ALMOST BI-LIPSCHITZ EMBEDDINGS AND ALMOST HOMOGENEOUS SETS 1. Introduction In ...

24 Pages·2009·0.33 MB·English

TRANSACTIONSOFTHE
AMERICANMATHEMATICALSOCIETY
Volume362,Number1,January2010,Pages145–168
S0002-9947(09)04604-2
ArticleelectronicallypublishedonAugust17,2009
ALMOST BI-LIPSCHITZ EMBEDDINGS AND ALMOST
HOMOGENEOUS SETS
ERICJ.OLSONANDJAMESC.ROBINSON
Abstract. Thispaperisconcernedwithembeddingsofhomogeneousspaces
into Euclidean spaces. We show that any homogeneous metric space can
be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-
Lipschitztowithinlogarithmiccorrections). Theimageofthissetisnolonger
homogeneous, but ‘almost homogeneous’. We therefore study the problem
of embedding an almost homogeneous subset X of a Hilbert space H into a
finite-dimensionalEuclideanspace. WeshowthatifX isacompactsubsetof
a Hilbert space and X−X is almost homogeneous, then, for N sufficiently
large, a prevalent set of linear maps from X into RN are almost bi-Lipschitz
betweenX anditsimage.
1. Introduction
Inthispaperweinvestigateabstractembeddingsbetweenmetricspaces, Hilbert
spaces, and finite-dimensional Euclidean spaces. Historically (starting with Bouli-
gand in [3]), attention has been on bi-Lipschitz embeddings. By weakening this to
almost bi-Lipschitz embeddings, we are able to obtain a number of new results.
Ametricspace(X,d)issaidtobe(M,s)-homogeneous (orsimplyhomogeneous)
if any ball of radius r can be covered by at most M(r/ρ)s smaller balls of radius ρ.
Since any subset of RN is homogeneous and homogeneity is preserved under bi-
Lipschitz mappings, it follows that (X,d) must be homogeneous if it is to admit
a bi-Lipschitz embedding into some RN (cf. comments in Hajl(cid:1)asz [6]). The As-
souad dimension of X, d (X), is the infimum of all s such that (X,d) is (M,s)-
A
homogeneous for some M ≥1.
Assouad[1]showedthat(X,d)ishomogeneousifandonlyifthesnowflakespaces
(X,dα) with 0 < α < 1 admit bi-Lipschitz embeddings into some RN (where N
depends on α). However, the three-dimensional Heisenberg group equipped with
its Carnot-Carath´eodory metric is homogeneous but cannot be embedded into any
Euclidean space in a bi-Lipschitz way (see Semmes [22]). Furthermore, there are
examplesduetoLaakso[13](seealsoLang&Plaut[12])ofhomogeneousspacesthat
do not even admit a bi-Lipschitz embedding into an infinite-dimensional Hilbert
ReceivedbytheeditorsSeptember15,2005and,inrevisedform,May27,2007.
2000 MathematicsSubjectClassification. Primary54F45,57N35.
Key words and phrases. Assouad dimension, Bouligand dimension, doubling spaces, embed-
dingtheorems,homogeneousspaces.
ThesecondauthorwasaRoyalSocietyUniversityResearchFellowwhenthispaperwaswritten,
and he would like to thank the Society for all their support. Both authors would like to thank
Professor Juha Heinonen for his helpful advice, and Eleonora Pinto de Moura for a number of
correctionsafterherclosereadingofthemanuscript.
(cid:1)c2009AmericanMathematicalSociety
Reverts to public domain 28 years from publication
145
146 ERICJ.OLSONANDJAMESC.ROBINSON
space. This paper starts with a simple result, based on Assouad’s argument, that
any homogeneous metric space admits an almost bi-Lipschitz embedding into an
infinite-dimensional Hilbert space.
The class of γ-almost L-bi-Lipschitz mappings f : (X,d) → (X˜,d˜) (or almost
bi-Lipschitz mappings for short) consists of all those maps for which there exists a
γ ≥0 and an L>0 such that
1 d(x,y)
(1.1) ≤d˜(f(x),f(y))≤Ld(x,y)
L slog(d(x,y))γ
for all x,y ∈ X such that x (cid:5)= y. Here slog(x) is the ‘symmetric logarithm’ of x,
defined as
slog(x):=log(x+x−1),
andsoanalmostbi-Lipschitzmapisbi-Lipschitztowithinlogarithmiccorrections.
