Ergodic Theory and Configurations in Sets of Positive ... - CiteSeerX

Ergodic Theory and Configurations in Sets of Positive Density1 Hillel Furstenberg Yitzhak Katznelson Benjamin Weiss

1

Introduction

We shall present here two examples from "geometric Ramsey theory" which illustrate how ergodic theoretic techniques can be used to prove that subsets of Euclidean space of positive density necessarily contain certain configurations. Specifically we will deal with subsets of the plane, and our results will be valid for subsets of "positive upper density". For any measurable subset E ⊂ R2 we let S range over all squares in the plane and we set (1.1)

D(E) = lim sup m(S ∩ E)/m(S), `(S)→∞

where `(S) denotes the length of a side of S . D(E) is the upper density of E and we shall be concerned with sets E having D(E) > 0. Our first result is

Theorem A. If E ⊂ R2 has positive upper density then ∃`0 such that for any ` > `0 one can find a pair of points x, y ∈ E with kx − yk = `.

We could say that the configuration {x, y} is congruent to the configuration {0, `} ⊂ R.

The next result deals with triangles.

Theorem B. Let E ⊂ R2 have positive upper density and let Eδ denote the points of distance < δ from E . Let u, v ∈ R2 , then ∃`0 such that for any ` > `0 and δ > 0 there exists a triple {x, y, z} ⊂ Eδ forming a triangle congruent to {0, `u, `v}. Theorem A answers a question posed by L. Szekely (1983). Since we announced this result it has been proved by other methods by J. Bourgain who has also given several refinements (Bourgain 1986) and also by Falconer and Marstrand (1986). Bourgain also has shown by an example that the result of Theorem B cannot be improved to finding triangles in E itself (instead of the "thickened" set of Eδ ). We shall reproduce this in Section 7. The underlying 1

Pages 184–198 in Mathematics of Ramsey Theory, Nesetril and Rödl (Eds), SpringerVerlag, 1989

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E RGODIC T HEORY AND C ONFIGURATIONS

idea in our proof of both Theorems A and B is the possibility of attaching to a measurable subset of the plane a measure preserving action of R2 on a particular measure space. If E ⊂ R2 is the subset
in question we shall obtain an "R2 measure preserving system" X, B, µ, Tu and a subset E˜ ⊂ X so that every "recurrence" of the set E˜ : ˜ ∩ T −1 E ˜ ∩ · · · ∩ T −1 E) ˜ >0 (*) µ(E u1 uk implies a "recurrence" of the thickened set Eδ : (**) Eδ ∩ (Eδ − u1 ) ∩ · · · ∩ Eδ − uk ) 6= ∅. Ergodic theory will enable us to establish results of the form (*) and the foregoing correspondence then guarantees the existence of points x ∈ Eδ with x + u1 , x + u2 , . . . , x + uk ∈ Eδ .

In the case of Theorem A we shall be able to pass from the existence of configurations in Eδ to the existence of (simpler) configurations in E . In the case of Theorem B we shall have to be satisfied with results regarding Eδ .

2

Correspondence Between Subsets of R2 and R2 -Actions

Let E ⊂ R2 be an arbitrary subset and set ϕ0 (u) = ϕ0E (u) = dist(u, E) = inf{ku − vk : v ∈ E}.

Where ku − vk denotes the euclidean metric in R2 . If ϕ(u) = min{ϕ0 (u), 1},

then ϕ(u) is a bounded uniformly continuous function on R2 with (2.1)

|ϕ(u1 ) − ϕ(u2 )| ≤ ku1 − u2 k.

The functions Ψv (u) = ϕ(u + v) form an equicontinuous family and have compact closure in the topology of uniform convergence over bounded sets in R2 . Denote this closure by X ; thus Ψ ∈ X if there is a sequence {vn } ⊂ R2 with ϕ(u + vn ) → Ψ(u)

uniformly for kuk < R, for each R < ∞. R2 acts on X with Tv Ψ(u) = Ψ(u + v) for Ψ ∈ X , and u, v ∈ R2 . The function ϕ belongs to X and its orbit {Tv ϕ}v∈R2 is dense in X . X is a compact metrizable space and we can identify Borel measures on X with functionals on C(X).

