A note on configurations in sets of positive density which occur at all large scales

0.22 MB·English

by  Ian D. Morris

A NOTE ON CONFIGURATIONS IN SETS OF POSITIVE
DENSITY WHICH OCCUR AT ALL LARGE SCALES
3
1 IAND.MORRIS
0
2

n
a
J Abstract. Furstenberg,KatznelsonandWeissprovedintheearly1980sthat
every measurablesubsetofthe planewithpositivedensity atinfinityhas the
7
property that all sufficiently large real numbers are realisedas the Euclidean
1
distancebetweenpointsinthatset. Theirproofusedergodictheorytostudy

translations onaspaceofLipschitzfunctions correspondingtoclosedsubsets
]
O of theplane,combined withameasure-theoretical argument. Weconsider an
alternativedynamicalapproachinwhichthephasespaceisgivenbythesetof
C measurablefunctions fromRd to[0,1],whichweviewasacompact subspace
. of L∞(Rd) in the weak-* topology. The pointwise ergodic theorem for Rd-
h
actions impliesthatwithrespecttoanytranslation-invariantmeasureonthis
t
a space, almost every function is asymptotically close to a constant function
m at large scales. This observation leads to a general sufficient condition for
a configuration to occur in every set of positive upper Banach density at all
[
sufficiently largescales, extending arecent theorem ofB.Bukh. Toillustrate

1 theuseofthiscriterionweapplyittoproveanewresultconcerningthree-point
v configurations inmeasurablesubsetsoftheplanewhichformtheverticesofa
9 triangle with specified area and side length, yielding a new proof of a result
0 relatedtoworkofR.Graham.
2 Keywordsandphrases: EuclideanRamseytheory,measurablesets,point-
4 wiseergodictheorem. MSCPrimary05D10,22A99,Secondary37A15,37A30.
.
1
0 1. Introduction and main results
3
1 Given a Lebesgue measurable subset A of Rd, let us define the upper density of
: A to be the quantity
v
m(A Q(0,t))
Xi d(A):=limsup ∩
m(Q(0,t))
t→∞
r
a where m denotes d-dimensional Lebesgue measure and Q(x,t) denotes the closed
solid cube in Rd with side length t, centre x, and sides oriented parallel to the
co-ordinate axes. Define the upper Banach density of A to be the quantity
m(A Q(x,t))
d∗(A):=limsup sup ∩ .
t→∞ x∈Rd m(Q(x,t))
ThisarticleismotivatedbythefollowingresultwhichwasprovedbyH.Furstenberg,
Y. Katznelson and B. Weiss [9] in response to a conjecture of L. A. Sz´ekely:
Theorem 1.1 (Furstenberg–Katznelson–Weiss). Let A R2 be a Lebesgue mea-
⊆
surable set such that d(A) > 0. Then for all sufficiently large real numbers t we
may find points x,y A which are separated by a Euclidean distance of precisely t.
∈
The proof given originally by Furstenberg, Katznelson and Weiss combines an
ergodic-theoretic argument based on translations of closed subsets of the plane
1
2 IAND.MORRIS
with a subsequent measure-theoretical argument. Shorter alternative proofs based
on demonstrating the positivity of the integral
χ (x)χ (x+ty)d(x,y)
A A
ZR2×S1
weresubsequentlypresentedbyK.FalconerandJ.Marstrand[5]andJ.Bourgain[3]
using techniques from geometric measure theory and Fourier analysis respectively,
and a probabilistic proof was recently given by A. Quas [16]; the last two of these
results require only the weaker condition d∗(A) > 0. Bourgain in fact proves the
followingstrongerresult: ifV isaconfigurationofdpointsinRd whichdonotliein
a(d 2)-dimensionalsubspace,andA Rd isameasurablesetwithpositiveupper
− ⊆
Banach density, then A contains a set isometric to tV for all sufficiently large real
numbers t. On the other hand, in the case where V consists of three evenly-spaced
colinearpoints Bourgainexhibitedanexample ofa positive-densitymeasurableset
A R3 which is free from isometric copies of tV at an unbounded set of scales t.