Although the bi-Lipschitz image of a homogeneous set is homogeneous, this is
not true for almost bi-Lipschitz images; they are, however, almost homogeneous:
we say that (X,d) is (α,β)-almost (M,s)-homogeneous if
(cid:1) (cid:2)
r s
(1.2) N (r,ρ)≤M slog(r)βslog(ρ)α
X ρ
for all 0 < ρ < r < ∞, where N (r,ρ) is the minimum number of balls of radius
X
ρ necessary to cover any ball of radius r. The Assouad (α,β)-dimension of X,
dα,β(X), is the infimum of all s such that X is (α,β)-almost (M,s)-homogeneous
A
for some M ≥1.
If X is a subset of a vector space, then one can define the set of differences
X −X:
X−X ={x −x : x ,x ∈X}.
1 2 1 2
Olson[17]showedthatgivenacompactX ⊂RN withd (X−X)=d,almostevery
A
projection of rank k >d provides an almost bi-Lipschitz embedding of X into Rk.
In this paper we show a similar result for compact subsets X of a Hilbert space: if
the set of differences1 X −X is almost homogeneous with dα,β(X −X)=d, then
A
‘most’linearmapsintoEuclideanspacesRk withk >dprovidealmostbi-Lipschitz
embeddings of X. More explicitly, if k > d, then the set of almost bi-Lipschitz
embeddingsintoRk isprevalentinthespaceofalllinearmapsintoRk,inthesense
of Hunt, Sauer & Yorke [9].
There is an unfortunate gap here. An almost homogeneous metric space has an
almostbi-LipschitzimagethatisanalmosthomogeneoussubsetofaHilbertspace.
However, our embedding theorem for a subset X of a Hilbert space requires that
not X itself, but the set X −X of differences, is almost homogeneous.
InSection2westatesomeelementarypropertiesofthe(α,β)-Assouaddimension
andshowthatanyalmosthomogeneousmetricspace(X,d)canbeembeddedintoa
Hilbert space in an almost bi-Lipschitz way. That such almost bi-Lipschitz images
1TheintroductionofaconditiononthedimensionofthesetX−Xofdifferences,ratherthanon
Xitself,iscommonintheliteratureonabstractembeddings. TheproofofMan˜´e’s1981embedding
theoremrequires the HausdorffdimensionofX−X tobe finite, acondition not ensuredby the
finitenessoftheHausdorffdimensionofX. Foias&Olson[4]andHunt&Kaloshin[11]treatthe
upperbox-countingdimension,whichisunusualinhavingthepropertythatdF(X)<∞implies
that dF(X−X) < ∞. [Recall that dF(X) = limsup(cid:1)→0logN(X,(cid:1))/(−log(cid:1)), where N(X,(cid:1)) is
theminimumnumberofballsofradius(cid:1)neededtocoverX.]
ALMOST BI-LIPSCHITZ EMBEDDINGS 147
of almost homogeneous spaces are again almost homogeneous is shown in Section
3. Section 4 treats the local versions of homogeneity and almost homogeneity.
Section5containsourmainresultonembeddingasubsetX ofaHilbertspacewith
X −X almost homogeneous, while in Section 6 we consider what is possible for
suchsubsetsknowingonlypropertiesofX. InSection7weexploretherelationship
betweendα,β(X)anddα,β(X−X). AfterSection8, wherewegive anexample ofa
A A
homogeneous subset of a Hilbert space that cannot be bi-Lipschitz embedded into
any Rk using any linear map, we finish with some interesting open problems.
Throughout the paper all Hilbert spaces are real.
2. Almost homogeneous metric spaces
Asdiscussedabove,wewillsaythatametricspace(X,d)is(α,β)-almost(M,s)-
homogeneous(orsimplyalmosthomogeneous)ifanyballofradiusrcanbecovered
by at most2
(cid:1) (cid:2)
r s
(2.1) N (r,ρ)≤M slog(r)βslog(ρ)α
X ρ
balls of radius ρ (with ρ < r), for some M ≥ 1 and s ≥ 0, where slog(x) =
log(x+x−1).
