Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric

Publ. Ma . 47 (2003), 237–259
COMPARISON OF HAUSDORFF MEASURES WITH
RESPECT TO THE EUCLIDEAN AND THE
HEISENBERG METRIC
Zol ´
an M. Balogh, Ma hieu Rickly and
F ancesco Se a Cassano∗
Abs ac
We compa e he Hausdo ff measu es and dimensions wi h espec
o he Euclidean and Heisenbe g me ics on he fi s Heisenbe g
g oup. The esul is a dimension jump desc ibed by wo inequali-
ies. The sha pness o ou es ima es is shown by examples. Mo e-
o e acompa ison be ween Euclidean and H- ec ifiabili y is gi en.
1. In oduc ion
In his pape we conside he Heisenbe g g oup H=H1=(R3,∗)
as a homogeneous g oup endowed wi h he le in a ian , homogeneous
Heisenbe g dis ance dHdefined as ollows. The g oup mul iplica ion
∗:H×H→His gi en by
(x, y, )∗(x,y,
):=(x+x,y+y, + +2(xy−yx)).(1.1)
His endowed wi h he homogeneous no m
pH:= ((x2+y2)2+ 2)1
4
(1.2)
i p=(x, y, )∈H, which induces he Heisenbe g dis ance
dH(p, p):=p−1∗pH.(1.3)
I is well-known ha he opological dimension o His 3, since H
coincides wi h R3as a smoo h mani old (see Lemma 2.1). On he o he
hand, he Hausdo ff measu es and dimensions o subse s o H≡R3wi h
espec o ei he dHo he Euclidean me ic dEcan be e y diffe en . Fo
2000 Ma hema ics Subjec Classifica ion. 28A78, 43A80.
Key wo ds. Hausdo ff measu es, Hausdo ff dimension, Heisenbe g me ic.
∗F.S.C. is suppo ed by GNAMPA o INDAM, p ojec “Analysis in me ic spaces
and subellip ic equa ions”, by MURST, I aly, and Uni e si y o T en o, I aly. Pa
o he wo k was done while F.S.C. was a isi o a he Uni e si y o Be n. He wishes
o hank he Ins i u e o Ma hema ics o i s hospi ali y.
238 Z. M. Balogh, M. Rickly, F. Se a Cassano
ins ance, he Hausdo ff dimension o (R3,d
H)is4,while he dimension
o a egula su ace in (R3,d
H)is3.Howe e , i we conside a egula
cu e in (R3,d
H), hen i may ha e Hausdo ff dimension bo h 1 and 2
(see [G o] and also [S ]).
The pu pose o his pape is o desc ibe in de ails his dimension
jump phenomenon be ween he Heisenbe g and he Euclidean Hausdo ff
dimensions on subse s o R3. Indeed we pe o m a compa ison be ween
he α-dimensional Hausdo ff measu es induced on R3by dHand dE,
which we espec i ely deno e wi h Hα
Hand Hα
E. Mo e p ecisely ou fi s
esul eads as ollows:
Theo em 1.1 (Dimension jump heo em).Le α≥0. Then
(i)
Hmin{2α,α+1}
HH
α
E,(1.4)
i.e. Hmin{2α,α+1}
His absolu ely con inuous wi h espec o Hα
E.
(ii)
Hmin{α,1+ α
2}
EH
α
H,(1.5)
i.e. Hmin{α,1+ α
2}
Eis absolu ely con inuous wi h espec o Hα
H.
Ou second esul shows ha he es ima es o Theo em 1.1 a e sha p:
Theo em 1.2 (Sha pness o he dimension jump).
(i) Gi en 0<α≤3, he e is a compac subse Aαo Hsa is ying
Hα
E(Aα)<∞and Hmin{2α,α+1}
H(Aα)>0.
(ii) Fo 0<α<2and α=4, he e is a compac subse Aαo H
sa is ying
Hα
H(Aα)<∞and Hmin{α,1+ α
2}
E(Aα)>0.
Fo 2≤α<4and 0<δ<1, he e is a compac subse Aα,δ o H
sa is ying
Hα
H(Aα,δ)=0 and Hmin{α,1+ α
2}−δ
E(Aα,δ)=H1+ α
2−δ
E(Aα,δ)>0.