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E RGODIC T HEORY AND C ONFIGURATIONS

Suppose now that D(E) > 0 so that there exists a sequence of squares Sn ⊂ R2 with `(Sn ) → ∞ and m(Sn ∩ E) → D(E), m(Sn )

(2.2)

with D(E) > 0. Using the sequence of squares {Sn } we shall define a probability measure on X . Namely refine the sequence {Sn } so that 1 lim k→∞ m(Snk )

Z

f (Tv ϕ)dm(v)

Snk

exists for every f ∈ C(X). Such a subsequence can be found since it can be found simultaneously for a countable dense set of functions f ∈ C(X). We now define a measure µ on X by: Z

(2.3)

1 k→∞ m(Snk )

f dµ = lim

X

Z

f (Tv ϕ)dm(v).

Snk

Set f0 (Ψ) = Ψ(0). By definition of the topology on X , f0 is a continuous function. We define E˜ ⊂ X by ˜ ⇐⇒ f0 (Ψ) = 0 ⇐⇒ Ψ(0) = 0. Ψ∈E ˜ is a closed subset of X and we have E Z ˜ (2.4) µ(E) = lim (1 − f0 (Ψ))l dµ(Ψ). l→∞ X

˜ ≥ D(E). Lemma 2.1. µ(E)

P ROOF : By (2.4), it suffices to show that for any l, Z

(1 − f0 (Ψ))l dµ(Ψ) ≥ D(E).

X

By (2.3) Z

1 (1 − f0 (Ψ)) dµ(Ψ) = lim k→∞ m(Snk ) X

Z

1 = lim k→∞ m(Snk )

Z

l

(1 − f0 (Tv ϕ))l dm(v)

Snk

(1 − ϕ(v))l dm(v).

Snk

Since ϕ(v) = 0 for v ∈ E , the last expression is at least m(Snk ∩ E) = D(E), k→∞ m(Snk ) lim

J

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E RGODIC T HEORY AND C ONFIGURATIONS

We now establish the correspondence between E and E˜ described in the Introduction.

Proposition 2.2. Let E ⊂ R2 and E˜ ⊂ X be as above. If for a k-tuple of vectors, {u1 , . . . , uk } we have ˜ ∩ T −1 E ˜ ∩ · · · ∩ T −1 E) ˜ > 0, µ(E u1 uk

(2.5) then for all δ > 0,

Eδ ∩ (Eδ − u1 ) ∩ · · · ∩ Eδ − uk ) 6= ∅ .

P ROOF : Define the function g(Ψ) on X by ( δ − f0 (Ψ), g(Ψ) = 0,

if f0 (Ψ) < δ, if f0 (Ψ) ≥ δ.

˜ and (2.5) implies that g(Ψ) is positive for Ψ ∈ E Z g(Ψ)g(Tu1 Ψ) · · · g(Tuk Ψ)dµ > 0.

In particular for some v , the integrand is > 0. Since g(Tw ϕ) > 0 ⇐⇒ ϕ(w) < δ ⇐⇒ w ∈ Eδ , this implies for some v , v ∈ Eδ , v + u1 ∈ Eδ , . . . , v + uk ∈ Eδ ,

and this is the assertion of the proposition.

3

J

Ergodic Averages for Subsets of R2

We use the following variant of the ergodic theorem.

Theorem 3.1. Let G be a locally compact abelian group and
let Tg , g ∈ G be a measure preserving action on a probability space X, B, µ . Let {mu } be a sequence (one parameter family, etc.) of probability measures on G such that (3.1)

ˆ γ 6= 0. lim m cu (γ) = 0, γ ∈ G,

u→∞

Denote by P the orthogonal projection of L2 (X) on the subspace of G-invariant elements. Then (3.2)

T (u) =

in the strong operator topology.