We now give some simple properties of the function slog.
Lemma 2.1. Given L > 0 and γ ≥ 0, there exist constants A ,B ,a ,b ,σ ∈
L L γ γ
(0,∞) independent of x such that
(p1) |log(x)|≤slog(x)≤log2+|log(x)|, in particular slog(2k)≤(1+|k|)log2,
(p2) A slog(x)≤slog(Lx)≤B slog(x),
L L
(p3) a slog(x)≤slog(xslog(x)γ)≤b slog(x),
γ γ
for all x≥0, and
(p4) if 2−(k+1) ≤x≤2−k, then slog(x)≥σslog(2−k).
Proof. (p1) is elementary. For (p2) consider the quotient function g : (0,∞) →
(0,∞) defined by
slog(Lx)
g(x)= .
slog(x)
Let a =inf{g(x):x∈(0,∞)} and b =sup{g(x):x∈(0,∞)}. Since
L L
limg(x)=1, lim g(x)=1, and 0<g(x)<∞ for x∈(0,∞),
x→0 x→∞
then both a and b are finite positive constants. The proof of (p3) is similar. For
L L
(p4) set x = 2−r with k ≤ r ≤ k+1. Since slog(x) = log(x+1/x) ≥ log2 and
slog(2−r) ≥ |log2−r| = |r|log2 from (p1), then slog(x) ≥ (1+|r|)/2. Therefore,
the estimate
slog(2−k) (1+|k|)log2
≤ ≤4log2
slog(x) (1+|r|)/2
gives (p4) with σ =1/(4log2). (cid:1)
2Forboundedmetricspaces(2.1)couldbereplacedby
(cid:1) (cid:2)
NX(r,ρ)≤M(cid:3) r s log(e+ρ−1)γ
ρ
(in terms of our current definition we would have M(cid:3) ≥ M and γ = α+β), while for compact
spaces the factor of e in the logarithm could also be dropped by considering only ρ≤ r ≤(cid:1) for
some(cid:1)>0(seeSection4). However,(2.1)allowsustotreatgeneralmetricspaces.
148 ERICJ.OLSONANDJAMESC.ROBINSON
We define the Assouad (α,β)-dimension of X, dα,β(X), to be the infimum of all
A
s for which X is (α,β)-almost (M,s)-homogeneous. When α = β = 0 we recover
the standard definition of a homogeneous space and the usual Assouad dimension.
WenoteherethatitisstraightforwardtoshowthattheAssouad(α,β)-dimension
satisfies the minimal properties we would ask for in a dimension, namely that
X ⊆Y ⇒ dα,β(X)≤dα,β(Y), dα,β(X∪Y)=max(dα,β(X),dα,β(Y)),
A A A A A
and dα,β(O)=n if O is an open subset of Rn. Furthermore,
A
(2.2) α ≥α and β ≥β ⇒ dα1,β1(X)≤dα2,β2(X).
1 2 1 2 A A
We now show that if (X,d) is almost homogeneous, then it can be embedded
into an infinite-dimensional Hilbert space in an almost bi-Lipschitz way. Key to
this result is the following proposition, which although not given explicitly in this
form, essentially occursin Assouad’s paper. Indeed, itis themain ingredient inhis
proof of the existence of bi-Lipschitz maps between (X,dα) and RN.
Proposition 2.2. Let (X,d) be an (α,β)-almost (M,s)-homogeneous metric space
and distinguish a point a∈X. Then there are constants A,B,C >0 such that for
every j ∈Z there exists a map φj :(X,d)→RMj, where Mj =C(1+|j|)α+β, with
φ (a)=0, and for every x ,x ∈X
j 1 2
(a1) 2−(j+1) <d(x ,x )≤2−j implies that (cid:11)φ (x )−φ (x )(cid:11)≥A, and
1 2 j 1 j 2
(a2) (cid:11)φ (x )−φ (x )(cid:11)≤BM min[1,2jd(x ,x )].
j 1 j 2 j 1 2
Proof. The proof follows exactly the steps in Assouad’s original paper (see also
Heinonen’s book [7] or lecture notes [8] for an account that is easier to follow)
which we outline very briefly here: if N is a maximal 2−j net in (X,d), then for
j
every x∈X
(cid:3) (cid:4)
card N ∩B(x,12·2−j) ≤ N (12·2−j,2−j−1)
j X
≤ 24Mslog(12·2−j)αslog(2−j−1)β
≤ C(1+|j|)α+β,
where the constant C is a product of M and c(cid:5)onstants app(cid:6)earing in(cid:5)Lemma 2.1(cid:6).
