a Xi :1009.0481 1 [ma h.CA] 2 Sep 2010
FURSTENBERG SETS FOR A FRACTAL SET OF
DIRECTIONS
URSULA MOLTER AND EZEQUIEL RELA
Abs ac . In his no e we s udy he beha io o he size o Fu s en-
be g se s wi h espec o he size o he se o di ec ions defining i . Fo
any pai α, β ∈(0,1], we will say ha a se E⊂R2is an Fαβ -se i
he e is a subse Lo he uni ci cle o Hausdo ff dimension a leas β
and, o each di ec ion ein L, he e is a line segmen ℓein he di ec-
ion o esuch ha he Hausdo ff dimension o he se E∩ℓeis equal
o g ea e han α. The p oblem is conside ed in he wide scena io o
gene alized Hausdo ff measu es, gi ing es ima es on he app op ia e di-
mension unc ions o each class o Fu s enbe g se s. As a co olla y o
ou main esul s, we ob ain ha dim(E)≥max α+β
2; 2α+β−1
o any E∈Fαβ. In pa icula we a e able o ex end p e iously known
esul s o he “endpoin ” α= 0 case.
1. In oduc ion
In his a icle we a e in e es ed in he s udy o dimension p ope ies o
Fu s enbe g se s associa ed o ac al se s o di ec ions. Le us in oduce he
de ini ion o ou objec o s udy. In he sequel, we will deno e wi h dim(E)
he Hausdo dimension o he se E.
De ini ion 1.1. Fo α, β in (0,1], a subse Eo R2will be called an Fαβ-se
i he e is a subse Lo he uni ci cle such ha dim(L)≥βand, o each
di ec ion ein L, he e is a line segmen ℓein he di ec ion o esuch ha he
Hausdo dimension o he se E∩ℓeis equal o g ea e han α.
This gene alizes he classical de ini ion o Fu s enbe g se s, when he
whole ci cle is conside ed as se o di ec ions. Fo L=S, which is a pa ic-
ula case o β= 1, we eco e he classical class Fαo α-Fu s enbe g se s,
and he bes known esul is
(1) max α+1
2; 2α≤γ(α)≤1
2+3
2α, α ∈(0,1].
whe e γ(α) = in {dim(E) : E∈Fα}. In [MR10] and [MR] he abo e
inequali ies a e p o ed in he gene al se ing o dimension unc ions, allowing
he ex ension o he endpoin α= 0 o some class o gene alized Fu s enbe g
se s.
Una oidable e e ences on his ma e a e [Wol99], [Wol03], [KT01] and
[Tao].
1991 Ma hema ics Subjec Classi ica ion. P ima y 28A78, 28A80.
Key wo ds and ph ases. Fu s enbe g se s, Hausdo ff dimension, dimension unc ion,
Kakeya se s.
This esea ch is pa ially suppo ed by G an s: PICT2006-00177, PIP 11220080100398
and UBACyT X149.
1
2 URSULA MOLTER AND EZEQUIEL RELA
The pu pose o his no e is o s udy how he pa ame e βa ec s he
bounds abo e. Mo eo e , by using gene al Hausdo measu es, we will
ex end he inequali ies (1) o he ze o dimensional case.
F om ou esul s we will de i e he ollowing p oposi ion.
P oposi ion 1.2. Fo any se E∈Fαβ, we ha e ha
(2) dim(E)≥max α+β
2; 2α+β−1, α, β > 0.
I is no ha d o p o e P oposi ion 1.2 di ec ly, bu we will s udy his
p oblem in a wide scena io and de i e i as a co olla y. We also ema k
ha ou esul s a e consis en wi h he ones in [Mi 02], whe e he au ho
p o es, essen ially, he second bound o he case α= 1, β∈(0,1].
The e is a na u al way o gene alize his p oblem by looking a dimension
unc ions ha a e no necessa ily powe unc ions ([Hau18]). Le us begin
wi h he no ion o dimension unc ions.
1.1. Dimension Func ions.
De ini ion 1.3. The ollowing class o unc ions will be called dimension
unc ions.
H:= {h: [0,∞)→[0 : ∞),non-dec easing, igh con inuous, h(0) = 0}.
The impo an subclass o hose h∈H ha sa is y a doubling condi ion
will be deno ed by Hd:
Hd:= {h∈H:h(2x)≤Ch(x) o some C > 0}.
Rema k 1.4. Clea ly, i h∈Hd, he same inequali y will hold (wi h some
o he cons an ) i 2 is eplaced by any o he λ > 1. We also ema k ha any
conca e unc ion i ially belongs o Hd. Also no e ha he mono onici y
o himplies ha C≥1.