Z

Tg dmu (g) → P

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E RGODIC T HEORY AND C ONFIGURATIONS

P ROOF : For f ∈ L2 (X), the function g 7→< Tg f, f > is positive-definite on G ˆ. and is in fact the Fourier transform of νf (γ)—the spectral measure of f on G R 2 ˆ , and Recall that kf k = dνf , that νP f is the part of νf carried by {0} ⊂ G (u) that T P f = P f for all u. Thus T (u) f − P f = T (u) (f − P f )

and by (3.1) kT

(u)

2

f − P fk =

Z

|m cu (γ)|2 d(νf − νP f ) → 0.

J

Theorem 3.2. Let Tz be an R2 -action on X, B, µ , and let P denote the or


thogonal projection L2 (X) onto the subspace of Tz -invariant functions. Then, for 0 ≤ α < β ≤ 2π , 1 lim R→∞ β − α

(3.3)

Z

β

TReiϑ dϑ = P

α

in the strong operator topology for L2 (X). P ROOF : Apply Theorem 3.1. The parameter u = R and mu is the normalized arc length in the arc Reiϑ , α ≤ ϑ ≤ β . To check condition (3.1) we write ζ = (ξ, η) = reiϕ and 1 m d R (ζ) = β−α

Z

β iR(ξ cos ϑ+η sin ϑ)

e

α

1 dϑ = β−α

Z

β

eiRr cos(ϑ−ϕ) dϑ.

α

Apply Van der Corput’s Lemma (Zygmund 1955) to obtain   −1/2 −1/2 m d (ζ) = O r R R

4

as

R → ∞.

J

First Application to Subsets of Positive Density in R2

Theorem 4.1. Let E ⊂ R2 be a subset of positive density, D(E) > 0. Let ε > 0 be given, as well as numbers 0 ≤ α1 < β1 < α2 < β2 < · · · < αN < βN ≤ 2π . Then for all sufficiently large R there exist points z0 , z1 , . . . , zN ∈ E such that, writing zj − z0 = rj eiϑj , 0 ≤ ϑj < 2π , we have

(i) |rj − R| < ε (ii) αj < ϑj < βj for j = 1, 2, . . . , N .

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E RGODIC T HEORY AND C ONFIGURATIONS

0 ∈ E satisfying P ROOF : If δ is small and we find points z00 , z10 , . . . , zN δ

(i’) rj0 = R (ii’) αj0 < ϑ0j < βj0 for j = 1, 2, . . . , N 0

with zj0 − z00 = rj0 eiϑj and with αj < αj0 < βj0 < βj , then there will be points z0 , z1 , . . . , zN ∈ E as required. We shall use Proposition 2.2 to show the existence of the desired configurations in Eδ by way of intersection properties of E˜ . To obtain the relevant properties of E˜ we use Theorem 3.2. The operator P in Theorem 3.2 is a positive self-adjoint operator, so that hP f, f i ≥ 0 for all f ∈ L2 (X); also, P 1 = 1. Taking f = 11A − µ(A) for a subset A ⊂ X , we deduce that hP 11A , 11A i ≥ µ(A)2 . This implies that for a.e. x ∈ A, P 11A (x) > 0. For if B = {x ∈ A : P 11A (x) = 0} we will have hP 11B , 11B i ≤ hP 11A , 11B i = 0, so that µ(B) = 0. Now apply Theorem 3.2 to the function 11E˜ to obtain for each j , j = 1, 2, . . . , N , 1 0 βj − αj0

(4.1)

Z

βj0

α0j

TReiϑ 11E˜ dϑ → P 11E˜

in L2 X, B, µ as R → ∞, where X, B, µ, Tu is the system attached to E ⊂ R2 . Since the function of the right in (4.1) is positive for almost every x ∈ E˜ , it follows that if R is sufficiently large, the expression on the left will also be ˜ positive for all x ∈ E˜ but for, say, a subset of measure < µ(2NE) . Hence for at least half of the points x ∈ E˜ , all N of the averages in (4.1), j = 1, 2, . . . , N are positive and the product





1 QN 1

(βj0 − αj0 )

Z

β10

α01

···

Z

0 N βN Y

α0N

(TReiϑj 11E˜ )dϑ1 · · · dϑN

1

is positive. Multiplying by 11E˜ and integrating over X , we conclude that for 0 , β 0 ), some ϑ1 ∈ (α10 , β10 ), ϑ2 ∈ (α20 , β20 ), . . . , ϑN ∈ (αN N ˜ ∩ T −1iϑ E ˜ ∩ · · · ∩ T −1iϑ E) ˜ > 0. µ(E Re 1 Re N

Proposition 2.2 now gives the desired result.