Thus, there exists a ‘colouring map’ κ :N → e ,...,e , where e ,...,e
j j 1 Mj 1 Mj
is the standard basis of RMj, such that κj(a)(cid:5)=κj(b) if d(a,b)<12·2−j. Let
(cid:7) (cid:5) (cid:6)
φ˜ (x)= max (2−2jd(x,a )),0 κ (a ).
j i j i
ai∈Nj
Note that 22−j < d(x ,x ) ≤ 23−j implies that φ˜ (x ) is orthogonal to φ˜ (x ). It
1 2 j 1 j 2
is then straightforward to show that the map φ (x) = φ˜ (x)−φ˜ (a) satisfies
j j+3 j+3
the properties given in the statement of the proposition. (cid:1)
Theorem2.3. Let(X,d)bean(α,β)-almost(M,s)-homogeneousmetricspaceand
H an infinite-dimensional separable Hilbert space. Then, for every γ >α+β+ 1,
2
there exists a map f :X →H and a constant L such that
1 d(x,y)
≤(cid:11)f(s)−f(t)(cid:11)≤Ld(x,y),
L slog(d(x,y))γ
i.e., f is γ-almost bi-Lipschitz.
ALMOST BI-LIPSCHITZ EMBEDDINGS 149
Proof. Let {e } be an orthonormal set of vectors in some Hilbert space. Let
j j∈Z (cid:8)
δ >1/2 and define f :(X,d)→ ∞j=−∞RMj ⊗ej (cid:14)H by
(cid:7)∞ 2−j
(2.3) f(x)= φ (x)⊗e ,
(1+|j|)δM j j
j=−∞ j
where the maps φ are those of Proposition 2.2. Since f(a) = 0, the upper bound
j
on (cid:11)f(s)−f(t)(cid:11) that we now prove will also show convergence of the series (2.3)
defining f. Let (x ,x ) be a pair of distinct points of X. Thus, there exists
1 2
l ∈ Z such that 2−(l+1) < d(x ,x ) ≤ 2−l. Note that for such a pair of points
1 2
(cid:11)φ (x )−φ (x )(cid:11)≥A. We have
l 1 l 2
(cid:7)∞ 2−2j (cid:11)φ (x )−φ (x )(cid:11)2
(cid:11)f(x )−f(x )(cid:11)2 = j 1 j 2
1 2 (1+|j|)2δ M2
j=−∞ j
(cid:7)∞
B2
≤ d(x ,x )2
(1+|j|)2δ 1 2
j=−∞
≤ c d(x ,x )2,
1 1 2
where the sum converges since 2δ >1.
The lower bound is straightforward, since
2−l 2−l
(cid:11)f(x )−f(x )(cid:11) ≥ (cid:11)φ (x )−φ (x )(cid:11) ≥ A
1 2 (1+|l|)δM l 1 l 2 (1+|l|)δM
l l
2−l d(x,y)
≥ c ≥ c .
2 (1+|l|)α+β+δ 2(1+|l|)α+β+δ
Since d(x,y)=2−r with l≤r <l+1 it follows using (p1) from Lemma 2.1 that
1+|l| 1+|l| 1+|l| 1
= ≥ ≥ ,
slog(d(x,y)) slog(2−r) (1+|r|)log2 2log2
and so
d(x,y)
(cid:11)f(x )−f(x )(cid:11)≥c .
1 2 3slog(d(x,y))α+β+δ
Taking L=max(c ,1/c ) finishes the proof. (cid:1)
1 3
We note here that if (X,d) is bounded, then there exists a k such that d(x ,x )
1 2
≤2k for all x ,x ∈X. In this case the definition of f in (2.3) can be simplified to
1 2
(cid:7)∞ 2−j
(2.4) f(x)= φ (x)⊗e ,
(1+|j|)δM j j
j=−k j
and will still provide a γ-almost bi-Lipschitz embedding.
