Table Of ContentQUANTITATIVE MULTIPLE RECURRENCE FOR TWO AND
THREE TRANSFORMATIONS
7
1 SEBASTIA´NDONOSOANDWENBOSUN
0
2
Abstract. Weprovidevariouscounterexamplesforquantitativemulti-
n
a plerecurrenceproblemsforsystemswithmorethanonetransformation.
J Weshowthat
7 Thereexistsanergodicsystem(X, ,µ,T ,T )withtwocommut-
1 2
• X
2 ing transformations such that for every 0 < ℓ < 4, there exists
A suchthat
] ∈X
S µ(A T nA T nA)<µ(A)ℓ foreveryn,0;
D ∩ 1− ∩ 2−
Thereexistsanergodicsystem(X, ,µ,T ,T ,T )withthreecom-
. • X 1 2 3
h muting transformations such that for every ℓ > 0, there exists
t A suchthat
a
∈X
m µ(A T nA T nA T nA)<µ(A)ℓ foreveryn,0;
∩ 1− ∩ 2− ∩ 3−
[
Thereexistsanergodicsystem(X, ,µ,T1,T2)withtwotransfor-
• X
1 mations generating a 2-step nilpotent group such that for every
v ℓ>0,thereexistsA suchthat
9 ∈X
3 µ(A∩T1−nA∩T2−nA)<µ(A)ℓ foreveryn,0.
1
8
0
.
1 1. Introduction
0
7
1.1. Quantitativerecurrence. ThePoincare´recurrencetheoremstatesthat
1
: for every measure preserving system (X, ,µ,T) and every A with
v X ∈ X
i µ(A) > 0, theset
X
n Z : µ(A T nA) > 0
−
r { ∈ ∩ }
a isinfinite. AquantitativeversionofitwasprovidedbyKhintchine[8],who
showed that for every ǫ > 0, the set n N : µ(A T nA) > µ(A)2 ǫ is
−
{ ∈ ∩ − }
syndetic,meaningthatit hasboundedgaps.
Multiplerecurrence problems refers to theones concerning the behavior
of the set A T nA T nA. In the case T = Ti for an ergodic transfor-
∩ 1− ···∩ d− i
mationT, Furstenberg [6]showedthattheset
n Z : µ(A T nA T dnA) > 0
− −
{ ∈ ∩ ···∩ }
2010MathematicsSubjectClassification. Primary:37A30.
Key words and phrases. Poincare´ recurrence, multiple recurrence, commuting
transformations.
ThefirstauthorissupportedbyFondecytIniciacio´nenInvestigacio´ngrant11160061.
1
2 SEBASTIA´NDONOSOANDWENBOSUN
isinfinite. ThisresultisnowknownastheFurstenbergMultipleRecurrence
Theorem.
The quantitative version of the multiple recurrence problems concerns
not only the positivity of the measure of the set A T nA T nA, but
∩ 1− ··· ∩ d−
alsohowfarfrom 0 thismeasureis. Moreprecisely,we ask:
Question1.1. Let(X, ,µ,T ,...,T )beameasurepreservingsystemand
1 d
X
F: [0,1] R beanon-negativefunction. Is theset
0
→ ≥
n Z : µ(A T nA T nA T nA) F(µ(A))
{ ∈ ∩ 1− ∩ 2− ∩···∩ d− ≥ }
syndeticforall A ?
∈ X
IftheanswertothequestionisaffirmativeforsomeT ,...,T and F,we
1 d
say that F is good for (T ,...,T ). Based on the result of Khintchine [8]
1 d
statingthat F(x) = x2 ǫ isgoodfor(T)(withasingleterm),anaturalcon-
jecturewouldbethatt−hefunction F(x) = xd+1 ǫ isgoodfor(T ,...,T ).
1 d
The case T = Ti was solved by Bergelso−n, Host and Kra [3]. They
i
showed that if (X, ,µ,T) is ergodic, then F(x) = xd+1 ǫ is good for
(T,T2,...,Td)foraXllǫ > 0ford = 2or3. Theyalsoshowe−dtwosurprising
phenomena. First, the hypothesis of ergodicity cannot be removed: there
existsa non-ergodic system(X, ,µ,T)such that F(x) = xℓ is not good for
(T,T2) for all ℓ > 0. SecondlyX, if d 4, then F(x) = xℓ is not good for
≥
(T,T2,...,Td)forall ℓ > 0.