J

Because of the approximative nature of Theorem 4.1 it is easily seen that the result will be valid for a set E if it is true for arbitrarily small thickenings Eδ . Thus it will be true if each Eδ , δ > 0, has positive upper density. This happens, for example, if E is a subset of the lattice Λ = cZ2 ⊂ R2 , c > 0, that has positive upper density in the lattice. This gives the following.

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E RGODIC T HEORY AND C ONFIGURATIONS

Theorem 4.2. Let E be a subset of positive upper density in the lattice cZ2 . Given ε > 0, and numbers 0 ≤ α1 < β1 < α2 < β2 < · · · < αN < βN ≤ 2π , then for all sufficiently large R there exist points z0 , z1 , . . . , zN ∈ E such that, writing zj − z0 = rj eiϑj , 0 ≤ ϑj < 2π , we have (i) |rj − R| < ε (ii) αj < ϑj < βj for j = 1, 2, . . . , N .

5

Proof of Theorem A

We can now prove Theorem A. P ROOF : Let E ⊂ R2 be measurable and with positive upper density. Denote by Qk the partition of R2 into squares of side 2−k determined by the lattice 2−k Z2 . A number β > 0 is eligible for Qk if the set of squares Q ∈ Qk in which the relative measure of E exceeds β has positive upper density. It is clear that if β is eligible for Qk it is eligible for Qk+l for l > 0, since every Q ∈ Qk in which the relative measure of E exceeds β contains at least one Q0 ∈ Qk+l with the same property. Define β ∗ = sup{β : ∃k such that β is eligible for Qk }; the positive upper density of E insures that β ∗ > 0. The key observation now is that if β < β ∗ but is very close to it and if we partition a typical square of Qk in which the relative measure of E exceeds β into its Qk+l subsquares, all of the 22l subsquares will have the property that the relative measure of E in them exceeds β ∗ /2. The proper order of quantifiers here is: for any l > 0 there exists βl < β ∗ such that the above is valid for β > βl . The observation is that if the relative measure of E in one of the subsquares is lower than the average, other subsquares have to compensate, but none of them, typically, can exceed the average by more than β ∗ − β . We now set N = 30(β ∗ )−2 , l >> log N, β ∗ > β > βl , k large enough so that β is eligible for Qk , and denote by F the set of lower left-hand corners of the squares Q ∈ Qk in which the relative density of E exceeds β , and are “typical” in the sense discussed above. F ⊂ 2−k Z2 and has there positive upper density. We apply Theorem 4.2 πj πj with ε 0 such that given any R > R0 , we have a configuration z0 , . . . , zN ∈ 2−k Z2 such πj that zj − z0 = rj eiϑj with |rj − R| < ε and |ϑj − 3N | < N12 . We denote by Q0 , . . . , QN the corresponding squares in Qk , and set Ej = E∩Qj , j = 0, . . . , N ; by the definition of F , m(Ej ) > βm(Qj ) = β2−2k , and for convenience we normalize all measures by a factor 2k so that m(Qj ) = 1 and m(Ej ) > β . Our final step is to evaluate the measure of the set of points in Q0 which are at distance R from some point in ∪N j=1 Ej and show that the measure exceeds

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E RGODIC T HEORY AND C ONFIGURATIONS