I one only looks a he powe unc ions, he e is a na u al o al o de
gi en by he exponen s. I we deno e wi h hα(x) = xα, hen hαis, in some
sense, smalle han hβi and only i α < β. In Hwe also ha e a na u al
no ion o o de , bu we can only ob ain a pa ial o de .
De ini ion 1.5. Le g, h be wo dimension unc ions. We will say ha gis
dimensionally smalle han hand w i e g≺hi and only i
lim
x→0+
h(x)
g(x)= 0.
We also ema k ha we will be pa icula ly in e es ed in he special sub-
class o dimension unc ions ha allows us o classi y ze o dimensional se s,
ha means, ha his in his class i i is smalle han any o he unc ions
xα,α > 0.
De ini ion 1.6. A unc ion h∈Hwill be called “ze o dimensional dimen-
sion unc ion” i h≺hα o any α > 0. We will deno e by H0 he subclass
o hose unc ions.
FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 3
As usual, he h-dimensional (ou e ) Hausdo measu e Hhwill be de ined
as ollows. Fo a se E⊆Rnand δ > 0, w i e
Hh
δ(E) = in (X
i
h(diam(Ei)) : E⊂∞
[
i
Ei,diam(Ei)< δ).
The h-dimensional Hausdo measu e Hho Eis de ined by
Hh(E) = sup
δ>0Hh
δ(E).
We ema k ha , e en hough hey would no lead o he exac same mea-
su es, we will conside unc ions g, h such ha he e exis cons an s c, C
wi h 0 < c ≤g(x)
h(x)≤C < ∞ o all x > 0 o be equi alen . In ha case we
w i e g≡h.
To measu e he “dis ance” be ween o dimension unc ions, we in oduce
he ollowing no ion:
De ini ion 1.7. Le g, h ∈Hwi h g≺h. De ine he “gap” be ween gand
has
(3) ∆(x) = h(x)
g(x).
F om his de ini ion and he de ini ion o pa ial o de , we always ha e
ha limx→0∆(x) = 0, and he e o e he speed o con e gence o ze o can
be seen as a no ion o dis ance be ween gand h.
Now we p esen he p oblem. Le us begin wi h he de ini ion o Fhg-se s.
Le hand gbe wo dimension unc ions. A se E⊆R2is a Fu s enbe g se
o ype hg, o an Fhg-se , i he e is a subse Lo he uni ci cle such ha
Hg(L)>0 and, o each di ec ion ein L, he e is a line segmen ℓein he
di ec ion o esuch ha Hh(ℓe∩E)>0.
No e ha his hypo hesis is s onge han he one used o de ine he
o iginal Fu s enbe g-αse s. Howe e , he hypo hesis dim(E∩ℓe)≥αis
equi alen o Hβ(E∩ℓe)>0 o any βsmalle han α. I we use he wide
class o dimension unc ions in oduced abo e, he na u al way o de ine Fh-
se s would be o eplace he pa ame e s β < α wi h wo dimension unc ions
sa is ying he ela ion h≺h. Bu equi ing E∩ℓe o ha e posi i e Hh
measu e o any h≺himplies ha i has also posi i e Hhmeasu e (Theo em
42, [Rog70]). The e o e, his de ini ion is he na u al gene aliza ion o he
F+
αβ class de ined below.
De ini ion 1.8. Fo each pai α, β in (0,1], a subse Eo R2will be called
an F+
αβ-se i he e is a subse Lo he uni ci cle such ha Hβ(L)>0 and,
o each di ec ion ein L, he e is a line segmen ℓein he di ec ion o esuch
ha Hα(ℓe∩E)>0.
Now, o he sake o cla i y in he p oo o ou esul s, we will pe o m he
same educ ion made in [MR10]. A s anda d pigeonhole a gumen allows us
o wo k wi h he ollowing de ini ion.
De ini ion 1.9. Le hand gbe wo dimension unc ions. A se E⊆R2
is a Fu s enbe g se o ype hg, o an Fhg-se , i he e is a subse Lo he
uni ci cle such ha Hg(L)>0 and, o each di ec ion ein L, he e is a line
4 URSULA MOLTER AND EZEQUIEL RELA
segmen ℓein he di ec ion o esuch ha Hh
δ(ℓe∩E)>1 o all δ < δE o
some δE>0 wi h δEdepending only on E.