In[7],FurstenbergandKatznelsonprovedamultiplerecurrencetheorem
for commuting transformations. For two commuting transformations, its
quantitative study was done by Chu [4] who proved that for every system
(X, ,µ,T ,T ) with two commuting transformations T and T (meaning
1 2 1 2
thatXT T = T T ), ergodic for T ,T , and every ǫ > 0, F(x) = x4 ǫ is
1 2 2 1 1 2
h i −
good for (T ,T ). Here it is worth to stress that the exponent for (T ,T )
1 2 1 2
is 4 while the exponent for (T,T2) is 3. Indeed, in the same paper, Chu
constructed an exampleshowing that F(x) = x3 is not good for (T ,T ). In
1 2
a later work, Chu and Zorin-Kranich [5] improved this example, showing
that F(x) = x3.19 isnotgoodfor(T ,T ).
1 2
Inthispaper,westudythebestcomponentℓ forwhich F(x) = xℓ isgood
for (T ,...,T ) in an ergodic system (X, ,µ,T ,...,T ) with commuting
1 d 1 d
X
transformations. The result of Chu and Zorin-Kranich [5] suggested that
the largest ℓ not good for (T ,T ) is between 3.19 and 4. We show that ℓ
1 2
can besufficientlycloseto 4:
Theorem 1.2. There exists a measure preserving system (X, ,µ,T ,T )
1 2
X
with commuting transformations T and T , ergodic for T ,T such that
1 2 1 2
forevery0 < ℓ < 4, F(x) = xℓ is notgoodfor(T ,T ), i.e.hthereiexistsa set
1 2
A suchthat
∈ X
µ(A T nA T nA) < µ(A)ℓ
∩ 1− ∩ 2−
QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS3
foreveryn , 0.
Churaisedanotherquestion[4]onwhetherF(x) = xℓisgoodfor(T ,T ,...,T )
1 2 d
inanergodicsystemwithdcommutingtransformationsford 3. Weshow
≥
thatthisisnotthecase:
Theorem 1.3. There exists a measure preserving system (X,µ,T ,T ,T )
1 2 3
withthreecommutingtransformationsT andT andT ,ergodicfor T ,T ,T
1 2 3 1 2 3
such that for every ℓ > 0, F(x) = xℓ is not good for (T ,T ,T ), i.eh. there i
1 2 3
existsa subset A suchthat
∈ X
µ(A T nA T nA T nA) < µ(A)ℓ
∩ 1− ∩ 2− ∩ 3−
foreveryn , 0.
If we relax the condition of commutativity of the transformations, the
natural condition to look at is nilpotency. Outside the abelian category,
we show that there is no polynomial quantitative recurrence even for two
transformationsT andT spanninga2-stepnilpotentgroup. Weshow
1 2
Theorem 1.4. There exists a measure preserving system (X, ,µ,T ,T )
1 2
X
such that T and T generate a 2-step nilpotent group T ,T , whose ac-
1 2 1 2
tions is ergodic and such that for every ℓ > 0, F(x) =hxℓ is niot good for
(T ,T ), i.e. thereexistssubset A suchthat
1 2
∈ X
µ(A T nA T nA) < µ(A)ℓ
∩ 1− ∩ 2−
foreveryn , 0.
1.2. Notationandconventions. Ameasurepreservingsystem(orasystem
for short) is a tuple (X, ,µ,T ,...,T ), where (X, ,µ) is a probability
1 d
X X
spaceand T ,...,T : X X are actionssuch that forall A ,1 i d,
1 d
T 1A andµ(T 1A) =→µ(A). Weuse T ,T ,...,T tode∈noXteth≤egr≤oup
i− ∈ X i− h 1 2 di
spanned by the transformations T ,...,T . We say that X is ergodic for
1 d
T ,...,T ifA ,T 1A = A forall1 i d impliesthatµ(A) = 0or1.
h 1 di ∈ X i− ≤ ≤
Forapositiveintegernumber N, thesubset 1,...,N isdenoted by[N].
{ }
2. Combinatorial constructions
In this section we study subsets of N2 and N3 satisfying special combi-
natorial conditions that help us construct the counter examples we need.
TheconstructionofsuchsetsisinspiredbythemethodsusedbySalemand
Spencer [11] and Behrend [2] in building “large” subsets of [N] with no
arithmetic progressions of length 3. The ways to make use of special sub-
setsinTheorems1.2,1.3and1.4aremotivatedbytheexamplesconstructed
inBergelson, Hostand Kra[3]and Chu [4].