1 − β . Once we do that we are done because it implies that this set must have positive intersection with E0 . Denote by Gj the subset of Q0 of points whose distance from some point in Ej is R. Gj is a union of circular arcs, intersection of Q0 with circles of radius R centered at points of Ej . Divide Qj into its Qk+l subsquares, and in each of the principal diagonal subsquares find a subset of Ej contained in a horizontal segment of full length and of relative linear measure equal to β . Denote the set so obtained by Ej0 and the corresponding subset of Q0 by G0j . We shall estimate the (planar) measures of G0j , and G0i ∩ G0j . G0j is a union of arcs from circles of radius R, and to estimate the measures

in question we approximate these arcs by line segments. The following is an approximate description of G0j . Through the lower-left and upper-right vertices of Q0 pass two lines orπj thogonal to the direction ϑ = 3N . Divide the strip formed by these lines into πj l 2 equal strips. Let Sj be the union of the lines also orthogonal to ϑ = 3N such l that Sj meets each of the 2 strips in a fixed proportion βj of the strip. The βj will be bounded from below. Then G0j is approximately Sj ∩ Q0 . Let P be a parallelogram formed by intersecting one of the narrow strips πj with one of the narrow strips corresponding to ϑ = corresponding to ϑ = 3N πi . We have 3N m(Sj ∩ P ) = βj m(P ),

m(Si ∩ P ) = βi m(P ),

m(Si ∩ Sj ∩ P ) = βi βj m(P ).

By choosing l very large we can assume that Q0 differs by an arbitrarily small amount from the union of parallelograms such as P. Hence we will have m(G0j ) ≈ βj

m(G0i ) ≈ βi

m(G0i ∩ G0j ) ≈ βi βj .

Let β 0 = inf βj . We now show that for l large (5.1)

m

[


G0j > 1 −

1 −δ N β 02

for arbitrarily small δ > 0. Set ϕi = βi − 11G0i on Q0P . We consider the situation m(G0i ) = βi and m(G0i ∩ G0j ) ≈ βi βj . On Q0 \ ∪G0j , ϕj ≥ N β 0 and so m Q0 \

[

X 2
G0j N 2 β 02 ≤ ϕi Q0 .

Now hϕi , ϕi iQ0 = βi − βi2 ,

hϕi , ϕj i = 0

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E RGODIC T HEORY AND C ONFIGURATIONS

P

so that

2 P P ϕi Q0 = βi − βi2 ≤ N and m Q0 \

[


G0j ≤

1 . N β 02

This would give m

[


G0j ≥ 1 −

1 N β 02

and, since l can be chosen very large, we have (5.1). Finally we note that from our construction β=

πj β cos 3N

cos

πj 3N

+ sin

πj 3N

≥

β 1 + tan β3

.

Hence, choosing N large after β has been prescribed, we can assure that some G0i meets E0 . This proves Theorem A. J

6

A Recurrence Property of R2-Actions

In this section we prove a certain recurrence result for R2 -actions from which Theorem B will follow. We begin with a lemma regarding mean values of vectors in a Hilbert space.

Lemma 6.1. Let {um } be a bounded sequence of vectors in a Hilbert space H. Assume that

(6.1)

γm

N 1 X hun , un+m i = lim N →∞ N n=1

exists, and that (6.2)

M 1 X lim γm = 0. M →∞ M m=1

Then

N 1 X un → 0 N 1

in the norm of H.

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E RGODIC T HEORY AND C ONFIGURATIONS

P ROOF : We choose M large so that the average in (6.2) is small and we choose N large with respect to M . Having done so the two expressions N 1 X un , N

N M 1 XX un+m MN

n=1

n=1 m=1

will be close, the vectors un being bounded. In general one has N N

1 X

2 1 X

yn ≤ kyn k2 ; N N n=1

n=1

so up to a small error N

1 X

2

un N n=1

will be bounded by N M N X

2 1 X 1 X

1

un+m = N M NM2 n=1

m=1

M X

hun+m1 , un+m2 i.

n=1 m1 ,m2 =1

Let N → ∞ and it is easily seen that this expression goes to 0.

J

The same proof yields the following uniform version.

Lemma 6.2. For each ξ in some index set Ξ let un (ξ) ∈ H, such that all the un (ξ) are uniformly bounded. Assume that for each m the limits N 1 X γm (ξ) = lim hun (ξ), un+m (ξ)i N →∞ N n=1

exist uniformly, and that M 1 X γm (ξ) = 0 M →∞ M

lim

m=1

uniformly. Then N 1 X un (ξ) → 0 N n=1

in H, uniformly in ξ .