3. Almost bi-Lipschitz images of sets
Since we can embed any almost homogeneous metric space into a Hilbert space
usinganalmostbi-Lipschitz map, itisnaturaltostudytheeffectofsuchmappings
on almost homogeneous spaces. Here we show that almost bi-Lipschitz images of
almost homogeneousmetricspacesarestillalmost homogeneous. Inparticularthis
impliesthatitisnecessarythatX bealmosthomogeneousifitistoenjoyanalmost
bi-Lipschitz embedding into some RN.
150 ERICJ.OLSONANDJAMESC.ROBINSON
Lemma 3.1. Let (X,d) be an (α,β)-almost (M,s)-homogeneous metric space and
φ : (X,d) → (X˜,d˜) a γ-almost L-bi-Lipschitz map. Then (φ(X),d˜) is an almost
homogeneous metric space with dα+γ,β+γ(X)≤dα,β+γ(φ(X))≤dα,β(X).
A A A
Proof. Increase L if necessary so that
(3.1) L2bγ(log2)γ ≥1,
wherehereandintherestoftheproofb=b ,withb theconstantoccurringin(p3)
γ γ
in Lemma 2.1; clearly φ remains γ-almost L-bi-Lipschitz under this assumption.
Take s > dα,β(X), 0 < ρ < r < ∞, and consider an arbitrary ball B (φ(x),r) of
A X˜
radius r in φ(X). Now, we have
B (φ(x),r)⊆φ{B (x,Lrbγslog(Lrbγ)γ)},
X˜ X
since using (p3) in Lemma 2.1
1 Lrbγslog(Lrbγ)γ rbγslog(Lrbγ)γ
≥ =r.
Lslog(Lrbγslog(Lrbγ)γ)γ [bslog(Lrbγ)]γ
By our choice of L in (3.1) and since ρ < r we have 0 < ρ/L < Lrbγslog(Lrbγ)γ,
and so we can cover B (x,Lrbγslog(Lrbγ)γ) by
X
N (Lrbγslog(Lrbγ)γ,ρ/L)
X (cid:1) (cid:2)
Lrbγslog(Lrbγ)γ s
≤M slog(Lrbγslog(Lrbγ)γ)βslog(ρ/L)α
ρ/L
(cid:1) (cid:2)
r s
≤c slog(r)β+γslog(ρ)α
1 ρ
balls of radius ρ/L (inX), wherec depends on M, L andthe constants appearing
1
in Lemma 2.1. Denote these balls by B (x ,ρ/L). Since
X i
φ{B (x ,ρ/L)}⊆B (φ(x ),ρ)
X i X˜ i
and B (φ(x),r) was arbitrary, it follows that
X˜
(cid:1) (cid:2)
r s
N (r,ρ)≤c slog(r)β+γslog(ρ)α
φ(X) 1 ρ
for any 0 < ρ < r < ∞. Thus φ(X) is (α,β +γ)-almost (c ,s)-homogeneous.
1
Taking the infimum over s>dα,β(X) yields dα,β+γ(φ(X))≤dα,β(X).
A A A
Byconsideringtheinversemapφ−1 :φ(X)→X similarly, oneobtainsthelower
bound dα,β+γ(φ(X))≥dα+γ,β+γ(X). (cid:1)
A A
CombinedwithLemma3.1,theembeddingresultofTheorem2.3showsthatany
almost homogeneous metric space (X,d) has an almost bi-Lipschitz image f(X)
that is an almost homogeneous subset of a Hilbert space.
Weendbynotingthatsincealmostbi-Lipschitzmapsare,infact,Lipschitz,then
foranyalmostbi-Lipschitzmapφtheupperbox-counting(‘fractal’)dimension(see
footnote1foradefinition)satisfiesd (φ(X))≤d (X). Moreover, itisnotdifficult
F F
to prove the following:
Lemma 3.2. Let (X,d) be a metric space and φ : (X,d) → (X˜,d˜) an almost
bi-Lipschitz map. Then d (φ(X))=d (X).
F F
ALMOST BI-LIPSCHITZ EMBEDDINGS 151
4. Aside: Compact spaces and local versions of
(almost) homogeneity
In this section we briefly discuss the local definitions of homogeneity and al-
most homogeneity, and the dimensions associated with them. While they agree for
compact spaces, they are distinct in general.