Weremarkthatthecombinatorialpropertiesstudiedinthissectionareof
independentinterest.
4 SEBASTIA´NDONOSOANDWENBOSUN
2.1. Corner-freesubsetsofN2. Thefirstcombinatorialproblemwestudy
ishowlargeasubsetΛ [N]2 can bewithouta“corner”.
⊆
Definition2.1. WesaythatasetΛ [N]2 iscorner-freeif(x,y),(x ,y)and
′
(x,y ) Λ and x y = x y impli⊆esthat x = x and y = y .
′ ′ ′ ′ ′
∈ − −
Wehave
Theorem2.2. Letν(N)denotethelargestcardinalityofcorner-freesubsets
of[N]2. Then
ν(N) > N2−4lolgoglo2g+Nǫ
as N forall ǫ > 0.
→ ∞
It is worth noting that Atjai and Szemere´di [1] had a similar estimate
for the largest cardinality of the set Λ [N]2 such that (x,y), (x ,y) and
′
(x,y ) Λ,x > x and x y = x y im⊆plies that x = x and y = y (with
′ ′ ′ ′ ′ ′
∈ − −
an additional but not essential assumption that x > x). Since our proof is
′
different from their method, we write it down for completeness. We thank
T.Zieglerforbringingusto thisreference.
Proof. Let1 d nbetwoparameterstobechosenlaterandassumethat
n isdivisible≪by d2≪. Let Λ betheset ofpoints(x,y) [(2d)n]2 such that the
followingconditionholds: expand x = x +x (2d 1∈)+ +x (2d 1)n 1
0 1 n 1 −
andy = y +y (2d 1)+ +y (2d 1)n 10 x,y− 2d··2· for0− i −n 1.
0 1 n 1 − i i
Consider the n pa−irs o·f·i·nteg−ers (x−,y ),(x≤,y ),..≤.,(x− ,y ).≤De≤fine−Λ
0 0 1 1 n 1 n 1
by(x,y) Λ ifand onlyifamongthepairs(x ,y ),(x ,y−),..−.,(x ,y ),
0 0 1 1 n 1 n 1
there are∈exactly n of them that are equal to (i, j) for all 0 i, j − d −1.
d2 ≤ ≤ −
RecallforasetS ofcardinalityk andk thatdividesk, k! isthenumber
of different patitions of S where each a′tom has exactly(kk/k′)e!kl′ements. Using
′
thisformula, wegetthat
n!
Λ = .
| | ((n )!)d2
d2
We claim that Λ satisfies the properties we are looking for. We first es-
timate the size of Λ. Set n = d2ω(d), where ω: N N is an increasing
→
function such that ω(d + 1) ω(d) = O(1), ω(d) and logω(d) 0 as
− logd → ∞ logd →
d . Forevery N N, pickd N such that
→ ∞ ∈ ∈
(2.1) (2d 1)d2ω(d) N < (2d +1)(d+1)2ω(d+1).
− ≤
By the Stirling formula, if d and n/d2 = ω(d) are large enough, we have
that
n! nn√2πne n 1 d2n d2d2ω(d)
> − = ,
(( n )!)d2 [(n/d2)n/d2 2π(n/d2)e n/d2]d2 Cd2 ≥ (γn)d2/2 (γω(d))d2/2
d2 − d2
p
QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS5
whereγ = 2πC2,andC isaconstantas closeto1 as wewant. So
(2.2)
N2 (2d+1)2(d+1)2ω(d+1) (2d+1)2(d+1)2ω(d+1)(γω(d))d2/2
log < log < log
L ! L d2d2ω(d)
| | | | d2
= 2(d +1)2ω(d+1)log(2d +1) 2d2ω(d)logd+ (logγ+logω(d))
− 2
= d2ω(d)(2log2+o(1)),
where in the last step we repeatedly used the properties of ω(d). On the
otherhand, by(2.1), wehave
logN d2ω(d)log(2d 1)
≥ −
and
loglogN < 2log(d +1)+logω(d +1)+loglog(2d +1),
whichimpliesthat
logN
(2.3) > d2ω(d)(1/2+o(1)).
loglogN
Combining(2.2)and(2.3), wehavethat
Λ > N2−4lolgoglo2g+Nǫ
| |
as N forallǫ > 0.