⊂
In this article we are interested in giving general conditions under which a con-
figurationorpropertymustbe satisfiedinallpositive-densitysetsatallsufficiently
large scales. B. Bukh [4, Theorem 8] has previously given a sufficient condition of
this kind which we now describe. Let us say that a property is a function P from
thesetM(Rd)ofallLebesguemeasurablesubsetsofRd to 0,1 . Weconsiderthat
A Rd haspropertyP ifP(A)=1anddoesnothaveprop{erty}P ifP(A)=0. We
⊂
shallsaythatAhaspropertyP atalllargescalesifP(t−1A)=1forallsufficiently
large real numbers t. The following definition paraphrases Bukh [4]:
Definition 1.2. We shall say that a property P has supersaturable complement
if there exists a function I : M(Rd) [0,+ ] which satisfies the following seven
P
→ ∞
axioms:
(i) There exists m(P)>0 such that if d(A)>m(P) then P(A)=1.
(ii) If A B then I (A) I (B) and P(A) P(B).
P P
⊆ ≤ ≤
(iii) If I (A)>0 then A has property P.
P
(iv) For all v Rd we have P(A)=P(A+v) and I (A)=I (A+v).
P P
∈
(v) There exists r > 0 such that if all points of A are at least distance r
1
away from all points of A , then I (A A ) I (A )+ I (A ) and
2 P 1 2 P 1 P 2
∪ ≥
P(A A )=max P(A ),P(A ) .
1 2 1 2
(vi) There∪exist ε > 0 {and a strictly }positive function f: (0,+ ) R such
∞ →
that if the set
x Rd: m(Q(x,δ) A)>(1 ε)m(Q(x,δ))
{ ∈ ∩ − }
has property P, then I (A) f(δ).
P
≥
(vii) If A Q(0,R) then
⊂
I (A) g (ε)I x Rd: m(Q(x,δ) A)>εm(Q(x,δ)) h (ε,δ)Rd
P P P P
≥ ∈ ∩ −
where g (ε)>(cid:0)(cid:8)0 and lim h (ε,δ)=0 for each fixe(cid:9)d(cid:1)ε.
P δ→0 P
Bukh’s work in fact considers necessary conditions for measurable sets to omit
certainstructuresasopposedtosufficientconditionsformeasurablesetstocontain
certain structures, and in respect of this the above description inverts the object
consideredbyBukh: P is apropertywith supersaturablecomplementinthe above
sense if and only if 1 P is a supersaturable property in the originalsense defined
−
CONFIGURATIONS IN MEASURABLE SETS 3
in [4]. Bukh obtains a number of interesting results concerning supersaturable
properties, of which we single out the following:
Theorem 1.3 (Bukh, [4, Theorem 8]). Let P: M(Rd) 0,1 be a property with
→{ }
supersaturable complement such that m(P)<1. If A Rd satisfies d(A)>0, then
⊂
A has property P at all large scales.
Bukhinfactprovesastrongerresultconcerningthe simultaneoussatisfactionof
afinitenumberofpropertiesP ,...,P withsupersaturablecomplementatwidely-
1 n
differing scales. Since our interest in this article is in direct extensions of Theorem
1.1 we restrict our attention to the case of a single property occuring at all large
scales. In this article we shall use ergodic theory to give a much weaker suffi-
cient condition for a configurationto appear at all sufficiently large scales in every
positive-density measurable set.
In order to state our results we require a few items of notation. We shall use
the symbol S to denote the set of all f L∞(Rd) such that 0 f 1 al-
d
∈ ≤ ≤
most everywhere, and we equip this set with the weak-* topology inherited from
L∞(Rd) L1(Rd)∗ withrespecttowhichitiscompactandmetrisable. Weusethe
symbol 1≃ S to refer to the almost surely constant function with value 1.
d
∈
Definition 1.4. We say that a property P: M(Rd) 0,1 is δ-mild, where
→ { }
δ (0,1], if it satisfies the following properties:
∈
(i) There exists an open set U S such that if χ U then P(A)=1.
d A
(ii) The open set U contains δ1⊂. ∈
(iii) For all v Rd we have P(A)=P(A+v).