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E RGODIC T HEORY AND C ONFIGURATIONS

We shall need the following notion. D EFINITION : An action of a locally compact abelian group
G by measure preserving transformations Tg of a measure space X, B, µ is a Kronecker action if X is a compact abelian group, µ = Haar measure on X , and we have a homomorphism τ : G 7→ X with τ (G) a dense subgroup of X and Tg (x) = τ (g) + x.


Theorem 6.3. If X, B, µ, Tg is an ergodic measure preserving action of an abelian group G then there is a map π : X 7→ Z where Z is a compact abelian group, and a Kronecker action Tg on Z so that Tg π(x) = π(Tg x) for x ∈ X . For every character χ on Z the function χ0 (x) = χ(π(x)) satisfies χ0 (Tg x) = χ(τ (g) + π(x)) = χ(τ (g))χ0 (x)

and so is an eigenfunction of the G-action, and, moreover, every eigenfunction of the G-action comes about this way. We refer the readers to Furstenberg (1981) for the proof of this. The next theorem is a consequence of the fact that the eigenvectors of the tensor product of two unitary operators are spanned by tensor products of the eigenvectors.

Theorem
6.4. Let T be a measure preserving transformation on the space

X, B, µ and let S be a measure preserving transformation on the space (Y, D, ν). If F ∈ L2 (X ×
Y, B × D, µ × ν) satisfies F (T x, Sy) = F (x, y) a.e., and if f ∈ L2 X, B, µ is orthogonal to all eigenfunctions of the transformation f 7→ T f where T f (x) = f (T x), then Z F (x, y)f (x)dµ(x) = 0

a.e. on Y . We now take G = R2 and we consider an ergodic R2 -action on a space
X, B, µ . The following proposition is also presented without proof. The proof is based on the notion of the spectral measure (class) of an R2 -action and the manner in which this determines the spectral measure for the restriction of the action of subgroups on R2 .

Proposition 6.5. Let X, B, µ, Tu denote an ergodic R2 -action. But for a


countable set of 1-dimensional subgroups of R, each subgroup Ru0 contains at
most a countable subset of elements v for which Tv is not ergodic on X, B, µ .

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E RGODIC T HEORY AND C ONFIGURATIONS




Lemma 6.6. Let
X, B, µ, Tu denote an ergodic action, and suppose Tv is

ergodic on X, B, µ . Then every eigenfunction of Tv is an eigenfunction for the R2 -action. P ROOF : Assume ϕ(Tv ) = λϕ(x); then for each u, ϕ(Tu Tv x) = ϕ(Tv Tu x) = λϕ(Tu x).

Since |λ| = 1, ϕ(Tu x)ϕ(x) is invariant under x 7→ Tv x and, by ergodicity of Tv , is a constant. Moreover |ϕ(x)| is constant by ergodicity so that ϕ(x) =

c ϕ(x)

and we conclude that ϕ(Tu x) = const ϕ(x).

J

The foregoing results are combined in the following.

Proposition 6.7. Let X, B, µ, Tv be an ergodic action of R2 and let v1 , v2


be such that Tv1 and Tv2 −v1 act ergodically. Let f, g be bounded measurable functions on X and suppose that f is orthogonal to all eigenfunctions of the R2 -action. Then for all w1 , w2 ∈ R2 N 1 X Tw1 +nv1 f Tw2 +nv2 g → 0 N

(6.3)

n=1


in L2 X, B, µ , uniformly in (w1 , w2 ).

P ROOF : This
will be an application of Lemma 6.2 with ξ = (w1 , w2 ), H = L2 X, B, µ , and un (ξ) = Tw1 +nv1 f Tw2 +nv2 g.