Following he in ui ion sugges ed by P oposi ion 1.2, one could conjec u e
ha i Ebelong o he class Fhg hen an app op ia e dimension unc ion o
Eshould be dimensionally g ea e han h2g
id and h√g(whe e id is he iden i y
unc ion). This will indeed be he case, and we will p o ide some es ima es
on he gap be ween hose conjec u ed dimension unc ions and a gene ic es
unc ion h∈H o ensu e ha Hh(E)>0. In addi ion we illus a e wi h
some examples. We will conside he wo esul s sepa a ely. Namely, o a
gi en pai o dimension unc ions g∈Hand h∈Hd, in Sec ion 3 we ob ain
su icien condi ions on a es dimension unc ion h∈H,h≻h2g
id o ensu e
ha Hh(E)>0 o any se E∈Fhg. In Sec ion 4 we conside he analogous
p oblem o h≻h√g. The nex sec ion summa izes some p elimina y esul s
o be used in ou p oo s and addi ional no a ion. Finally, in Sec ion 5 we
b ie ly discuss he app op ia e no ion o size o he se o di ec ions de ining
he Fu s enbe g classes.
2. P elimina ies
In his sec ion we include some p elimina y and echnical esul s needed
in he sequel. We will use he no a ion A.B o indica e ha he e is a
cons an C > 0 such ha A≤CB, whe e he cons an is independen o A
and B. By A∼Bwe mean ha bo h A.Band B.Ahold. As usual, by
aδ-co e ing o a se Ewe mean a co e ing o Eby se s Uiwi h diame e s
no exceeding δ.
In Sec ion 3 he main ool will be an L2es ima e o he Kakeya maximal
unc ion o gene al measu es. Fo an in eg able unc ion on Rn, he Kakeya
maximal unc ion a scale δwill be Kδ( ) : Sn−1→R,
Kδ( )(e) = sup
x∈Rn
1
|Tδ
e(x)|ZTδ
e(x)| (x)|dx e ∈Sn−1,
whe e Tδ
e(x) is a 1 ×δ- ube (by his we mean a ube o leng h 1 and c oss
sec ion o adius δ) cen e ed a xin he di ec ion e.
The es ima e we need is he main esul o [Mi 02]. The e he au ho
p o es (Theo em 3.1) he ollowing.
P oposi ion 2.1. Le µbe a Bo el p obabili y measu e on Ssuch ha
µ(B(x, )) .ϕ( ) o some non-nega i e unc ion ϕ o all ≪1. De-
ine he Kakeya maximal ope a o Kδas usual:
Kδ( )(e) = sup
x∈Rn
1
|Tδ
e(x)|ZTδ
e(x)| (x)|dx, e ∈Sn−1.
Then we ha e he es ima e
(4) kKδk2
L2(R2)→L2(S,dµ).C(δ) = Z1
δ
ϕ(u)
u2du.
Rema k 2.2. I should be no ed ha i we choose ϕ(x) = xs, hen we
ob ain as a co olla y ha
(5) kKδk2
L2(R2)→L2(S,dµ).δs−1.
FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 5
In he special case o s= 1, he bound has he known loga i hmic g ow h:
kKδk2
L2(R2)→L2(S,dµ)∼log(1
δ).
This esul will be used in Sec ion 3, whe e he hypo heses imposed on
a se E o being an Fhg se gua an ee, ia F os man’s lemma, ha he e
exis s a p obabili y measu e µon he se o di ec ions Lwi h µ(B ).g( )
o any ball B (see [Ma 95]). Le us ema k ha (5) sugges s ha he
cons an C(δ) plays, in he gene al case, he ole o g
id (δ).
In Sec ion 4 we pe o m a mo e combina o ial kind o p oo . We in oduce
he no ion o δ-en opy o a se Ein he nex de ini ion
De ini ion 2.3. Le E⊂Rnand δ∈R>0. The δ-en opy o Eis he
maximal possible ca dinali y o a δ-sepa a ed subse o E. We will deno e
his quan i y wi h Nδ(E).
The main idea is o ela e he δ-en opy o some no ion o size o he se .
Clea ly, he en opy is essen ially he Box dimension o he Packing dimen-
sion o a se (see [Ma 95] o [Fal03] o he de ini ions) since bo h concep s
a e de ined in e ms o sepa a ed δballs wi h cen e s in he se . Howe e , o
ou p oo we will need o ela e he en opy o a se o some quan i y ha
has he p ope y o being (in some sense) s able unde coun able unions.