∈
We shall say that a property is mild if it is δ-mild for every δ >0.
The main result of this article is the following:
Theorem 1.5. Let P: M(Rd) 0,1 be a δ-mild property. If the upper Banach
density of A Rd is equal to δ→the{n A}has property P at all large scales.
⊆
Theorem1.1mayeasilybededucedfromTheorem1.5viathefollowingproposi-
tion,whichimpliesthatthepropertyofcontainingtwopointsatEuclideandistance
1 from one another is a mild property. Proposition1.6 is provedin 5 below. Here
§
and throughout the article we let σ denote normalised one-dimensional Lebesgue
measure on S1 R2.
⊂
Proposition 1.6. The set
U:= g S : g(x)g(x y)dσ(y)dx>0
2
∈ −
(cid:26) ZZ (cid:27)
is open and contains δ1 for every δ (0,1].
∈
It is not difficult to see that δ-mild properties are weaker than properties with
supersaturable complement in several respects: for example, the property that A
has a translate A+v such that 1 < m((A+v) [0,1]d) < 3 is clearly 1-mild —
4 ∩ 4 2
for example, we could define U := f S : 1 < f < 3 — but does not
{ ∈ 2 4 [0,1]2 4}
satisfyDefinition1.2(ii). Ontheotherhand,ifP isapropertywithsupersaturable
R
complement such that m(P) < 1 then it is necessarily mild. Given f S and a
d
∈
property P with supersaturable complement, let us say that P(f) =1 if for every
Lebesgue measurable function g: Rd [0,1] which is almost everywhere equal
→
4 IAND.MORRIS
to f we have P( x: g(x)>0 ) = 1. It follows from [4, Lemma 12] that the set
U := f S : P({f) = 1 is o}pen, and by Definition 1.2(i) it follows that δ1 U
d
{ ∈ } ∈
for every δ (0,1]. Thus if a property has supersaturable complement then it is
∈
mild, but the converse implication is not true in general.
The followingapplicationofTheorem1.5givesasomewhatlesscontrivedexam-
ple of a property which is mild but does not have supersaturable complement:
Theorem 1.7. For every v S1 R2 let v⊥ denote the point on S1 which lies
∈ ⊂
anticlockwise from v at a distance of one quarter-circle. For every M > 0 the set
of all g S such that the integral
2
∈
∞
(1.1) e−sg(x)g(x+y)g x+2αy⊥+sy dsdσ(y)dx
ZR2ZS1Z0
is nonzero for every α (0,M] contains (cid:0)an open neighbo(cid:1)urhood of δ1 for every
∈
δ (0,1]. In particular, the property P defined by P (A) = 1 if and only if for
M M
∈
each α (0,M] we may find points x,y,z A which form the vertices of a triangle
∈ ∈
with area α and in which at least one side has length exactly 1 is a mild property.
The proof of Theorem 1.7 is given in 6 below. The reader should not have
§
difficulty in constructing for each sufficiently large r >0 a pair of measurable sets
A ,A R2 such that every point of A is separated from every point of A by at
1 2 1 2
⊂
least distance r, and such that A A has the property P defined in Theorem
1 2 1
∪
1.7 but each of A and A individually does not. This shows that the property
1 2
described in Theorem 1.7 does not have supersaturable complement since it does
not satisfy Definition 1.2(v).
In addition to generalising Theorem 1.1 and illustrating the separation between
mild properties and properties with supersaturable complement, this result yields
the following direct corollary:
Corollary 1.8. Let A R2 be a Lebesgue measurable set such that d∗(A)>0, and
⊆
let α be any positive real number. Then there exist points x,y,z A which form
∈
the vertices of a triangle of area α.