A metric space (X,d)is said tobe locally (M,s)-homogeneous (or simply locally
homogeneous) if there exists an (cid:9) > 0 such that any ball of radius r < (cid:9) can be
covered by at most M(r/ρ)s smaller balls of radius ρ. The constant (cid:9) for a locally
homogeneous space may be interpreted as a small scale beneath which the set may
beviewedashomogeneous. InthiscaseM maydependon(cid:9),whichinturndepends
on the units of measurement used in the definition of the metric.
Movahedi-Lankarani [16] defined the metric (or ‘Bouligand’) dimension as
(cid:9) (cid:10)
logN (r,ρ)
(4.1) d (X)= lim lim sup X :0<ρ<r <(cid:9) and r >tρ .
B (cid:6)→0t→∞ log(r/ρ)
This dimension, d (X), is the infimum of all s such that (X,d) is locally (M,s)-
B
homogeneous for some M ≥1.
Here we give a simple example that shows that the concepts of homogeneous
and locally homogeneous are indeed different. Let H be a Hilbert space with
orthonormal basis given by {e } . Define
n n∈N
1
X ={ρ e :n∈N}, where ρ =1− .
n n n n
If (X,d) is (M,s)-homogeneous for some M and s, then
(cid:1) (cid:2)
2n−1 s
(4.2) N (ρ ,ρ )≤M(ρ /ρ )s =M ≤M.
X 2n n 2n n 2n−2
However, each ball B(0,ρ ) contains the n points
2n
{0}∪{ρ e :n<k <2n}
k k
which are mutually more than a distance ρ apart. Therefore N (ρ ,ρ ) ≥ n.
n X 2n n
Taking n large enough shows that (4.2) cannot hold, and so (X,d) is not homoge-
neous. On the other hand, (X,d) is locally homogeneous for any (cid:9)<1.
Note that if (X,d) is compact, then the notions of homogeneous and locally
homogeneous are equivalent (see Olson [17]). Thus d (X) = d (X) for compact
A B
spaces X.
Aswithhomogeneousspaces,thereisasimilarlydistinctnotionoflocally (α,β)-
almost(M,s)-homogeneous. Thismeansthereissome(cid:9)>0suchthat(2.1)holdsfor
all 0<ρ<r <(cid:9). Similar argumentstothosegiveninOlson [17]show thattheno-
tions of almost homogeneous and locally almost homogeneous are equivalent when
(X,d) is compact. Define the local Assouad (α,β)-dimension of X, dα,β(X), to be
B
the infimum of all s such that (X,d) is locally (α,β)-almost (M,s)-homogeneous
for some (cid:9)>0 and M ≥1.
Let(X,d)beametricspace. Ingeneraldα,β(X)≤dα,β(X). Bothdα,β anddα,β
B A A B
are invariant under a rescaling of the metric. Thus, the metric space (X˜,d˜), where
X˜ =X andd˜=ηdforsomeη >0,hasdα,β(X˜)=dα,β(X)anddα,β(X˜)=dα,β(X).
A A B B
Note that
dα+θβ,(1−θ)β(X)≤dα,β(X)≤d(1−θ)α,θα+β(X)
B B B
152 ERICJ.OLSONANDJAMESC.ROBINSON
for 0≤θ ≤1. Moreover, if X is compact, then
d (X)≤dα,β(X)=dα,β(X),
F A B
where d (X) denotes the fractal or upper box-counting dimension.
F
Wenoteherethatd shareswithd theusualpropertiesofdimensiondiscussed
B A
in Section 2, along with the monotonicity property in (2.2).
5. Embedding Hilbert subsets X with X −X homogeneous
In this section we prove our main result, in which we take a subset X of a
Hilbert space, assume that X −X is almost homogeneous, and obtain an almost
bi-Lipschitz embedding into a finite-dimensional space.
Our argument is essentially a combination of that of Olson [17], who treated
a subset X of a Euclidean space with d (X − X) finite, and that of Hunt &
A
Kaloshin [11], who considered a subset of a Hilbert space with finite upper box-
counting (‘fractal’) dimension. The key to combining these successfully is Lemma
5.3, below.