One choice is he e o e he no ion o Hausdo con en , which enjoys he
needed p ope ies: i is an ou e measu e, is ini e, and e lec s he en opy
o a se in he ollowing manne . Recall ha he g-dimensional Hausdo
con en o a se Eis de ined as
(6) Hg
∞(E) = in (X
i
g(diam(Ui) : E⊂[
i
Ui).
No e ha he g-dimensional Hausdo con en Hg
∞is clea ly no he same
han he g-dimensional Hausdo measu e Hg. In ac , hey a e he measu es
ob ained by applying Me hod I and Me hod II (see [Ma 95]) espec i ely o
he p emeasu e ha assigns o a se A he alue g(diam(A)).
Fo u u e e e ence, we s a e he ollowing es ima e o he δ-en opy o
a se wi h posi i e g-dimensional Hausdo con en as a lemma.
Lemma 2.4. Le g∈Hand le Abe any se . Le Nδ(A)be he δ-en opy
o A. Then Nδ(A)≥Hg
∞(A)
g(δ).
P oo . Le {xi}N
i=1 be a maximal δ-sepa a ed subse . By maximali y, we
can co e Awi h balls B(xi, δ). The e o e, o he g-dimensional Hausdo
con en Hg
∞, we ha e he bound
(7) Hg
∞(A)≤
N
X
iHg
∞(B(xi, δ)) ≤Ng(δ)
and i ollows ha Nδ(A)≥N≥Hg
∞(A)
g(δ).
O cou se, his esul is meaning ul when Hg
∞(A)>0. We will use i in
he case Hg(A)>0 which is equi alen o Hg
∞(A)>0. Fo a de ailed s udy
o he p ope ies o Hgand Hg
∞see [Del02] and [Del03].

6 URSULA MOLTER AND EZEQUIEL RELA
No e ha he lemma abo e only equi es he ini eness and he subaddi-
i i y o he Hausdo con en . The ele an ea u e ha will be needed in
ou p oo is he σ-subaddi i i y, which is a p ope y ha he Box dimension
does no sha e.
Now we in oduce he ollowing no a ion and a echnical lemma.
De ini ion 2.5. Le b={bk}k∈Nbe a dec easing sequence wi h lim bk= 0.
Fo any amily o balls B={Bj}wi h Bj=B(xj; j), j≤1, and o any
se E, we de ine
(8) Jb
k:= {j∈N:bk< j≤bk−1},
and
(9) Ek:= E∩[
j∈Jb
k
Bj.
In he pa icula case o he dyadic scale b={2−k}, we will omi he supe -
sc ip and deno e
(10) Jk:= {j∈N: 2−k< j≤2−k+1}.
The nex lemma in oduces a echnique used in [MR10] o decompose he
se o all di ec ions.
Lemma 2.6. Le Ebe an Fhg-se o some h,g∈Hwi h he di ec ions in
L⊂Sand le a={ak}k∈N∈ℓ1be a non-nega i e sequence. Le B={Bj}
be a δ-co e ing o Ewi h δ < δEand le Ekand Jkbe as abo e. De ine
Lk:= e∈S:Hh
δ(ℓe∩Ek)≥ak
2kak1.
Then L=∪kLk.
The p oo ollows di ec ly om he summabili y o a.
3. The Kakeya ype bound
In his sec ion we p o e a gene alized e sion o he announced bound
dim(E)≥2α+β−1 o E∈Fαβ. We ha e he ollowing heo em.
Theo em 3.1 (hg →h2g
id ).Le g∈H,h∈Hdbe wo dimension unc ions
and le Ebe an Fhg-se . Fo δ > 0, le C(δ)be as in (4). Fo any h∈H
such ha X
kqh2(2−k)C(2−k)
h(2−k)<∞,Hh(E)>0.
P oo . Le E∈Fhg and le {Bj}j∈Nbe a co e ing o Eby balls wi h
Bj=B(xj; j). We need o bound Pjh(2 j) om below. Since his non-
dec easing, i su ices o ob ain he bound
(11) X
j
h( j)&1
o any h∈Hsa is ying he hypo hesis o he heo em.
De ine a={ak}by a2
k=h2(2−k)C(2−k)
h(2−k). Also de ine, as in he p e ious
sec ion, o each k∈N,Jk={j∈N: 2−k< j≤2−k+1}and Ek=
FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 7
E∩∪j∈JkBj. Since by hypo hesis a∈ℓ1, we can apply Lemma 2.6 o ob ain
he decomposi ion o he se o di ec ions as L=SkLkassocia ed o his
choice o a.
We will apply he maximal unc ion inequali y o a weigh ed union o
indica o unc ions. Fo each k, le Fk=[
j∈Jk
Bjand de ine he unc ion
:= h(2−k)2kχFk.