→ ∞
Nowwe showthat Λ iscorner-free. Supposethat(x,y), (x ,y)and (x,y )
′ ′
belong to Λ and that x y = x y . Expand w = w + w (2d 1) +
′ ′ 0 1
+ w (2d 1)n 1,0 − w ,...,−w d 1 for w = x,y,x ,y .−Since
n 1 − 0 n 1 ′ ′
··· − − ≤ − ≤ −
0 x,y,x ,y d 1 forevery0 i n 1,wehavethatnecessarily
≤ i i ′i ′i ≤ − ≤ ≤ −
(2.4) yi − xi = y′i − x′i
forall0 i n 1.
If y ≤x =≤ (−d 1) for some 0 i n 1, then by the construction of
i i
Λ,weh−ave x −= d −1andy = 0. B≤y (2≤.4),−y x = y x = (d 1), and
so x = d 1iand y−= 0. Thierefore x = x an′id−y′i= y .i− i − −
N′iowsu−pposetha′it (d 1) y xi h′i impliiesth′iat(x,y) = (x ,y )for
all0 i n 1forso−me−(d ≤1)i −hi ≤d 2. Weshowthiatiy x′i=′ih+1
i i
also i≤mpl≤ies−that (x,y) =−(x ,−y ) fo≤r al≤l 0 −i n 1. By the c−onstruction
of Λ, the number oif thie pair′is (′ix,y) such≤that≤0 −y x h is the same
i i i i
≤ − ≤
as thenumber of the pairs (x,y ) such that 0 y x h, and is the same
i ′i ≤ ′i − i ≤
as the number of the pairs (x ,y) such that 0 y x h. Therefore,
′i i ≤ i − ′i ≤
6 SEBASTIA´NDONOSOANDWENBOSUN
by induction hypothesis, if y x = h + 1, we have y x h + 1 and
y x h+1. So i − i ′i − i ≥
i − ′i ≥
y x = (y x)+(y x ) (y x)
′i − ′i ′i − i i − ′i − i − i
(h+1)+(h+1) (h+1) = h+1 = y x.
i i
≥ − −
By(2.4), wehavethaty x = y x = h+1 = y x,whichimpliesthat
x = x and y = y and w′ie−arie donie−. ′i i − i
i ′i i ′i
Weconcludethat
ν(N) Λ > N2−4lolgoglo2g+Nǫ
≥ | |
as N forallǫ > 0. (cid:3)
→ ∞
2.2. ThreepointfreesubsetsofN3. Westudyanothercombinatorialprob-
leminthissection.
Definition 2.3. Let Λ be asubset of[N]3. We say that Λ is three pointfree
if(x,y,z ),(x,y ,z),(x ,y,z) Λ impliesthat x = x ,y = y ,z = z .
′ ′ ′ ′ ′ ′
∈
In particular, (x,y,z ) and (x,y,z) Λ implies that z = z . so Λ contains
′ ′
at most one point on each line parall∈el to Z-axis. Similarly, Λ contains at
mostonepointalonganylineparallelto the X orY-axis.
Remark 2.4. To a three point free set Λ [N]3 we can associate a N N
matrixA(Λ) = (a ) . Todoso,weloo⊆kattheline (i, j,k): k [N]×. If
i,j i,j [N]
thereisapointofΛin∈ suchalineweseta tobetheu{niqueinteg∈erin}[N]
i,j
suchthat(i, j,a ) Λ. Ifthereisnopointinsuchalinewejustseta = 0.
i,j i,j
IfΛ isathreepoin∈tfree set,thematrix A(Λ)has thefollowingproperties:
(1) For every k [N], k appears at most once in each row and each
columnof A(∈Λ).
(2) Foreveryk [N], ifa = k and a = k thena = a = 0.
i,j i,j i,j i,j
∈ ′ ′ ′ ′
Conversely, if A is a N N matrix that satisfies conditions (1) and (2),
then there is a three point f×ree set Λ [N]3 such that A = A(Λ). The set Λ
is just (i, j,a ) : a , 0 and note t⊆hat the cardinality of Λ is the number
i,j i,j
ofnon-{zero entries of A(Λ}).
The combinatorial problem we study is how large such a set three point
free set can be. It is clear that if Λ [N]3 is a three point free set, then
Λ N2. Weshowthatin fact Λ can⊆besufficiently“close”to N2:
| | ≤ | |
Theorem 2.5. Let w(N) denote the largest cardinalityof a three point free
subsetof[N]3. Then
w(N) > N2−4lologglo2g+N2ǫ
as N forall ǫ > 0.