Rema k 1.1.We conjec u e ha gi en 2 ≤α<4, he e is a compac
subse Aαo Hsuch ha
Hmin{α,1+ α
2}
E(Aα)=H1+ α
2
E(Aα)>0 and Hα
H(Aα)<∞.
Howe e , we ha e no ye been able o cons uc such se s.
Compa ison o Hausdo Measu es 239
The echnique in ol ed in he p oo o Theo em 1.1 is based on an op-
imal co e ing o Heisenbe g balls by smalle Euclidean balls and ice -
e sa. This kind o mu ual co e ings ha e al eady been p oposed in [G o].
The p oo o Theo em 1.2 elies on some mo e delica e a gumen s in ol -
ing ecen esul s on he size o Can o - ype and cha ac e is ic se s o
egula su aces in he me ic spaces (R3,d
H) and (R3,d
E) (see [Ba1],
[Ba2]).
Ano he s iking ea u e o he Heisenbe g g oup is he lack o m- ec-
ifiable subse s in he me ic space (H,d
H) acco ding o he classical
no ion o ec ifiabili y in a me ic space due o Fede e [Fe]. Indeed, i
has been p o ed ha (H,d
H)ispu ely m-un ec ifiable o m=2,3,4
(see [AK, Theo em 7.2]). This has led o a mo e in insic defini ion
o ec ifiabili y in (H,d
H), he 3-dimensional H- ec ifiabili y, in oduced
in [FSSC1]. We will use ou esul s om Theo em 1.2 o compa e he
classical no ion o 2-Euclidean ec ifiabili y in (R3,d
E) o he no ion o
3-dimensional H- ec ifiabili y (see Theo em 5.1).
The pape is o ganized as ollows: in Sec ion 2 we ecall some basic
ac s abou he Heisenbe g g oup and he Hausdo ff measu es defined
on i . In Sec ion 3 we p o e ou fi s heo em. The second heo em is
p o ed in Sec ion 4. In he las sec ion we apply ou esul s o compa e
2- ec ifiable subse s o (H,d
E) and 3-dimensional H- ec ifiable subse s
o (H,d
H).
Acknowledgemen s. We hank Pe i Ma ila o use ul discussions
on he subjec .
2. Basic defini ions and p elimina y esul s
In his sec ion, we b iefly es a e he basic defini ions and esul s
needed in he es o he a icle.
We use he model o he fi s Heisenbe g g oup H=H1wi h unde ly-
ing space R3and g oup ope a ion gi en by (1.1). No ice ha p−1=−p
and ha 0 = (0,0,0) is he uni elemen o he g oup H. The dila ion
by is he au omo phism δ :H→Hgi en by
δ (x, y, ):=( x, y, 2 ).(2.1)
Fo p∈H he le ansla ion by pis he au omo phism lp:H→H
defined by
lp(p):=p∗p.
240 Z. M. Balogh, M. Rickly, F. Se a Cassano
The 3-dimensional Lebesgue measu e L3in R3≡His a bi-in a ian
Haa measu e o he g oup and sa isfies L3(δ (A)) = 4L3(A) whene e
A⊆Hand >0 (see [FoS , P oposi ion 1.2 (c)]).
The Lie algeb a o His spanned by he le in a ian ec o fields
X=∂
∂x +2y∂
∂ ,Y=∂
∂y −2x∂
∂ ,T=∂
∂ .
The Lie algeb a s uc u e o His de e mined by he only non- i ial
commu a o ela ion [X,Y ]=−4T.
The ec o fields Xand Yspan a ec o bundle, he so-called ho izon-
al bundle HH, whe e HHp:= span{X(p),Y(p)} o all p∈H, which can
be canonically iden ified wi h a ec o subbundle o he angen ec o
bundle TH≡TR3.
The ollowing ela ionship be ween he dis ances dEand dHin R3can
be easily e ified.
Lemma 2.1. Le Abeabounded subse o (R3,d
E)and le b≥1be a
bound o Awi h espec o dE. Then he e is a posi i e cons an c=c(b)
such ha o p, p∈Awe ha e
1
cdE(p, p)≤dH(p, p)≤c(dE(p, p))1
2.(2.2)
In pa icula he iden i y map Id: (R3,d
E)→(R3,d
H)is a homeomo -
phism.
