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A no e on opological dimension, Hausdo measu e, and ec i iabili y
© 2020 Ame ican Ma hema ical Socie y
Accep ed e sion (Final d a )
Da id, Guy C.; Le Donne, En ico
Da id, G. C., & Le Donne, E. (2020). A no e on opological dimension, Hausdo measu e, and
ec i iabili y. P oceedings o he Ame ican Ma hema ical Socie y, 148(10), 4299-4304.
h ps://doi.o g/10.1090/p oc/15051
2020
A NOTE ON TOPOLOGICAL DIMENSION,
HAUSDORFF MEASURE, AND RECTIFIABILITY
GUY C. DAVID AND ENRICO LE DONNE
Abs ac . We gi e a su icien condi ion o a gene al compac me -
ic space o admi an n- ec i iable piece, as a consequence o a ecen
esul o Da id Ba e. Le Xbe a compac me ic space o opologi-
cal dimension n. Suppose ha he n-dimensional Hausdo measu e o
X,Hn(X), is ini e. Suppose u he ha he lowe n-densi y o he
measu e Hnis posi i e, Hn-almos e e ywhe e in X. Then Xcon ains
an n- ec i iable subse o posi i e Hn-measu e. Mo eo e , he assump-
ion on he lowe densi y is unnecessa y i one uses ecen ly announced
esul s o Cs¨o nyei-Jones.
1. In oduc ion
The pu pose o his no e is o eco d a consequence, o gene al me ic
spaces, o a ecen esul o Ba e [2]. We p o e he ollowing ac :
Theo em 1.1. Le Xbe a compac me ic space o opological dimension n.
Suppose ha he n-dimensional Hausdo measu e o X,Hn(X), is ini e.
Suppose u he ha
(1) lim in
→0
Hn(B(x, ))
n>0 o Hn-a.e. x∈X.
Then Xcon ains an n- ec i iable subse o posi i e Hn-measu e.
Mo eo e , assump ion (1) is unnecessa y i one uses ecen ly announced
esul s o Cs¨o nyei-Jones.
The use in Theo em 1.1 o he esul s o Cs¨o nyei-Jones a ises pu ely
h ough ou use o he wo k o Ba e [2] (Theo em 2.1 below), and does no
di ec ly appea in any o he p oo s he e. See Ba e’s discussion jus below
[2, Theo em 1.1] o de ails conce ning he announcemen o Cs¨o nyei-Jones
and he dependence o Theo em 2.1 on hem.
When Xis a subse o some Euclidean space, Theo em 1.1, wi hou as-
suming (1) o he esul s o Cs¨o nyei-Jones, appea s o al eady be known
Da e: July 17, 2018.
2010 Ma hema ics Subjec Classi ica ion. P ima y: 28A75. Seconda y: 28A78, 30L99.
G.C.D was suppo ed by he Na ional Science Founda ion unde G an no. NSF DMS-
1758709. E.L.D. was pa ially suppo ed by he Academy o Finland (g an 288501 ‘Ge-
ome y o subRiemannian g oups’ and by g an 322898 ‘Sub-Riemannian Geome y ia
Me ic-geome y and Lie-g oup Theo y’) and by he Eu opean Resea ch Council (ERC
S a ing G an 713998 GeoMeG ‘Geome y o Me ic G oups’).
1
2 GUY C. DAVID AND ENRICO LE DONNE
(see [16, p. 880]), as a consequence o he Besico i ch-Fede e p ojec ion
heo em. Fo gene al me ic spaces, he Besico i ch-Fede e heo em is un-
a ailable [3], bu Ba e’s wo k [2] se es as ou eplacemen . As a gene al
ule, su icien condi ions o inding ec i iabili y in an abs ac me ic space
a e much a e han o subse s o Euclidean space, whe e ools such as p o-
jec ion and densi y heo ems a e a ailable.
When n= 1, Theo em 1.1 (wi hou assuming (1) o elying on he esul s
o Cs¨o nyei-Jones) is a consequence o he ac ha con inua o ini e H1-
measu e a e Lipschi z images o [0,1] (see, e.g., [15, Lemma 3.7]), bu his
pa icula ac does no ex end o n > 1.
We now ecall some backg ound: Fo compac me ic spaces, he com-
monly used no ions o opological dimension (Lebesgue co e ing dimen-
sion, la ge/s ong induc i e dimension, and small/weak induc i e dimen-
sion) ag ee. We e e he eade o [14, Sec ions I.4 and II.5] o his ac
and he ele an de ini ions. Fo Hausdo measu e and dimension, we e e
he eade o [9, Chap e 8].
An Hn-measu able subse Eo a me ic space Xis called n- ec i iable i
Hn(E
∞
[
i=1
i(Fi)) = 0
whe e Fia e measu able subse s o Rnand i:Fi→Xa e Lipschi z maps.
By a heo em o Ki chheim [12, Lemma 4], one can equi alen ly ake i o
be bi-Lipschi z mappings.