→ ∞
QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS7
Proof. Let n,d, N and Λ begivenin theproofofTheorem 2.2. Define
V := (x,y,z): (x,y) Λ [N]3.
∈ ⊆
n o
and
V := (x,y,z) V: x+y+z = s .
s
∈
n o
Wehavethat forbigenough N,
3N 3
− Vs = V = N N2−4lolgoglo2g+Nǫ.
| | | | ·
Xs=0
So thereexists0 s 3N 3 such that
≤ ≤ −
Vs |V| = N2−4lolgoglo2g+Nǫ/3 > N2−4lologglo2g+N2ǫ
| | ≥ 3N
provided N is large enough. We verify that V is three point free. Suppose
s
that(x,y,z ), (x,y ,z)and (x ,y,z)belongto V . Then in particularwe have
′ ′ ′ s
that (x,y), (x,y ) and (x ,y) belong to Λ and s x y = z = s x y. So
′ ′ ′ ′
x x = y y. SinceΛ iscorner-free, wehave−tha−t x = x and−y = y−. This
′ ′ ′ ′
im−pliesthat−z = s x y = s x y = zand weconcludethatV isthree
′ ′ s
− − − −
pointfree.
It followsimmediatelythat
w(N) Vs N2−4lologglo2g+N2ǫ
≥ | | ≥
providedthat N is largeenough. (cid:3)
3. Nilsystems andaffine nilsystems
In all that follows, we make use of the class of nilsystems, specially of
affinenilsystemsand webriefly introducethem.
3.1. Affine nilsystems with a single transformation. Let G be a group.
For a,b G, [a,b] := aba 1b 1 denotes the commutator of a and b and
− −
∈
for A and B subsets ofG, and [A,B]denotes the group generated by all the
commutators [a,b] for a A and b B. The commutator subgroups are
defined recursively as G ∈= G and G∈ = [G ,G], j 1. We say that G is
1 j+1 j
d-stepnilpotent ifG = 1 . ≥
d+1
Let G a d-step nilpotent{L}ie group and Γ be a discrete a cocompact sub-
group ofG. Thecompact manifoldG/Γ is a d-stepnilmanifold. The group
G acts onG/Γas left translationsandthereis auniqueprobabilitymeasure
µ which invariant under such action (called the Haar measure). A dynam-
ical system of the form (G/Γ, (G/Γ),µ,T ,...,T ) is called a nilsystem,
1 n
where (G/Γ) is the Borel σ-aBlgebra of G/Γ, and each T is given by the
i
rotationBby afixed τ G, i.e. T : G/Γ G/Γ,x τ x.
i i i
∈ → 7→
8 SEBASTIA´NDONOSOANDWENBOSUN
An important class of such systems are the affine nilsystems. The affine
nilsytems for a single transformation are defined as follows. Let α Td
and let A be a d d unipotent integer matrix (i.e. (A I)d = 0).∈Let
T: Td Td be th×e affine transformation x Ax+α. Le−t G be the group
→ 7→
oftransformationsof Td generated by A and thetranslationsof Td. That is,
every element g G is a map x Aix+β for somei Z and β Td. The
group G acts tran∈sitively on Td 7→and we may identify t∈his space w∈ith G/Γ,
whereΓisthestabilizerof0,whichconsistsofthepowersofA. Thesystem
(Td,µ d,T) is called an affine nilsystem (here µ is the Haar measure on T).
⊗
Propertiessuchas transitivity,minimality,ergodicityand uniqueergodicity
are equivalentfor asystemin this class and thiscan bechecked by looking
attherotationinducedby αontheprojectionTd/Ker(A I) [10].