A proof of Corollary 1.8 based on Szemer´edi’s theorem was previously given by
R. L. Graham [11]. Graham’s result also has the stronger feature that the vectors
x z and y z may be chosen parallel to the co-ordinate axes.
− −
2. Dynamical formulation and technical results
Inthetraditionofearlierergodic-theoreticinvestigationsoftranslation-invariant
combinatorial structures our proof of Theorem 1.5 operates by investigating the
translation dynamics on a phase space comprising a compactification of the set of
indicator functions of the sets of interest. In Furstenberg, Katznelson and Weiss’
investigations of subsets of Zd (see for example [6, 8, 10]) the set of all indica-
tor functions Zd 0,1 is already compact in the infinite product topology on
→ { }
Zd
0,1 andso no enlargementofthis spaceof functions is necessary. Furstenberg,
K{atz}nelsonandWeiss’investigationofsubsetsofRd,ontheotherhand,substitutes
for the indicator function Rd 0,1 of a measurable set a Lipschitz continuous
function Rd [0,1] given by→th{e dis}tance to the closure of the set in question,
→
and equips the set of such functions with the compact-uniform topology. In this
articlewetakethealternativeapproachofgrantingthe setofmeasurablefunctions
Rd 0,1 the topologywhichitinherits asa subsetofL∞(Rd) L1(Rd)∗ inthe
→{ } ≃
CONFIGURATIONS IN MEASURABLE SETS 5
weak-* topology. (As Furstenberg, Katznelson and Weiss’ approach to subsets of
Rd does not distinguish between sets with different closures, so our approach does
not distinguish between sets which agree up to measure zero.) We remark that in
the context of functions Zd 0,1 these two approaches would be indistinguish-
→{ }
able,since inthat environmentthe infinite product topology,the compact-uniform
topology and the weak-* topology inherited from L1(Zd)∗ are all coincident.
For each v Rd we define a function : S S by ( f)(x) =f(x+v), and
v d d v
for each t > 0∈we also define a map :TS →S by ( fT)(x) = f(tx). It is not
t d d t
Z → Z
difficult to verify that each and each is a homeomorphism,that v is an
v t v
actionofRd,that = T foreveryZtandv,andthat converges7→unTiformly
ZtTv TtvZt Tvn
to in the limit as v v. We denote the collection of maps simply by ,
v n v
andTsay that a set B S→is -invariant if B =B for all v RTd. Finally, letTus
d v
define the upper Bana⊂ch densTity of a functioTn f S to be the∈quantity
∈
1
d∗(f):=limsup sup f(x)dx
t→∞ v∈Rd m(Q(v,t))ZQ(v,t)
which is of course analogous to the upper Banach density of a set: if f = χ for
A
some Lebesgue measurable set A Rd then d∗(A) = d∗(χ ). Theorem 1.5 is a
A
⊆
corollary of the following:
Theorem 2.1. Letf S andlet V S bea -invariant open set which contains
d d
the constant function d∈∗(f)1. Then ⊂f V foTr all sufficiently large t.
t
Z ∈
ToobtainTheorem1.5foragivenδ-mildpropertyP itsufficestoapplyTheorem
2.1 to the set V := v∈RdTvU and function f := χA, noticing that ZtχA = χt−1A
for all t > 0. We derive Theorem 2.1 using a dynamical argument in two parts:
S
the first part characterises the density of a set f in terms of space averages over
S with respect to translation-invariantmeasures, and the second shows that with
d
respect to any ergodic translation-invariant measure, almost every element of S
d
is approximately constant at large scales in a precise sense. In order to describe
these results we require some further definitions. Let us use the symbol to
denote the set of all Borel probability measures on S , which we equip witMh the
d
weak-* topology arising from that set’s identification with a subset of C(S )∗ via
d
theRieszrepresentationtheorem. ItfollowsfromtheBanach-Alaoglutheoremand
the separability of C(S ) that is compact and metrisable in this topology. We
d
let denote the set of all M-invariant Borel probability measures on S , which
T d
M T
isanonemptyclosedsubsetof . Thefollowingtheoremcharacterisesthe density
M
d∗(f) in terms of translation-invariantmeasures:
Theorem 2.2. Let f S and define X := f: v Rd . Then
d f v
∈ {T ∈ }
(2.1) d∗(f)=sup h(x)dµ(h): µ and µ(X )=1
T f
(ZZ[0,1]d ∈M )
and this supremum is attained by an ergodic measure.