In he sequel, BE(p, ) and BH(p, ) will deno e he closed ball o
adius >0 cen e ed a pin (H,d
E) and (H,d
H) espec i ely. ·will
deno e he Euclidean no m on Rn o n=2,3.
Le us now ecall he defini ion o Hausdo ff and sphe ical measu es
on a me ic space (X,d). Gi en a subse Ao X, he diame e o Ais
diamd(A):=sup{d(a, a)|a, a∈A}.
We w i e diam ins ead o diamdwhen he e is no isk o con usion, and
we le diamE:= diamdEand diamH:= diamdH. Obse e ha o p∈H
and >0weha e
diamH(BH(x, )) = 2 .(2.3)
Compa ison o Hausdo Measu es 241
Fo 0 ≤s<∞,0<δ≤∞and A⊆X,wele
Hs
d,δ(A):=in ∞

n=1
(diam(En))s|En⊆X, diam(En)≤δ, A ⊆
n∈N
En
and
Ss
d,δ(A):=in ∞

n=1
diam(B(xn,
n))s|diam(B(xn,
n)) ≤δ,
A⊆
n∈N
B(xn,
n),
whe e B(x, ) deno es he closed ball o adius cen e ed a x. Since 0 <
δ1≤δ2clea ly implies Hs
d,δ2(A)≤H
s
d,δ1(A) and Ss
d,δ2(A)≤Ss
d,δ1(A), we
may define he s-dimensional Hausdo ff measu e Hs
don (X,d) and he
s-dimensional sphe ical measu e Ss
don (X,d)by
Hs
d(A):=lim
δ↓0Hs
d,δ(A)
and
Ss
d(A):=lim
δ↓0Ss
d,δ(A)
espec i ely.
We will w i e Hs
E,Ss
Eins ead o Hs
dE,Ss
dEand Hs
H,Ss
Hins ead o
Hs
dH,Ss
dHand we will gene ally conside hese measu es on R3.By
abuse o no a ion we shall also w i e Hs
E o he s-dimensional Hausdo ff
measu e on (Rm,d
E) o m=1,2.
The Hausdo ff dimension o a se A⊆(X,d)is
H−dimd(A):=in {s≥0|H
s
d(A)=0}.
In he ollowing p oposi ion we collec some gene al p ope ies o
Hausdo ff measu es ha will be needed below.
P oposi ion 2.2.
(i) Le (X,d)be a me ic space. Then Hs
dand Ss
da e egula Bo el
(ou e ) measu es and
Hs
d(A)≤Ss
d(A)≤2sHs
d(A)
o all A⊆X,s≥0.

242 Z. M. Balogh, M. Rickly, F. Se a Cassano
(ii) Le (Xi,d
i)(i=1,2)be wo me ic spaces and le :(X1,d
1)→
(X2,d
2)be a L-Lipschi z con inuous map, i.e.
d2( (x), (y)) ≤L·d1(x, y)(∀x, y ∈X1).
Then
Hs
d2( (A)) ≤Ls·Hs
d1(A)
o all A⊆X1,s≥0.
(iii) Le (X,d)be a me ic space and le d (x, y):=(d(x, y)) o x, y ∈
Xand ∈(0,1). Then d is a dis ance on Xand
Hs
d(A)=H
s
d (A)
o all A⊆X,s≥0and ∈(0,1).
(i ) Le (X,d)beame ic space and le Y⊆X. Deno e by dY he
me ic on Yinduced by d. Then
Hs
d(A)=Hs
dY(A)
o all A⊆Y,s≥0.
( ) P ope ies (ii) and (iii) hold i we eplace he Hausdo ff mea-
su e Hs
dby he sphe ical measu e Ss
d.
P oo : All hese s a emen s a e classical and can be ound in [Fa], [Fe]
o [Ma].
Rema k 2.1.Using (ii) o he abo e p oposi ion, one easily sees ha i d1
and d2a e wo bi-Lipschi z equi alen me ics on X, hen Hs
d1and Hs
d2
a e mu ually absolu ely con inuous measu es. In pa icula , i A⊆X
hen H−dimd1(A)=H−dimd2(A).