A subse Eo a me ic space Xis called pu ely n-un ec i iable i i con ains
no n- ec i iable subse s o posi i e Hn-measu e.
I a compac me ic space Xhas opological dimension n, hen i is a
well-known ac (see, e.g., [9, Theo em 8.15]) ha Hn(X)>0, al hough ce -
ainly Xmay ha e in ini e n-dimensional Hausdo measu e o e en Haus-
do dimension s ic ly la ge han n, as is he case o classical ac als.
Thus, Theo em 1.1 says ha in he ex emal si ua ion, one mus see some
Euclidean s uc u e in he space.
Rela ed esul s, in which a combina ion o n-dimensional opological be-
ha io and n-dimensional measu e heo e ic beha io implies some ype o
ec i iabili y, can be ound, o example, in [5,7,8,11,16]. These esul s yp-
ically employ mo e quan i a i e assump ions o ob ain mo e quan i a i e
conclusions han ou Theo em 1.1.
I is easy o see ha he assump ions o Theo em 1.1 (including (1)) do
no imply n- ec i iabili y o he whole space X. Fo example, Xmay be he
disjoin union o he uni ball in Rnwi h a me ic space ha is a pu ely
n-un ec i iable Can o se o posi i e n-dimensional Hausdo measu e.
Mo e su p ising is ha he assump ions o Theo em 1.1 do no imply
n- ec i iabili y e en i one assumes ha Xis a compac n-dimensional opo-
logical mani old. In he appendix o [19], Schul and Wenge cons uc a
compac opological n-sphe e wi h Hn(X)<∞ ha con ains a pu ely n-
un ec i iable subse o posi i e measu e.
TOPOLOGICAL DIMENSION, HAUSDORFF MEASURE, AND RECTIFIABILITY 3
On he o he hand, Theo em 1.1 implies ha e e y open ball in a compac
n-mani old wi h ini e Hn-measu e con ains an n- ec i iable subse o posi-
i e Hn-measu e. No e ha he e exis such mani olds wi h no bi-Lipschi z
embedding in o any Euclidean space [13,18].
As a inal ema k, we poin ou wo well-known, pu ely un ec i iable ex-
amples ha con as wi h Theo em 1.1. Fo one, conside he closed uni
ball Bin he Heisenbe g g oup, which is a compac me ic space o opolog-
ical dimension 3 and Hausdo dimension 4. This me ic space Bis pu ely
4-un ec i iable, as one can show wi h a s anda d “blowup” a gumen . In
ac , Bis also pu ely 2- and 3-un ec i iable, bu his is mo e di icul o
es ablish (see [1, Theo em 7.2]).
Fo a second example, conside any compac me ic space (X, d) o opo-
logical dimension mand Hausdo dimension n≥m, and le Y= (X, dp)
o some p∈(0,1). Then Yis a compac me ic space o opological di-
mension mand Hausdo dimension n/p > m ha is pu ely k-un ec i iable
o each k∈N. Indeed, i E⊆Rkis compac and :E→ (E)⊂Yis a
bi-Lipschi z map, hen blowing up :E→ (E), in he G omo -Hausdo
sense, a a poin o densi y o Eyields a bi-Lipschi z embedding o Rkin o
a me ic space o he o m (Z, dp). This is impossible, as such a space can
con ain no ec i iable cu es. Fo mo e on such blowup a gumen s, we e e
he eade o [6, Chap e s 8-9].
Acknowledgmen s. The au ho s a e g a e ul o commen s om Gio anni
Albe i, Luigi Amb osio, Ma hew Badge , Da id P eiss, and Raanan Schul.
2. P oo o Theo em 1.1
Gi en a me ic space Xand m∈N, le Lip1(X, m) deno e he space o
bounded, 1-Lipschi z unc ions :X→Rm, equipped wi h he sup emum
dis ance, which we deno e dis . This is a comple e me ic space, and hence
esidual subse s (in he sense o Bai e ca ego y) a e dense.
The p oo o Theo em 1.1 is based on he ollowing ecen esul .
Theo em 2.1 (Ba e [2, Theo em 1.1]).Le Xbe a comple e, pu ely n-
un ec i iable me ic space wi h Hn(X)<∞. Suppose u he ha (1) holds.
Then he se o all ∈Lip1(X, m)wi h Hn( (X)) = 0 is esidual.
Mo eo e , assump ion (1) is unnecessa y i one uses ecen ly announced
esul s o Cs¨o nyei-Jones.
This has he ollowing easy consequence.
Co olla y 2.2. Le Xbe a compac , pu ely n-un ec i iable me ic space wi h
Hn(X)<∞and sa is ying (1).
Le g:X→[0,1]nbe con inuous. Then he e is a sequence o Lipschi z
unc ions i:X→[0,1]n ha con e ge o gin he sup emum dis ance and
sa is y Hn( i(X)) = 0 o all i∈N.