−
3.2. Affine nilsystems with several transformations. When we consider
different affine transformations T : Td Td, x A x + α, where A is
i i i i
unipotentforeveryi = 1,...,n,wecans→tillregard7→thesystem(Td, (Td),µ d
⊗
B
, T ,...,T ) as a nilsystemas long as the matrices commute. Indeed, letG
1 n
be the group of transformations of Td generated by the matrices A ,...,A
1 n
and the translations of Td. Then every element g G is a map x
A(g)x+β(g), where A(g) = Am1 Amn,m ,...,m Z∈and β(g) Td. 7→
1 ··· n 1 n ∈ ∈
A simple computation shows that if g ,g G, the commutator [g ,g ]
1 2 1 2
is the map x x+(A(g ) I)β(g )+(A(g )∈ I)β(g ) and thus is a trans-
1 2 2 1
7→ − −
lation of Td. In the other hand, if g G and β Td, then [g,β] is the
translation x x + (A(g) I)β. It ∈follows that∈the iterated commuta-
7→ −
tor [ [[g ,g ],g ] g ] belongs to Td and is contained in the image of
1 2 3 k
··· ···
(A(g ) I) (A(g ) I). Ifkislargeenough,thisproductistrivial. SoGis
3 k
a nilpo−tent·L··ie group−. The torus Td can be identified withG/Γ, where Γ is
thestabilizerof0,which isthegroupgenerated bythematrices A ,...,A .
1 n
Wereferto(Td, (Td),µd,T ,...,T )asanaffinenilsystemwithntransfor-
1 n
B
mations. It is worth noting that the transformations T and T commute if
i j
(A I)α = (A I)α inTd.
i j j i
− −
By a theoremfrom Leibman[9], weget:
Proposition 3.1. Let (Td, (Td),µd,T ,...,T ) be an affine nilsystem with
1 n
B
ntransformations. Thenthepropertiesoftransitivity,minimality,ergodicity
anduniqueergodicityunder theactionof T ...,T areequivalent.
1 n
h i
Recall that (Td, (Td),µd,T ,...,T ) is minimal under T ...,T if the
1 n 1 n
B h i
orbit closure of every x Td under T ...,T is Td (in this paper we do
1 n
∈ h i
notneed theconceptsoftransitivityand uniqueergodicity). Werefer to[9]
foramodernreference on nilmanifoldsandnilsystems.
QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS9
4. Proofs ofthemain theorems
Weare nowready to provethemaintheorems.
ProofofTheorem 1.2. Let α,β R Q be rationally independent numbers.
Let X = T6 withtransformations∈ \
T (x ,x ,x ;y ,y ,y ) = (x +α,x +β,x ;y + x ,y ,y + x + x + x )
1 1 2 3 1 2 3 1 2 3 1 1 2 3 1 2 3
and
T (x ,x ,x ;y ,y ,y ) = (x ,x +β,x +α;y ,y + x ,y + x + x + x ).
2 1 2 3 1 2 3 1 2 3 1 2 3 3 1 2 3
Wehavethat(X, (T6),µ µ µ µ µ µ,T ,T )isanaffinenilsystem
1 2
B ⊗ ⊗ ⊗ ⊗ ⊗
withtwotransformations,whereµ istheHaarmeasureon T. Noticethat
T T (x ,x ,x ;y ,y ,y ) = T T (x ,x ,x ;y ,y ,y )
1 2 1 2 3 1 2 3 2 1 1 2 3 1 2 3
= (x +α,x +2β,x +α;y + x ,y + x ,y + x + x + x +α+β).
1 2 3 1 1 2 3 3 1 2 3
We first claim that the system is minimal and ergodic. To see this, let
(x ,x ,x ;y ,y ,y ) and (x ,x ,x ;y ,y ,y ) T6. It is not hard to see that
1 2 3 1 2 3 ′1 ′2 ′3 ′1 ′2 ′3 ∈
(x ,x ;x ,y ,y ,y ) belongs to the orbit closure of (x ,x ,x ;y ,y ,y ) un-
′1 ′2 3 ′1 2 ′3 1 2 3 1 2 3
der the transformation T . We also have that (x ,x ,x ;y ,y ,y ) is in the
1 ′1 ′2 ′3 ′1 ′2 ′3
orbit closure of (x ,x ,x ,y ,y ,y ) under the transformation T (here we
′1 ′2 3 ′1 2 ′3 2
use the fact that for fixed x , the transformation (x ,x ,y ,y ) (x +
1 2 3 2 3 2
β,x + α,y + x ,y + x + x + x ) is minimal in T4). We conc7→lude that
3 2 3 3 1 2 3
(x ,x ,x ,y ,y ,y )isintheorbitclosureof(x ,x ,x ;y ,y ,y )under T ,T .
′1 ′2 ′3 ′1 ′2 ′3 1 2 3 1 2 3 h 1 2i
Sincesincethepointsarearbitrary,thesystemisminimalandhenceergodic
byProposition3.1.