Using he homogenei y and le in a iance o dH, one has
Hs
H(lp(A)) = Hs
H(A) and Hs
H(δ (A)) = sHs
H(A)
o any A⊆H,p∈H,s≥0 and >0. Mo eo e i is easy o see ha
H4
H=cL3 o a sui able posi i e cons an c.
In Subsec ion 4.1 we will make use o he ollowing c i e ion gi ing a
lowe bound on he Hausdo ff dimension o a se in a me ic space.
Lemma 2.3. Le (X,d)beasepa able me ic space, le A⊆Xand le
s>0. Suppose he e is a measu e µon Xwi h he ollowing p ope ies:
(i) µ(A)>0.
(ii) µ(B(x, )∩A)≤cµ s o all x∈Xand 0< ≤R, whe e cµ,
R>0a e fixed cons an s.
Then Hs
d(A)>0.
The p oo is a simple exe cise.
Compa ison o Hausdo Measu es 243
3. P oo o he dimension jump heo em
In he ollowing we p o e he absolu e con inui y o H2α
Hand Hα+1
H
wi h espec o Hα
E(P oposi ions 3.2 (i) and 3.6 (i) below) and he
absolu e con inui y o Hα
Eand H1+ α
2
Ewi h espec o Hα
H(P opo-
si ions 3.2 (ii) and 3.6 (ii) below). Indeed, no ice ha om P opo-
si ion 2.2 (i) we can indiffe en ly es ima e he Hausdo ff o sphe ical
measu es. We s a wi h a echnical lemma:
Lemma 3.1. A subse Ao His bounded wi h espec o dEi and only
i i is bounded wi h espec o dH. Mo e p ecisely, i bE≥1is a bound
o Awi h espec o dE, hen 21
4bEis a bound o Awi h espec o dH,
and i bH≥1is a bound o Awi h espec o dH, hen 21
2b2
His a bound
o Awi h espec o dE.
The p oo is le o he eade .
P oposi ion 3.2. The Hausdo ff measu es sa is y he ollowing absolu e
con inui y p ope ies (α≥0a bi a y):
(i) Hα
HH
α
2
E.
(ii) Hα
EH
α
H.
P oo : (i) Suppose ha H
α
2
E(A)=0and deno e by √dE:R3×R3→
[0,+∞) he dis ance √dE(p, p):=(dE(p, p))1
2i p,p∈R3. Then by
P oposi ion 2.2 (iii) wi h =1
2we also ge 0 = H
α
2
E(A)=Hα
√dE(A).
Le An:= A∩BE(0,n). Applying Lemma 2.1 wi h A=An, he e is a
posi i e cons an cndepending only on n, such ha o all p, p∈Anwe
ha e
1
cn
dE(p, p)≤dH(p, p)≤cndE(p, p).(3.1)
Le d(n)
Hand d(n)
Edeno e he es ic ions o he dis ances dHand
√dE o An.By he igh inequali y in (3.1) and P oposi ion 2.2 (ii) wi h
X1=X2=An, = Id, d1=d(n)
E,d2=d(n)
Hand L=cn
Hα
d(n)
H
(An)≤cα
nHα
d(n)
E
(An).
On he o he hand om P oposi ion 2.2 (i )
Hα
H(An)=Hα
d(n)
H
(An)≤cα
nHα
d(n)
E
(An)=Hα
√dE(An)=0.
Then, by passing o he limi when n→∞,wege Hα
H(A)=0.
244 Z. M. Balogh, M. Rickly, F. Se a Cassano
The s a emen (ii) can be p o ed by simila a gumen s using he le
inequali y in (3.1).
In he ollowing we shall p o e Sα+1
HSα
Eand S1+ α
2
ESα
H, c . P opo-
si ion 3.6 below. This is conside ably mo e complica ed han P oposi-
ion 3.2. The p oo is based on mo e in ol ed co e ing lemmas ela ing
he Euclidean and Heisenbe g balls. The s a emen s o hese co e ing
lemmas can be ound in [G o, 0.6.C].
Be o e going in o de ails we need one mo e echnical lemma in o de
o con ol he dis o ion o he shape o Euclidean balls unde g oup
ansla ion.