Mo eo e , assump ion (1) is unnecessa y i one uses ecen ly announced
esul s o Cs¨o nyei-Jones.
4 GUY C. DAVID AND ENRICO LE DONNE
P oo . Le hibe a sequence o Li-Lipschi z unc ions con e ging o gin he
sup emum dis ance. (The exis ence o such a sequence is a consequence o
he S one-Weie s ass heo em, o see [17, Lemma 2.4] o a simple di ec
p oo .) Thus L−1
ihi∈Lip1(X, n).
By Theo em 2.1, we can ind, o each i∈N, a Lipschi z unc ion gi∈
Lip1(X, n) sa is ying
dis (L−1
ihi, gi)< L−1
i2−iand Hn(gi(X)) = 0.
Conside he 1-Lipschi z e ac ion :Rn→[0,1]ngi en by
(2) (x1, x2, . . . , xn) = (ψ(x1), ψ(x2), . . . , ψ(xn)),
whe e
ψ( ) = 




0 < 0
0≤ ≤1
1 > 1.
Las ly, se
i= ◦(Ligi).
Since is Lipschi z and Hn(gi(X)) = 0, we ha e Hn( i(X)) = 0 o all
i∈N. Fu he mo e,
dis ( i, g) = dis ( ◦(Ligi), ◦g)≤dis (Ligi, g)<2−i+ dis (hi, g)→0.

To p o e Theo em 1.1, we will also need some opological in o ma ion.
De ini ion 2.3. Le :X→Ybe a con inuous map be ween me ic spaces.
A poin y∈Yis called a s able alue o i he e is  > 0 such ha y∈g(X)
o e e y con inuous g:X→Ywi h dis (g, )< .
Some basic and well-known ac s abou s able alues o mappings o [0,1]n
a e collec ed in he ollowing lemma.
Lemma 2.4. Le Xbe a me ic space and le ybe a s able alue o a
con inuous map :X→[0,1]n. Then
(i) y /∈∂([0,1]n),
(ii) yis a s able alue o g o each con inuous g:X→[0,1]nwi h
dis (g, )su icien ly small, and
(iii) (X)con ains an open neighbo hood o yin [0,1]n.
P oo . Pa (i) is simple and explained in [10, Example VI 4]. Pa (ii) is
an immedia e consequence o he de ini ion o s able alue.
Fo pa (iii), ecall he 1-Lipschi z e ac ion :Rn→[0,1]nde ined in
(2). No e ha maps Rn [0,1]non o he bounda y o [0,1]n.
Le ybe a s able alue o :X→[0,1]n, wi h pa ame e  > 0. Then, by
pa (i), y∈(0,1)n. We claim ha (X) con ains B(y, )∩[0,1]n. Conside
any y0∈B(y, )∩[0,1]n. The o mula
h(x) = (x+y−y0)

TOPOLOGICAL DIMENSION, HAUSDORFF MEASURE, AND RECTIFIABILITY 5
de ines a con inuous map om [0,1]n o i sel such ha h(y0) = yand
|h(x)−x|<  o all x∈[0,1]n.
Conside he map g:X→[0,1]nde ined by g=h◦ . Then dis ( , g)< ,
so g(x) = y o some x∈X. The e o e,
( (x) + y−y0) = y.
Since yis no on he bounda y o [0,1]n, we mus ha e (x) + y−y0=y,
i.e., (x) = y0.
The ollowing heo em is he second main ing edien in he p oo o The-
o em 1.1.
Theo em 2.5 (Theo em III.1 o [14]).Le Xbe a compac me ic space
o opological dimension n. Then he e is a con inuous map g:X→[0,1]n
wi h a s able alue.
The echnique o using s able alues o ind some ec i iable s uc u e in
a me ic space was used by Da id and Semmes [6, Sec ion 12.3] and Bonk
and Kleine [4] in simila con ex s.
P oo o Theo em 1.1. Le Xbe a compac me ic space o opological di-
mension nand Hn(X)<∞. We claim ha Xcon ains an n- ec i iable
subse o posi i e measu e. Suppose, o he con a y, ha Xis pu ely n-
un ec i iable.
Then, by Theo em 2.5, he e is a con inuous map g:X→[0,1]nwi h a
s able alue y. By Co olla y 2.2, he e is a sequence io Lipschi z maps
om X o [0,1]n ha con e ge o gin he sup emum dis ance and sa is y
(3) Hn( i(X)) = 0
o all i∈N.
On he o he hand, by Lemma 2.4(ii), when i∈Nis su icien ly la ge, he
map imus also ha e yas a s able alue. In ha case, i(X) con ains an
open subse o [0,1]n, by Lemma 2.4(iii). This con adic s (3). 
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6 GUY C. DAVID AND ENRICO LE DONNE
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(Da id) Depa men o Ma hema ical Sciences, Ball S a e Uni e si y, Muncie,
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