Let N N to be chosen later and Λ [N]3 be a three point free set. For
∈ ⊆
(a,b,c) [N]3, denote
∈
a a 1 b b 1 c c 1
Q = , + , + , + T3
a,b,c
N N 2N × N N 2N × N N 2N ⊆
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
andlet B = Q , A = T3 B. Then
(a,b,c) Λ a,b,c
∪ ∈ ×
Λ
µ µ µ µ µ µ(A) = | | .
⊗ ⊗ ⊗ ⊗ ⊗ 8N3
Ontheotherhand,
µ µ µ µ µ µ(A T nA T nA)
⊗ ⊗ ⊗ ⊗ ⊗ ∩ 1− ∩ 2−
= 1 (y ,y ,y )1 (y ,y ,y )1 (y ,y ,y )dµ µ(x ,x ,x ;y ,y ,y ),
Z B 1 2 3 B ′1 2 ′3 B 1 ′2 ′3 ⊗···⊗ 1 2 3 1 2 3
T6
where
n n n
y = y +nx + α, y = y +nx + α, y = y +n(x +x +x )+ (α+β).
′1 1 1 2! ′2 2 2 2! ′3 3 1 2 3 2!
10 SEBASTIA´NDONOSOANDWENBOSUN
Suppose that the product of the functions inside the integral is nonzero.
Then we may assume that (y ,y ,y ) Q , (y ,y ,y ) Q and
1 2 3 ∈ a,b,c ′1 2 ′3 ∈ a′,b,c′
(y ,y ,y ) Q for some (a,b,c), (a ,b,c ) and (a,b ,c ) that belong
to1Λ.′2Si′3nce∈Λ ias,b′t,hc′ree point free, we have′ that′a = a , b ′= b′ and c = c .
′ ′ ′
Thisimpliesthat
n n n 1 1
nx + α, nx + α, y +n(x + x + x )+ (α+β) ( , ).
1 2 3 1 2 3
2! 2! 2! ∈ −2N 2N
Therefore,
µ µ µ µ µ µ(A T nA T nA)
⊗ ⊗ ⊗ ⊗ ⊗ ∩ 1− ∩ 2−
= 1 (y ,y ,y )1 (y ,y ,y )1 (y ,y ,y )dµ µ(x ,x ,x ;y ,y ,y )
Z B 1 2 3 B ′1 2 ′3 B 1 ′2 ′3 ⊗···⊗ 1 2 3 1 2 3
T6
1 1 Λ
= 1 (y ,y ,y )dµ(y )dµ(y )dµ(y ) = | | .
N3 Z B 1 2 3 1 2 3 N3 · 8N3
T3
Wehavethatµ µ µ µ µ µ(A T nA T nA) = Λ Λℓ = µ(A)ℓ
⊗ ⊗ ⊗ ⊗ ⊗ ∩ 1− ∩ 2− 8|N|6 ≤ (8|N|3)ℓ
ifand onlyif
3
(4.1) ℓ 1+ .
≤ 3+ log(8) log(Λ)
| |
log(N) − log(N)
ByTheorem 2.5, wecan takeΛ ofcardinalitylargerthan N2 ǫ and thusthe
−
right hand side in (4.1) can be as close to 4 as we want. This finishes the
proof. (cid:3)
ProofofTheorem 1.4. Letα R Q. Let X = T3 withtransformations
∈ \
T (x,y,z) = (x+α,y+ x,z)
1
and
T (x,y,z) = (x+α,y,z+ x).
2
It is an affine nilsystem with two transformations. We first claim that
the system (X, (T3),µ µ µ,T ,T ) is minimal and ergodic, where µ is
1 2
B ⊗ ⊗
the Haar measure on T. To see this, take (x ,y ,z ) T3 and note that the
0 0 0
∈
closureoftheorbitofthispointunderthetransformationT isT T z .
1 0
× ×{ }
Let(x,y,z) T3 bean arbitrarypointandnoticethatthispointiscontained
∈
intheorbitclosureof(x ,y,z )underthetransformationT . So theclosure
0 0 2
of the orbit of (x ,y ,z ) under T ,T is T3. We conclude that the system
0 0 0 1 2
h i
isminimaland thusergodicbyProposition3.1.
It is easy to verify that [T ,T ](x,y,z) = (x,y+α,z α) commuteswith
1 2
−
T and T . So T andT generatea 2-stepnilpotentgroup.
1 2 1 2