Lemma 3.3. Le Abeabounded subse o (R3,d
E)wi h bound b≥0.
Fo >0,p=(x, y, )∈R3and p=(x,y,
)∈Awe ha e
lp(BE(p, )) ⊆BE(lp(p),(2b+1) ).
Mo eo e i lp(p)+(x,y,
)∈lp(BE(p, )) hen (x,y)≤ .
P oo : Le (x+x,y+y, + )∈BE(p, ).
lp(x+x,y+y, + )
=(x+x+x,y+y+y,
+ +  + 2((x+x)y−x(y+y)))
=lp(p)+(x,y,
 +2(xy−xy)).
Now (x,y)=(x,y)≤ and
(x,y,
 +2(xy−xy))2
=x2+y2+|  +2(xy−xy)|2
≤x2+y2+(| |+2p·(x,y,
))2
≤(x,y,
)2+4b(x,y,
)·| |+4b2(x,y,
)2
≤ 2+4b 2+4b2 2=(2b+1)
2 2.
The claim ollows.
The ollowing s a emen exhibi s a close- o-op imal co e ing o a
Euclidean ball by Heisenbe g balls.
Compa ison o Hausdo Measu es 245
P oposi ion 3.4. Le Abe abounded subse o (R3,d
E)wi h bound b≥
0. Then he e exis s N=N(b)∈Nsuch ha o any Euclidean
ball BE(p, )wi h p∈Aand 0< <1we can find Heisenbe g balls
BH(p1, ),...,B
H(pk, )sa is ying:
(i) BE(p, )⊆k
i=1 BH(pi, ).
(ii) k≤N
.
Rema k 3.1.No ice ha (ii) yields a sha p con ol on he numbe k
o Heisenbe g balls needed o co e he Euclidean ball in e ms o i s
adius .
P oo : I is clea ly enough o show ha , gi en a bounded subse Ao
(R3,d
E) wi h bound b≥0, he e is N=N(b)∈Nsuch ha o p∈A
and 0 < <1wecan find BH(p1, ),...,B
H(pk, ) sa is ying:
BE(p, /2) ⊆
k

i=1
BH(pi, ) and k≤N
.
The idea is o use a g oup ansla ion o mo e he Euclidean ball o
he o igin and o pe o m he co e ing he e wi h balls cen e ed on he
e ical axis O .Fo p=(0,0, )∈O and >0, le us deno e by
BH,∞(p, ) he se o poin s p=(x,y,
)∈Hsa is ying (x,y)≤
and | − |≤ 2. Hence BH,∞(p, )isafla box o heigh 2 2cen e ed
a p∈Hand i s o hogonal p ojec ion along he e ical axis is a disk
o adius .Nowle p∈A,0< <1 and le k:= [4b+2
]+1. By
Lemma 3.3 we can co e lp−1(BE(p,
2)) wi h kboxes BH,∞(pi,
2), whe e
i∈{1,...,k}and (pi)i∈{1,...,k}a e sui able poin s on he e ical axis.
The eade can easily see ha BH,∞(pi,
2)⊆BH(pi, ) o 1 ≤i≤k,
hus
BE(p, /2) ⊆lpk

i=1
BH(pi, )=
k

i=1
lp(BH(pi, )) =
k

i=1
(BH(p∗pi, )).
Finally, k < 4b+3
=4b+3and he claim is p o ed wi h N:= [4b+
3]+1.
The ollowing s a emen is in some sense he coun e pa o P oposi-
ion 3.4. I p esen s a close- o-op imal co e ing o a Heisenbe g ball by
Euclidean balls.
252 Z. M. Balogh, M. Rickly, F. Se a Cassano
Rema k 4.1.Assume ha he su ace Sis gi en as he g aph o a C1
unc ion :U→Ro e adomain U⊆R2, i.e. S={(x, y, (x, y)) |
(x, y)∈U}. Then p=(x, y, )∈Sis a cha ac e is ic poin i and only i
∂
∂x(x, y)=2yand ∂
∂y(x, y)=−2x.
In wha ollows we conside cha ac e is ic se s C(S)onsuch g aphs.