We have hun (ξ), un+m (ξ)i =

Z

Tw1 +nv1 f Tw1 +nv1 +mv1 f¯ Tw2 +nv2 g Tw2 +nv2 +mv2 g¯dµ

=

Z

(Tw1 −w2 f Tw1 −w2 +mv1 f¯)Tvn2 −v1 (gTmv2 g¯)dµ.

Since Tv2 −v1 acts ergodically Z N 1 X n Tv2 −v1 (gTmv2 g¯) → gTmv2 g¯ dµ N n=1

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E RGODIC T HEORY AND C ONFIGURATIONS

and this expression is independent of (w1 , w2 ). Hence N 1 X hun (ξ), un+m (ξ)i → γm N n=1

where Z

Tw1 −w2 f Tw1 −w2 +mv1 f¯dµ Z Z ¯ = f Tmv1 f dµ gTmv2 g¯dµ

γm =

and this convergence is uniform in (w1 , w2 ). Finally ZZ M 1 X γm → f (x)g(y)F (x, y)dµ(x)dµ(y) M 1

where F (x, y) = lim

1 X¯ m f (Tv1 x)¯ g (Tvm2 y) M

which is well defined by the ergodic theorem. Now by Lemma 6.6 since f is orthogonal to all eigenfunctions of the R2 -action, it is orthogonal to all eigenfunctions of Tv1 . But clearly F (Tv1 x, Tv2 y) = F (x, y)

so that we may apply Theorem 6.4 to conclude that Z

and hence that

f (x)F (x, y)dµ(x) = 0

M 1 X γm → 0. M 1

This yields the proposition.

J

Let X, B, µ, Tv denote an ergodic R2 -action and let (Z, Tv ) represent the
Kronecker factor of X, B, µ, Tv . We have a map π : X 7→ Z which defines a "disintegration" of the measure µ to measures µz , z ∈ Z , with µz supported for each z by π −1 (z). The map


f 7→

Z

f dµz = fˆ(z)

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E RGODIC T HEORY AND C ONFIGURATIONS

takes L2 (X) to L2 (Z). If we lift functions on Z to X then the foregoing map represents the projection of L2 (X) to L2 (Z) ◦ π ⊂ L2 (X) and by Theorem 6.3, L2 (Z) ◦ π is the subspace of L2 (X) spanned by eigenfunctions of {Tv }. It follows that for each f ∈ L2 (X), f − f ◦ π is orthogonal to the subspace of L2 (X) spanned by eigenfunctions. From this it is easy to deduce from the foregoing proposition:

Theorem 6.8. Let X, B, µ, Tv be an ergodic
R2 -action; let (Z, Tv ) denote


ˆ denote the correits Kronecker factor, and let f, g, h ∈ L∞ X, B, µ . If fˆ, gˆ, h
sponding functions on Z , and if Tv1 and Tv2 −v1 act ergodically on X, B, µ , then N Z 1 X f (Tw1 +nv1 x)g(Tw2 +nv2 x)h(x)dµ N n=1 N Z 1 X ˆ fˆ(Tw1 +nv1 ξ)ˆ − g (Tw2 +nv2 ξ)h(ξ)dξ N Z n=1

converges to 0 as N → ∞, uniformly in (w1 , w2 ). The main result of this section is the following theorem. In formulating this theorem we identify R2 with C and we can then multiply elements of C.


Theorem 6.9. Let X, B, µ, Tz be a measure preserving action of C. Let A ∈ B with µ(A) > 0 and let ω ∈ C. There exists l0 such that for all l > l0 there will exist z ∈ C with |z| = l for which −1 µ(A ∩ Tz−1 A ∩ Tωz A) > 0.