In line with the treatment in Sauer, Yorke & Casdagli [21] and in Hunt &
Kaloshin [11], our main theorem is expressed in terms of prevalence. This concept,
which generalises the notion of ‘almost every’ from finite to infinite-dimensional
spaces, was introduced by Hunt, Sauer & Yorke [9]; see their paper for a detailed
discussion.
Definition 5.1. A Borel subset S of a normed linear space V is prevalent if there
exists a compactly supported probability measure µ such that µ(S+v)=1 for all
v ∈V. In particular, if S is prevalent, then S is dense in V.
Note that if we set Q = supp(µ), then Q can be thought of as a ‘probe set’,
which consists of ‘allowable perturbations’ with which, given a v ∈ V, we ‘probe’
and test whether v+q ∈S for almost every q ∈Q.
Since we will use it below, and for its historical importance, we quote Hunt &
Kaloshin’sresulthere,inaformsuitableforwhatfollows. GivenasetX,itsupper
box-counting (‘fractal’) dimension is defined as
logN(X,(cid:9))
d (X)=limsup ,
F (cid:6)→0 −log(cid:9)
where N(X,(cid:9)) denotes the minimum number of balls of radius (cid:9) necessary to cover
X. Also, its thickness exponent, τ(X), is
logd(X,(cid:9))
(5.1) τ(X)=limsup ,
(cid:6)→0 −log(cid:9)
where d(X,(cid:9)) is the minimum dimension of all finite-dimensional subspaces, V, of
B such that every point of X lies within (cid:9) of V. We note here for later use that
τ(X)≤d (X).
F
Theorem 5.2 (Hunt & Kaloshin). Let X be a compact subset of a Hilbert space
H, D an integer with D > d (X −X), and τ(X) the thickness exponent of X. If
F
θ is chosen with
D(1+τ(X)/2)
θ > ,
D−d (X−X)
F
ALMOST BI-LIPSCHITZ EMBEDDINGS 153
then for a prevalent set of linear maps L:H →RD there exists a c>0 such that
c(cid:11)x−y(cid:11)θ ≤|Lx−Ly|≤(cid:11)L(cid:11)(cid:11)x−y(cid:11) for all x,y ∈X;
in particular these maps are injective on X.
Wenoteherethatd (X−X)≤2d (X),sothatforzerothicknesssetswithfinite
F F
box-countingdimensiononecanchooseanyD >2d (X)andθ >D/(D−2d (X)).
F F
5.1. Construction of the probability measure µ for a given X. We now
applythedefinitionofprevalencegivenaparticularcompactsubsetX ofourHilbert
space H such that X −X is (α,β)-almost (M,s)-homogeneous.
For some fixed N, let V be the set of linear functions L : H → RN. We now
construct a compactly supported probability measure µ on V (as required by the
definition of prevalence) that is carefully tailored to the particular set X. The key
to this is the following result.
Lemma5.3. SupposethatX isacompact(α,β)-almost(M,s)-homogeneoussubset
of H. Then there exists a sequence of nested linear subspaces U with U ⊆U ,
n n n+1
dimU ≤C(1+n)α+β+1,
n
and
1
(cid:11)P x(cid:11)≥ (cid:11)x(cid:11) for all x∈X with (cid:11)x(cid:11)≥2−n,
n 8
where P is the orthogonal projection onto U .
n n
Proof. Consider the collection of shells
(cid:11) (cid:12)
∆ = x∈X :2−(j+1) ≤(cid:11)x(cid:11)≤2−j .
j
Since ∆ ⊂B(0,2−j) it can be covered using
j
N (2−j,2−(j+3))≤8sM(log2)2(1+|j|)β(4+|j|)α ≤c (1+|j|)α+β :=M
X 2 j
balls of radius 2−(j+3), where c is independent of j. We choose the centres
(cid:11) (cid:12) 2
u(j) Mj of these balls so that (cid:11)u(j)(cid:11)≥2−(j+2).
i i
i=1
Since X is compact, X ⊂B(0,2k) for some k sufficiently large, and so
(cid:13)n (cid:5) (cid:6)
∆ = x∈X : (cid:11)x(cid:11)≥2−n .