The observation that the upper density of a function (or rather, the character-
istic function of a set) is positive if and only if an associated ergodic average is
positive for at least one ergodic measure is a staple of ergodic Ramsey theory and
arises in numerous works on the topic such as [1, 6, 7]. The fact that the upper
Banach density is exactly characterised by a supremum over ergodic measures in
themannerofTheorem2.2seemstoberelativelyunremarked,thoughweareaware
6 IAND.MORRIS
of [14, Lemma 1]. In the immediate context of subsets of Rd the above result is
not dissimilar to [9, Lemma 2.1], although that result is prevented from being an
equation by the possibility that a measurable set may have strictly lower density
than its closure.
The following result relating a space average of an invariant measure µ to the
behaviour of µ-typical elements at large scales is a straightforward consequence of
thepointwiseergodictheoremappliedtothedynamicsoftheaction onthephase
space S . T
d
Theorem 2.3. Let µ be an ergodic measure. Then
T
∈M
µ f S : lim f = h(x)dxdµ(h) 1 =1.
d t
( ∈ t→∞Z ZZ[0,1]d !· )!
LetusbrieflyindicatethederivationofTheorem2.1fromTheorems2.2and2.3.
Let U, δ and f be given. By Theorem 2.2 there exists an ergodic measure µ such
that µ(X ) = 1 and h(x)dµ(h) = d∗(f). It follows from Theorem 2.3 that
f [0,1]d
there exists h X such that lim h = d∗(f)1. Since d∗(f)1 belongs to the
openset U we∈havef RRh U for altl→su∞ffiZcitently larget. Since U is open andh X
t f
it follows that for eaZch s∈uch t there exists v Rd such that f U, and∈since
t v
f = f and −1U=U it follows tha∈t f U for aZll sTuch∈t as required.
TtvZt ZtTv Ttv Zt ∈
Thefollowingtwosectionscomprisethe proofsofTheorems2.3and2.2,andthe
twosectionssubsequenttothosecontaintheproofsofProposition1.6andTheorem
1.7. ThroughoutthesesectionswewillfrequentlyuseTonelliandFubini’stheorems
without comment.
3. Proof of Theorem 2.3
While it is perhaps more natural to state Theorem 2.2 before Theorem 2.3, the
proof of the former requires the latter so we shall prove the second theorem first.
The following general ergodic theorem due to E. Lindenstrauss [13] is convenient
for our argument(though see remark below). Here and throughout,the expression
A B denotes the symmetric set difference (A B) (B A).
△ \ ∪ \
Theorem 3.1 (Lindenstrauss). Suppose that Γ is a locally compact, second count-
able amenable group which acts bi-measurably on the left on a probability space
(X, ,µ) by measure-preserving transformations, and let m denote Haar measure
B
on Γ. Suppose that (F ) is a sequence of compact subsets of Γ with the following
n
two properties: firstly, for every nonempty compact set K Γ
⊂
m(F KF )
n n
lim △ =0,
n→∞ m(Fn)
and secondly, there exists a constant C >0 such that for all n 2
≥
n−1
m F−1 F Cm(F ).
n i!≤ n
i=1
[
Suppose finally that the action of Γ is ergodic. Then for every f L1(µ)
∈
1
µ x X: lim f(gx)dm(g)= fdµ =1.