We collec he esul s om [Ba2] needed in he sequel in he ollowing
heo em:
Theo em 4.2. Gi en 1<α<2, he e exis s a compac subse Qαo
Q:= [−1
2,1
2]×[−1
2,1
2]and a C1,1smoo h map :Q→R, such ha :
(i) 0 <Hα
E(Qα)<∞.
(ii) {(x, y, (x, y)) |(x, y)∈Qα}⊆C(S), whe e S={(x, y, (x, y)) |
(x, y)∈Q}.
(iii) Fo any s≥0, he e is a cons an c=c(s)such ha Hs
H(A)≤
cHs
E(A)whene e A⊆C(S).
Rema k 4.2.(i) and (ii) a e shown in he p oo o Theo em 1.4 in [Ba2].
(iii) is essen ially Case 2 o Theo em 1.1 in [Ba2] whe e H
H,∞( espec-
i ely H
E,∞) has been eplaced by H
H( espec i ely H
E). The p oo
gi en he e only needs e y mino changes in ou case. The compu a-
ions can be ound in [Ri].
2. Case 1 <α<2. Ou se Aαis ob ained om he cha ac e is ic
se o a su ace in he ollowing way: Le Qαand :Q→Rbe as in
Theo em 4.2. Conside he mapping F:Q→R3gi en by F(x, y)=
(x, y, (x, y)) and se Aα:= F(Qα). Obse e ha F:(Qα,d
E)→
(Aα,d
E)isabi-Lipschi z mapping and he e o e
1
kHα
E(Qα)≤H
α
E(Aα)≤kHα
E(Qα)
o some k>0. This shows in pa icula ha Hα
E(Aα)>0. We apply
Theo em 4.2 o es ima e Hα
H(Aα). Since Aα⊆C(S), i ollows ha
Hα
H(Aα)≤cHα
E(Aα)≤ckHα
E(Qα).
3. Case α=2.Gi en 0 <δ<2 a bi a y, conside he se Q2−δand
he map F om he Case 2 abo e. Then o Aα,δ := F(Q2−δ), we ha e
H2−δ
E(Aα,δ)>0 and H2
H(Aα,δ)=0byTheo em 4.2.

Compa ison o Hausdo Measu es 253
4. Case 2 <α<4. Gi en 0 <δ<1 a bi a y, conside he map F
and he se F(Q2−δ) om 2. abo e. Ou se Aα,δ will be a p oduc o
F(Q2−δ) wi h he Can o se Cα−2
2lying in he e ical axis:
Aα,δ := {(x, y, (x, y)+ )|(x, y)∈Q2−δ, ∈Cα−2
2}.
The es ima e H1+ α
2−δ
E(Aα,δ)>0isagain a consequence o he Euclidean
p oduc s uc u e o Aα,δ.Tosee his, conside he map
g:Q2−δ×Cα−2
2→Aα,δ;g(x, y, ):=F(x, y)+(0,0, ).
gis a bijec ion, and he in e se is
g−1:Aα,δ →Q2−δ×Cα−2
2;g−1(x, y, ):=(x, y, )−(0,0, (x, y)).
I is easy o see ha g:(Q2−δ×Cα−2
2,d
E)→(Aα,δ,d
E)isbi-Lipschi z.
Hence, i is enough o show ha
H1+ α
2−δ
E(Q2−δ×Cα−2
2)>0.
Bu H2−δ
E(Q2−δ)>0 and H
α−2
2
E(Cα−2
2)>0, by P oposi ion 4.1 (ii) and
Theo em 4.2, so
H1+ α
2−δ
E(Q2−δ×Cα−2
2)=H(2−δ)+( α−2
2)
E(Q2−δ×Cα−2
2)>0
by [Ma, Theo em 8.10 (1)].
Nex , we p o e ha Hα
H,∞(Aα,δ)=0. This is a mo e difficul ask
since we ha e o conside co e ings wi h Heisenbe g balls. In ou a gu-
men i is c ucial ha he shape o Heisenbe g balls emains unchanged
unde e ical ansla ions. Now emembe ha 0 <λ<1is ela ed
o Cα−2
2by he equa ion λα−2
2=1
2and le c:= (1 + λ−1
2)22α−2.