j
j=−k
Let Pn be(cid:11)the orthogonal projection onto the line(cid:12)ar subspace Un spanned by the
collection u(j) :j =−k,...,n and i=1,...,M . Then the dimension of U is
i j n
bounded by c (1+n)α+β+1 using the same estimate as in (6.1). Moreover, for
3
every x ∈ ∆ there exists u(j) such that x = u(j)+v, where (cid:11)v(cid:11) ≤ 2−(j+3). Since
j i i
(cid:11)P (cid:11)=1 and (cid:11)P u(cid:11)=(cid:11)u(cid:11) for u∈U , then
n n n
1
(cid:11)P x(cid:11)=(cid:11)P (u(j)+v)(cid:11)≥(cid:11)P u(j)(cid:11)−(cid:11)P v(cid:11)≥2−(j+2)−2−(j+3) ≥ (cid:11)x(cid:11).
n n i n i n 8
(cid:1)
154 ERICJ.OLSONANDJAMESC.ROBINSON
Applying this lemma to X −X there are subspaces U with dimU ≤ d :=
k k k
c(1+k)α+β+1 such that (cid:11)P z(cid:11)≥(cid:11)z(cid:11)/8 for all z ∈X−X with (cid:11)z(cid:11)≥2−k. Let S
k k
denote the closed unit ball in U . Clearly any φ ∈ S induces a linear functional
k k
Lφ on H via the definition Lφ(u) = (φ(cid:14),u), where (·,·) is the inner product in H.
Let ζ >0 be fixed and define C =1/ ∞ k−1−ζ. We now define the probe set
ζ k=1
(5.2) (cid:15) (cid:16)
(cid:7)∞
Q= (l ,...,l ): l =L , where φ =C k−1−ζφ with φ ∈S .
1 N n φn n ζ nk nk k
k=1
WecanidentifySj withtheunitballBdj inRdj,andwedenotebyλj theprobability
measureonS thatcorrespondstotheuniformprobabilitymeasureonB . Welet
j dj
µ be the probability measure on Q that results from choosing each φ randomly
nk
with respect toλdk. Note that Q is√a compact subset of V and that all elements of
Q have Lipschitz constant at most N.
Before proving our main theorem we will prove a key estimate on µ. Although
the argument is essentially the same as that in Hunt & Kaloshin [11], our version
isalittlemoreexplicitandweinclude ithereforcompleteness. Theestimaterelies
on the following simple inequality.
Lemma 5.4. If x∈Rj and η ∈R, then
λ {ω ∈B : |η+(ω·x)|<(cid:9)}≤cj1/2(cid:9)|x|−1,
j j
where c is a constant that does not depend on η or j.
Proof. Let xˆ=x/|x|. This follows immediately from the estimate
(cid:5) (cid:6)
λ {ω ∈B : |η+(ω·x)|<(cid:9)} ≤ λ ω ∈B :|ω·xˆ|<(cid:9)|x|−1
j j j (cid:17) j
= Ωj−1 2 min((cid:6)|x|−1,1)(1−ξ2)(j−1)/2dξ
Ω
j 0
≤ Ωj−1 2(cid:9)|x|−1,
Ω
j
where Ω =πj/2Γ(j/2+1) is the volume of the unit ball in Rj. (cid:1)
j
Lemma 5.5. If x∈H and f ∈V, then
µ{L∈Q: |(L−f)(x)|<(cid:9)}≤c(d1/2k1+ζ(cid:9)(cid:11)P x(cid:11)−1)N
k k
for every k ∈N, where c is a constant independent of f and k.
Proof. Given k ∈N, let J be the index set J =N\{k} and define
(cid:18)(cid:19) (cid:20)
N
B = B .
dj
j∈J
Given α=((αnj)j∈(cid:5)J)Nn=1 ∈B fixed, define (cid:6)
A = (φ )N : |(η +k−1−ζφ )(x)|<(cid:9) for all n ,
α nk n=1 n nk
where (cid:7)
η (x)=C j−1−ζα (x)−f (x).
n ζ nj n
j∈J
By Lemma 5.4 there is a constant c independent of α, f and k such that
λN(A )≤c(d1/2k1+ζ(cid:9)(cid:11)P x(cid:11)−1)N.
dk α k k