(cid:18)(cid:26) ∈ n→∞m(Fn)ZFn Z (cid:27)(cid:19)
CONFIGURATIONS IN MEASURABLE SETS 7
We require only the case in which Γ = Rd acts continuously on a compact
metrisable space by homeomorphisms, in which case the measurability hypotheses
are satisfied trivially.
Proof of Theorem 2.3. Let us define a rational cube to be a compact set Q Rd
⊆
whichisequaltotheCartesianproductofdclosedintervalswithrationalendpoints
andequal,nonzerolengths,anddenotethesetofallrationalcubesbyQ. Weclaim
that for every Q Q,
∈
(3.1) µ f S : lim χ f =m(Q) h(x)dxdµ(h) =1.
d Q n
( ∈ n→∞Z Z ZZ[0,1]d )!
Before we provethe claim let us show that the truth of the claim implies the truth
of the theorem. Firstly we observe that for any fixed rationalcube Q and function
f S we have for all t>1
d
∈
1 1
χ f χ f = f f
(cid:12)(cid:12)(cid:12)(cid:12)Z QZt −Z QZ⌊t⌋ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1tdmZ(ttQQ −t⌊Qt⌋)d+Z⌊t⌋Q1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1 m( t Q)
≤ td △⌊ ⌋ t d − td ⌊ ⌋
(cid:18)⌊ ⌋ (cid:19)
m(Q t−1 t Q)+dt−1m(Q).
≤ △ ⌊ ⌋
Since this expression converges to zero as t it follows that the truth of the
→ ∞
claim implies
µ f S : lim χ f =m(Q) h(x)dxdµ(h) =1
d Q t
( ∈ t→∞Z Z ZZ[0,1]d )!
for every rational cube Q Q. The set Q being countable, this in turn implies
∈
µ f S : lim χ f =m(Q) h(x)dxdµ(h) for all Q Q =1,
d Q t
( ∈ t→∞Z Z ZZ[0,1]d ∈ )!
and since the linear span of the set of all characteristic functions of rational cubes
is dense in L1(Rd) it follows by a simple approximation argument that
µ f S : lim ϕ f = h(x)dxdµ(h) ϕ for all ϕ L1(Rd) =1.
d t
( ∈ t→∞Z Z ZZ[0,1]d Z ∈ )!
Since this is by definition equivalent to the statement
µ f S : lim f = h(x)dxdµ(h) 1 =1
d t
( ∈ t→∞Z ZZ[0,1]d · )!
we conclude that the truth of the claim implies the truth of the theorem.
Let us now prove the claim. For the remainder of the proof we fix a rational
cube Q Rd with side lengthr andcentre pointv. Letus write (u ,...,u ) :=
1 d ∞
max u⊂,..., u for all (u ,...,u ) Rd, and define F := nQ| for all n |1. If
1 d 1 d n
{| | | |} ∈ ≥
8 IAND.MORRIS
K Rd is any compact set then clearly
⊆
m(F (K+F )) m(nQ (K+nQ))
n n
lim △ = lim △
n→∞ m(Fn) n→∞ ndm(Q)
m(Q (n−1K+Q))
= lim △ =0.
n→∞ m(Q)
If n 2 and u F F for k 1,...,n 1 then clearly u n(v +r), so
k n ∞ ∞
≥ ∈ − ∈{ − } | | ≤ | |
for all n 2 we have
≥
n−1 (2v +2r)d
m F F 2dnd(v +r)d = | |∞ m(F ).
k n ∞ n
k=1 !− !≤ | | (cid:18) m(Q) (cid:19)
[
Thesequence(F )thereforesatisfiesthe requirementsofTheorem3.1withrespect
n
to the group Γ = Rd. Let Φ: S R be the functional Φ(f) := f which
d → [0,1]d
is continuous by the definition of the topology on S , and let denote the Borel
σ-algebra on S . Applying Theorem 3.1 to the medasure spacBe (SR, ,µ), group
d d
Γ=Rd, action v and function Φ we obtain B
v
7→T
1
(3.2) µ f S : lim Φ( f)dv = h(x)dxdµ(h) =1.
d v
( ∈ n→∞m(nQ)ZnQ T ZZ[0,1]d )!
