

28 28 BERNARD HOST AND BRYNA KRA of th absolut valu of th avrags on I of A(n) := ψ ɛ (n ɛ) h ɛ (T n ɛ x ɛ ) dµ (d+1,r) k (x) is boundd by ɛ [r] δ + C r ɛ [r] ɛ [d+1] ɛ [r] ψ ɛ I,k+r. Proof. W writ n = (m 1,..., m r 1, p) and m = (m 1,..., m r 1 ). Th xprssion to b avragd can b rwrittn as A (m, p) = ψ ɛ (m ɛ) ɛ [r 1] r ɛ [r] ψ ɛ (m ɛ+p) ɛ [d+1] ɛ [r] h ɛ (T m ɛ+pɛr x ɛ ) dµ (d+1,r) k (x), whr in th trm m ɛ, w only us th first r 1 coordinats of ɛ. For m Z r 1, w writ Φ m (p) = ψ ɛ (m ɛ + p) = σ m ɛ ψ ɛ (p) r ɛ [r] r ɛ [r] whr σ is th shift on l (Z). For x Z (d+1,r) k, w also writ H(x) = h ɛ (x ɛ ) ɛ [d+1] ɛ [r] and for vry δ > 0, w lt C = C(δ) b associatd to this continuous function on X (d+1,r) k as in Proposition 3.5. W hav ψ ɛ (m ɛ + p) h ɛ (T m ɛ+pɛr x ɛ ) r ɛ [r] and thus limsup j Av p Ij ɛ [d+1] ɛ [r] = Φ m (p)h ( (T r [d+1] ) p ((T [d+1] 1 ) m 1... (T [d+1] r 1 ) m r 1 x) ) r ɛ [r] ψ ɛ (m ɛ + p) ɛ [d+1] ɛ [r] h ɛ (T m ɛ+pɛr x ɛ ) for vry m and vry x Z (d+1,r) k. Taking th intgral, limsup Av p Ij A (m, p) δ + C Φ m I,k+1 j δ + C Φ m I,k+1

29 for vry m. Thrfor NIL-BOHR SETS OF INTEGERS 29 Itr limsup Av I,n1,...,n r A(n) Itr limsup Av I,n1,...,n r 1 lim j Av p Ij A (m, p) δ + C Itr limsup Av I,m1,...,m r 1 Φ m I,k+1 ( δ + C Itr limsup Av I,m1,...,m r 1 Φ m 2r 1 I,k+1 ) 1/2 r 1. By Proposition 4.3 in [9], th last limsup is actually a limit and is boundd by ψ ɛ I,k+r Itration. r ɛ [r] Proposition 6.3. Lt k 1, 0 r d and h ɛ, ɛ [d + 1], ɛ [r], b 2 d+1 2 r boundd masurabl functions on Z k. Lt φ ɛ, ɛ [r], b 2 r continuous functions on X. For n Z r, dfin and B(n) = r ɛ [r] A(n) = φ ɛ (T n ɛ x 0 ) ɛ [r 1] ɛ [r] φ ɛ (T n ɛ x 0 ) E(φ ɛ Z k+r 1 )(x ɛ ) ɛ [d+1] ɛ [r] ɛ [d+1] ɛ [r] h ɛ (T n ɛ x ɛ ) dµ (d+1,r) k (x) h ɛ p k+r 1,k (x ɛ ) dµ (d+1,r 1) k+r 1 (x). Thn th itratd limit of th avrags of A(n) B(n) is zro. Proof. W rmark that B(n) dpnds only on n 1,..., n r 1. By (18), it suffics to prov th rsult in th cas that th functions h ɛ ar continuous. W can also assum that φ ɛ 1 for vry ɛ [r]. Lt δ > 0 b givn and lt C b as in Lmma 6.2. For ach ɛ with r ɛ [d+1], lt φ ɛ b a continuous function on Z k+r 1 with φ ɛ 1, such that E(φ ɛ Z k+r 1 ) φ ɛ is sufficintly small. W hav that ( φ ɛ (T n k+r 1 ): n Z) (φ ɛ (T n x 0 ): n Z) I,k+r δ/2 r 1 C for vry ɛ. This follows from Proposition 3.6.

30 30 BERNARD HOST AND BRYNA KRA By Lmma 6.2 th itratd limsup of th absolut valu of th avrags on I of A(n) φ ɛ (T n ɛ x 0 ) ɛ [r 1] r ɛ [r] φ ɛ (T n ɛ k+r 1 ) ɛ [d+1] ɛ [r] h ɛ (T n ɛ x ɛ ) dµ (d+1,r) k (x) is boundd by 2δ. W rwrit th scond trm in this diffrnc as ɛ [r 1] φ ɛ (T n ɛ x 0 ) r ɛ [r] φ ɛ (T n ɛ k+r 1 ) ɛ [d+1] ɛ [r] h ɛ p k+r 1,r (T n ɛ x ɛ ) dµ (d+1,r) k+r 1 (x) and rmark that th first product in this last xprssion dpnds only on n 1,..., n r 1. By dfinition of th masurs and continuity of th functions φ ɛ, th avrags in n r on I of th abov xprssion convrgs to ɛ [r 1] φ ɛ (T n ɛ x 0 ) r ɛ [r] φ ɛ (x ɛ ) ɛ [d+1] ɛ [r] h ɛ p k+r 1,k (x ɛ ) dµ (d+1,r 1) k+r 1 (x). By (18) again, for vry n 1,..., n r 1 th diffrnc btwn this xprssion and B(n) is boundd by δ. Th announcd rsult follows. Proposition 6.4. Lt k 1 and lt f ɛ, ɛ [d + 1], ɛ, b 2 d+1 1 continuous functions on X. Thn th itratd avrags for n = (n 1,..., n d, n d+1 ) Z d+1 on I of (19) f ɛ (T n ɛ x 0 ) =ɛ [d+1] convrg to (20) =ɛ [d+1] E(f ɛ Z d )(x ɛ ) dµ (d+1) d (x). Proof. For notational convninc w dfin f to b th constant function 1.