Gi en 0 <<cλ
2, using he compac ness o F(Q2−δ) and he ac
ha H2
H(F(Q2−δ)) = 0, we can co e F(Q2−δ) wi h Heisenbe g balls
(BH(pn,
n))n∈{1,...,N}in such a way ha N
n=1(2 n)2< /c.
Fo each n∈{1,...,N}, le kn∈Nsuch ha λkn+1
2≤2 n<λ
kn
2
(2 n<(/c)1
2<λ o n=1,...,N by cons uc ion). Fo n=1,...,N
and ln=1,...,2kn, le
Bn,ln:= {(x, y, + )|(x, y, )∈BH(pn,
n),
∈Ikn,ln},
whe e Ikn,lna e he in e als appea ing in he cons uc ion o he Can o
se Cα−2
2. No ice ha Aα,δ ⊆n,lnBn,ln.Inwha ollows we es ima e
254 Z. M. Balogh, M. Rickly, F. Se a Cassano
he Heisenbe g diame e o Bn,ln: Le (x, y, ),(x,y,
)∈BH(pn,
n)
and le 1,
2∈Ikn,ln. Then
dH((x, y, + 1),(x,y,
+ 2))
=(x−x, y−y,( − )+( 2− 1)−2xy+2xy)H
=(x−x, y−y,( − )−2xy+2xy)∗(0,0,
2− 1)H
≤dH((x, y, ),(x,y,
)) + | 2− 1|1
2
≤2 n+λkn
2.
This compu a ion gi es diamH(Bn,ln)≤2 n+λkn
2.Now:

n,ln
(diamH(Bn,ln))α
=
n,ln
(diamH(Bn,ln))2(diamH(Bn,ln))α−2
≤
n,ln
(2 n+λkn
2)2(2 n+λkn
2)α−2≤
n,ln
((1 + λ−1
2)2 n)2(2λkn
2)α−2
=
n,ln
(1 + λ−1
2)2(2 n)22α−21
2kn
=
n
(1 + λ−1
2)2(2 n)22α−2
=(1+λ−1
2)22α−2
n
(2 n)2=c
n
(2 n)2<c
c=.
Hence, o an a bi a y 0 <<cλ
2, he e exis s a co e ing (Bn,ln)n,ln
(n=1,...,N,ln=1,...,2kn)o Aα,δ such ha n,ln
(diamH(Bn,ln))α<
.
5. Case α=4.This, again, is i ial. Take A4:= BE(0,1). Then
0<H3
E(A4)<∞and 0 <H4
H(A4)<∞.
5. An applica ion o H- ec ifiabili y
In his sec ion we will compa e he classical no ion o 2-Euclidean
ec ifiabili y and he no ion o H- ec ifiabili y in oduced in [FSSC1].
Le us ecall he classical no ion o m- ec ifiabili y in a gene al me ic
space due o Fede e ([Fe, 3.2.14]): Gi en a me ic space (X, d) and a
Compa ison o Hausdo Measu es 255
posi i e in ege m,wesay ha a Bo el se E⊆Xis m- ec ifiable i he e
exis s a coun able collec ion o Lipschi z maps i:Ai⊆(Rm,d
E)→
(X,d) such ha
Hm
dE ∞

i=1
i(Ai)=0.
An m-dimensional Euclidean ec ifiable se is an m- ec ifiable subse o
(Rn,d
E).
Rema k 5.1.I is well known ha m-dimensional Euclidean ec ifiabili y
is equi alen o he equi emen ha Hm
dE-almos all o he se can be
co e ed by a sequence o C1m-g aphs o Rn(see [AFP, Chap e 2,
Sec ion 2.9]).
I u ned ou howe e ha his no ion o ec ifiabili y is no app o-
p ia e in he se ing o he Heisenbe g g oup endowed wi h he Heisen-
be g me ic. Indeed, Amb osio and Ki chheim (see [AK, Theo em 7.2])
p o ed ha he Heisenbe g g oup (H,d
H)ispu ely m-un ec ifiable o
m=2,3,4, i.e. ha o any Lipschi z map
:A⊆(Rm,d
E)→(H,d
H)
one has Hm
H( (A)) = 0. This lack o ec ifiable se s in he classical sense
sugges s ha mo e in insic defini ions o ec ifiabili y could be use ul
ins ead. To his aim, in [FSSC1], an in insic defini ion o ec ifiabili y
in he Heisenbe g g oup was in oduced in he codimension one case.