It remains only to show that this is equivalentto the claimed expression(3.1). Let
us define
+ := nQ x, − := nQ x
Qn − Qn −
x∈[[0,1]d x∈\[0,1]d
for every n 1. When nr 1 is non-negative the sets + and − are rational
≥ − Qn Qn
cubes with side length respectively nr+1 and nr 1 such that − nQ +.
− Qn ⊂ ⊂ Qn
Inparticularwehavem( + −)=(nr+1)d (nr)d <2d(nr)d−1 foralln 1/r.
Qn \Qn − ≥
For each n 1 we have
≥
Φ( f)dv = f(x+v)dxdv = f(v)dvdx
v
T
ZnQ ZnQZ[0,1]d Z[0,1]dZnQ−x
and therefore in particular
1 1 1
f(v)dv Φ( f)dv f(v)dv
v
m(nQ)ZQ−n ≤ m(nQ)ZnQ T ≤ m(nQ)ZQ+n
and
1 1 1 1
f(v)dv χ f = χ f f(v)dv
nQ Q n
m(nQ)ZQ−n ≤ m(nQ)Z m(Q)Z Z ≤ m(nQ)ZQ+n
for all n 1/r. Since
≥
1 m( + −)
0 limsup f(v)dv f(v)dv lim Qn \Qn =0
≤ n→∞ m(nQ)(cid:18)ZQ+n −ZQ−n (cid:19)≤n→∞ ndm(Q)
we deduce that
1 1
lim Φ( f)dv χ f =0
v Q n
n→∞ m(nQ) T − m(Q) Z
(cid:12) ZnQ Z (cid:12)
(cid:12) (cid:12)
for every f S . We(cid:12) conclude that (3.2) implies (3.1) and the(cid:12) theorem is proved.
∈ d (cid:12) (cid:12) (cid:3)
CONFIGURATIONS IN MEASURABLE SETS 9
Remark. As analternativeto Lindenstrauss’ergodic theoremit wouldalso have
been possible for us to use a much earlier theorem due to A. Tempelman ([17], see
also[2,18]). Tempelman’sresulthowevercannotbeapplieddirectlytothesequence
(nQ) since it requires the additional hypotheses that the sequence (F ) must be
n
nestedandeachF mustincludethe origin. ToapplyTempelman’s theoreminour
n
context we would have to rewrite χ as a linear combination up to measure zero
nQ
of2d characteristicfunctionsofclosedcuboidsnK eachcontainingtheorigininits
i
boundary, apply Tempelman’s theorem individually to all of the sequences (nK ),
i
and then sum up to obtain the result (3.2). For example, to treat the case d = 1
βn
in this manner we would study the expression Φ( f)dv by writing it as the
αn Tv
βn αn
difference Φ( f)dv Φ( f)dv.
0 Tv − 0 Tv R
R R
4. Proof of Theorem 2.2
Let us use the notation (X ) to denote the set of all -invariant Borel
T f
probability measures on S wMhich give full measure to the compTact nonempty set
d
X . By the Krylov-Bogolioubov Theorem (X ) is nonempty. We claim that
f T f
M
given any µ (X ) we may find an ergodic measure µˆ (X ) such that
T f T f
∈M ∈M
(4.1) h(x)dxdµˆ(h) h(x)dxdµ(h).