P ROOF : One first shows by a standard argument that it suffices to treat the case of an ergodic C-action. By Proposition 6.5 we can find z1 so that Tz1 and T(ω−1 )z1 are both ergodic. We will later impose a further restriction on z1 which will be consistent with the present restriction. Set f = 11A and consider fˆ(ξ) defined on Z . We see that fˆ(ξ) is a nonnegative function which is strictly positive on a set of positive measure of Z . There will be a neighborhood W of the identity of Z such that if w1 , w2 ∈ W , Z

fˆ(ξ)fˆ(ξ + w1 )fˆ(ξ + w2 )dξ > a > 0

for some appropriate a. Define a homomorphism σ : C 7→ Z × Z by σ(z) = (τ (z), τ (ωz))

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E RGODIC T HEORY AND C ONFIGURATIONS

(see the definition of a Kronecker action), and let Ω = σ(C). A non-trivial character on Ω restricts to a non-trivial character on C which has the form 0 χ(z) = ei N 2L n=1

whenever N > L, for all w1 ∈ C. Now apply Theorem 6.8 to deduce that there exists N0 so that if N > N0 N 1 X a −1 µ(A ∩ Tw−1 A ∩ Tωw A) > , 1 +nz1 1 +nωz1 N 3L n=1

for all w1 ∈ C. In particular, for each w1 ∈ C, ∃n ≤ N0 with −1 µ(A ∩ Tw−1 A ∩ Tωw A) > 1 +nz1 1 +nωz1

a . 3L

Our G-actions are required to be continuous in the sense that for any measurable set A ⊂ X , ε > 0, ∃ a neighborhood of the identity V ⊂ G so that for a g ∈ V , µ(Tg A4A) < ε. Take ε < ∗L . Then if w0 is sufficiently close to w1 + nz1 we will have (6.4)

−1 µ(A ∩ Tw−1 0 A ∩ Tωw 0 A) >

a a a − = . 3L 4L 12L

To prove the theorem, choose w1 a vector length 1 which is orthogonal to z1 . Then for large l n2 |z1 |2 |lw1 + nz1 | = l + =l+O |lw1 + nz1 | + l

  1 l

for n in restricted range. It follows that for large l we can find w0 with |w0 | = l so that (6.6) is true. This proves the theorem. J

16

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E RGODIC T HEORY AND C ONFIGURATIONS

Proof of Theorem B

Theorem B is an immediate consequence of the foregoing theorem and Proposition 2.2. For E ⊂ R2 we form the R2 -action described in Section 2. Consider u, v ∈ R2 as complex numbers and with v = ωu, and apply Theorem 6.9 to ˜ and this ω . For each z with A=E ˜ ∩ T −1 ET ˜ −1 E) ˜ >0 µ(E z ωz

and for any δ > 0 there exist α, β, γ ∈ Eδ with β − α = z , γ − α = ωz and so the triangle {α, β, γ} is congruent to {0, z, ωz} which is congruent to {0, lu, lv} with l = |z|/|u|. We conclude with the example given by J. Bourgain (1986) which shows that the configurations of Theorem B may not exist in E itself. 1 Let E = {(x, y) ∈ R2 : ∃n with |x2 + y 2 − n| < 10 }, and let the "triangle" of Theroem B be {0, u, 2u}. It is easily checked that E has (uniform) density 1/5. Since for vectors v 0 , v 00 , kv 0 + v 00 k2 + kv 0 − v 00 k2 = 2kv 0 k2 + 2kv 00 k2

and so kv 0 k2 + kv 00 k2 − 2k

(7.1) 0

v 0 + v 00 2 v 0 − v 00 2 k = 2k k , 2 2

00

then if v 0 , v”, v +v ∈ E , the expression to the left of (7.1) differs from an 2 0 00 2 1 2 integer by less than 5 and so 2k v −v 2 k cannot be 2 an odd integer. This means 0 00 that kv − v k does not attain all possible large values in such a triple.

References Bourgain, J. (1986): A Szemerédi type theorem for sets of positive density in Rk . Isr. J. Math. 54, 307–316 Falconer, K.J., Marstrand, J.M. (1986): Plane sets with positive density at infinity contain all large distances. Bull. Lond. Math. Soc. 18, 471–474 Furstenberg, H. (1981): Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton, NJ Székely, L.A. (1983): Remarks on the chromatic number of geometric graphs. In: Fiedler, M. (ed.): Graphs and other combinatorial topics. B.G. Teubner, Leipzig, pp. 312–315 (Teubner-Texte Math., Vol. 59) Zygmund, A. (1955): Trigonometric series. Dover Publications.