This was success ully used o s udy he s uc u e o he se s o in insic
fini e pe ime e (see [FSSC1], [FSSC2], [FSSC3]).
The idea in [FSSC1]was o eplace he images o Lipschi z mappings
in Fede e ’s defini ion by su aces gi en as le el se s o H-diffe en iable
unc ions. This led o he ollowing defini ion:
Defini ion 5.1. We shall say ha S⊆His 3-dimensional H- ec ifiable
i he e exis s a sequence o H- egula hype su aces (Si)i∈Nsuch ha
H3
HS 
i∈N
Si=0.
In he abo e defini ion he no ion o H- egula hype su ace appea s.
He e S⊆R3≡His called an H- egula hype su ace i i is locally he
le el se o an H- egula unc ion :H→R.H- egula i y o means
ha he ho izon al g adien
∇H := (X ,Y ): H→R2
256 Z. M. Balogh, M. Rickly, F. Se a Cassano
o exis s, is con inuous and non- anishing on S.Inpa icula , i Sis
a Euclidean C1 egula su ace wi hou cha ac e is ic poin s, hen Sis
H- egula .
I was p o ed in [FSSC1] ha he essen ial bounda y o a se Eo
locally fini e H-pe ime e in His 3-dimensional H- ec ifiable.
The ollowing compa ison esul be ween 2-dimensional Euclidean ec-
ifiabili y and 3-dimensional H- ec ifiabili y is in o de :
Theo em 5.1.
(i) Each 2-dimensional Euclidean ec ifiable se S⊆R3≡His 3-di-
mensional H- ec ifiable.
(ii) The e a e 3-dimensional H- ec ifiable se s S⊆R3≡H ha a e
no 2-dimensional Euclidean ec ifiable.
P oo : (i) By Rema k 5.1, gi en a 2-dimensional Euclidean ec ifiable
subse S⊆R3,wecan assume wi hou loss o gene ali y ha
S=N∪
i∈N
Si
whe e H2
E(N)=0andSi⊆R3is a Euclidean C1 egula hype su ace.
Le ˜
Si:= Si C(Si), whe e C(Si) deno es he se o cha ac e is ic poin s
o Si. Then ˜
Siis an H- egula hype su ace, and
S=˜
N∪
i∈N
˜
Si
wi h
˜
N=N∪
i∈N
C(Si).
By Theo em 1.1 wi h α=2weha e H3
H(N)=0. Acco ding o a
esul om [Ba2], i Sis a Euclidean C1 egula hype su ace, hen
H3
H(C(S)) = 0. This gi es H3
H(C(Si)) = 0 o all i∈N. Hence
H3
H(˜
N)=0and consequen ly Sis 3-dimensional H- ec ifiable.
(ii) Gi en 0 <δ<0.5, we know by Theo em 1.2 (ii) ha he e is a
compac se N=A3,δ ⊆Hsuch ha
H3
H(N)=0 bu H2.5−δ
E(N)>0.
The e o e Nis 3-dimensional H- ec ifiable by defini ion bu no 2-dimen-
sional Euclidean ec ifiable since i s Hausdo ff dimension in (R3,d
E)is
s ic ly bigge han 2.
Compa ison o Hausdo Measu es 257
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Compa ison o Hausdo Measu es 259
Zol ´an M. Balogh:
Ins i u e o Ma hema ics
Uni e si y o Be n
Sidle s asse 5
3012 Be n
Swi ze land
E-mail add ess:[email p o ec ed]
Ma hieu Rickly:
Ins i u e o Ma hema ics
Uni e si y o Be n
Sidle s asse 5
3012 Be n
Swi ze land
E-mail add ess:[email p o ec ed]
F ancesco Se a Cassano:
Dipa imen o di Ma ema ica
Uni e si `adiT en o
Via Somma i e 14
38050 Po o (T en o)
I aly
E-mail add ess:[email p o ec ed]
Rebu el 4 de juny de 2002.