≥
ZZ[0,1]d ZZ[0,1]d
By the definition of the topology of S the function h h is a continuous
d 7→ [0,1]d
mapfromS toR. Thefunctionν h(x)dxdν(h)isthusacontinuousaffine
d 7→ [0,1]d R
functionalonthemetrisabletopologicalspace (X ),whichisacompactconvex
subspace of the locally convex topoloRgRical spaMceTC(Sf)∗ equipped with its weak-*
d
topology. By applying a suitable version of Choquet’s theorem to the measure µ
(see e.g. [15, p.14]) it follows that there exists a Borel probability measure P on
(X ) such that
T f
M
h(x)dxdν(h)dP(ν) = h(x)dxdµ(h)
ZZZ[0,1]d ZZ[0,1]d
and P gives full measure to the set of extreme points of (X ), which is defined
T f
M
to be the set of all elements of (X ) which may not be written as a strict
T f
linear combination of two distincMt elements of (X ). Consequently P gives
T f
M
nonzero measure to the set of all µˆ (X ) which are extreme points and
T f
∈ M
also satisfy (4.1), and in particular this set is nonempty. Choose any measure µˆ
belongingto this set. Ifµˆ is notergodic,thenthere existsa measurablesetA S
d
such that 0 < µˆ(A) < 1 and A = A up to measure zero for all v Rd. In⊂this
v
T ∈
instancewemaythenwriteµˆasastrictlinearcombinationofthedistinctmeasures
ν ,ν (X ) defined by ν (B):=µˆ(A)−1µˆ(B A), ν (B):=µˆ(A)−1µˆ(B A)
1 2 T f 1 2
for Bo∈reMlsets B S . By virtue ofits being an ext∩reme point µˆ cannotbe wri\tten
d
⊆
as strict linear combination of distinct invariant measures and it follows that µˆ is
ergodic. This completes the proof of the claim.
Let us now prove the theorem. Using the weak-* continuity of the functional
g g we obtain
7→ [0,t]d
R 1 1
d∗(f)=limsup sup f =limsup sup g.
t→∞ v∈Rd m([0,t]d)Z[0,t]dTv t→∞ g∈Xf td Z[0,t]d
10 IAND.MORRIS
Givenµ (X ),chooseanergodicmeasureµˆ (X )suchthat(4.1)holds.
T f T f
∈M ∈M
Using Theorem 2.3 we find that for µˆ-almost-every g we have g X and
f
∈
1
lim g = lim g = h(x)dxdµˆ(h) h(x)dxdµ(h).
t→∞td Z[0,t]d t→∞Z[0,1]dZt ZZ[0,1]d ≥ZZ[0,1]d
Since µ is arbitrary it follows by combining these two expressions that d∗(f) is
greater than or equal to the right-hand member of (2.1).
Let us now prove the opposite inequality. Let (t )∞ be a sequence of real
n n=1
numbers tending to infinity and (v )∞ a sequence of vectors in Rd such that
n n=1
d∗(f)=lim t−d f. Withoutlossofgeneralityweassumethatt >1
n→∞ n [0,tn]dTvn n
for every n. For each n 1 define a Borel probability measure on X by
R ≥ f
1
µ := δ du.
n td Tvn+uf
n Z[0,tn]d
Since is compact and metrisable we may choose a strictly increasing sequence
M
ofintegers(n )∞ and measureµ suchthat lim µ =µ. For eachn 1
j j=1 ∈M j→∞ nj ≥
we have
h(x)dµ (h)= f(x)dx du
ZZ[0,1]d n Z[0,tn]d Z[0,1]dTvn+u !
= f(x+u)dudx
Tvn
Z[0,1]dZ[0,tn]d
f(w)dw
≥ Tvn
Z[0,tn−1]d
where we have used the fact that [0,t 1]d [0,t ]d x for all x [0,1]d. Since
n n
− ⊂ − ∈
we additionally have for each n 1
≥
1 m([0,t ]d [0,t 1]d) 2d
n n
f(u)du f(u)du \ − <
tdn (cid:12)(cid:12)Z[0,tn−1]dTvn −Z[0,tn]dTvn (cid:12)(cid:12)≤ tdn tn
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
it follows that
h(x)dµ(h)= lim h(x)dµ (h)
ZZ[0,1]d j→∞ZZ[0,1]d nj
1
liminf f(u)du=d∗(f).
≥ n→∞ tdn Z[0,tn−1]dTvn