We will use he L2no m es ima es o he maximal unc ion. We can
compu e di ec ly he L2no m o :
k k2
2=h2(2−k)22kZ∪JkBj
dx
.h2(2−k)22kX
j∈Jk
2
j
.h2(2−k)#Jk,
since j≤2−k+1 o j∈Jk. The e o e
(12) k k2
2.#Jkh2(2−k).
The same a gumen s used in he p oo o Theo em 3.1 in [MR10] allows us
o ob ain a lowe bound o he maximal unc ion. Essen ially, he maximal
unc ion is poin wise bounded om below by he a e age o o e he ube
cen e ed on he line segmen ℓe o any e∈Lk. The e o e, we ha e he
ollowing bound o he (L2, µ) no m. He e, µis a measu e suppo ed on L
ha obeys he law µ(B(x, )≤g( ) o any ball B(x, ) gi en by F os man’s
lemma.
(13) kK2−k+1 ( )k2
L2(dµ)&a2
kµ(Lk) = µ(Lk)h2(2−k)C(2−k)
h(2−k).
Combining (13) wi h he maximal inequali y (4), we ob ain
µ(Lk)h2(2−k)C(2−k)
h(2−k).kK2−k+1 ( )k2
2.C(2−k+1)k k2
2≤C(2−k)k k2
2.
We also ha e he bound (12), which implies ha
µ(Lk)
h(2−k).#Jk.
Now we a e able o es ima e he sum in (11). Le hbe a dimension
unc ion sa is ying he hypo hesis o Theo em 3.1. We ha e
X
j
h( j)≥X
k
h(2−k)#Jk
&X
k
µ(Lk)≥µ(L)>0.

Co olla y 3.2. Le Ean F+
αβ-se . I his any dimension unc ion sa is ying
(14) h(x)≥Cx2α+β−1logθ(1
x)
8 URSULA MOLTER AND EZEQUIEL RELA
o θ > 2, hen Hh(E)>0.
P oo . I ollows di ec ly, since in his case we ha e C(δ).δβ−1, and he e-
o e he sum in Theo em 3.1 is
X
ksh2(2−k)C(2−k)
h(2−k).X
ks2−k2α2−k(β−1)
h(2−k)
≤X
ks2−k(2α+β−1)
(2−k)2α+β−1logθ(2k)
=X
k
1
kθ
2
<∞.

Rema k 3.3. No e ha he bound dim(E)≥2α+β−1 o E∈Fαβ ollows
di ec ly om his las co olla y.
4. The combina o ial bound
In his sec ion we deal wi h he bound hg →h√g, which is he signi ican
bound nea he endpoin α=β= 0 and gene alizes he bound dim(E)≥
β
2+α o E∈Fαβ. No e ha he second bound in (2) is meaningless
o small alues o αand β. We will conside sepa a ely he cases o h
being ze o dimensional o posi i e dimensional. In he nex heo em, he
addi ional condi ion on h e lec s he posi i i y o he dimension unc ion.
We belie e ha i would be help ul o ci e, wi hou he p oo s, wo ele-
an lemmas used in [MR10].
The i s is a “spli ing lemma”, which says ha a linea se wi h posi i e
h-dimensional mass can be spli ed in o wo well sepa a ed linea subse s.
Lemma 4.1. Le h∈H,δ > 0,Ian in e al and E⊆I. Le η > 0be such
ha h−1(η
8)< δ and Hh
δ(E)≥η > 0. Then he e exis wo subin e als I−,
I+ ha a e h−1(η
8)-sepa a ed and wi h Hh
δ(I±∩E)&η.
The second lemma is he combina o ial ing edien in he p oo o bo h
Theo em 4.3 and Theo em 4.6. This lemma p o ides an es ima e on he
numbe o lines wi h ce ain sepa a ion ha in e sec wo balls o a gi en
size.
Lemma 4.2. Le b={bk}k∈Nbe a dec easing sequence wi h lim bk= 0.
Gi en a amily o balls B={B(xj; j)}, we de ine Jb
kas in (8) and le
{ei}Mk
i=1 be a bk-sepa a ed se o di ec ions. Assume ha o each i he e a e
wo line segmen s I+
eiand I−
eilying on a line in he di ec ion ei ha a e
sk-sepa a ed o some gi en skDe ine Πk=Jb
k×Jb
k×{1, .., Mk}and Lb
kby
Lb
k:= (j+, j−, i)∈Πk:I−
ei∩Bj−6=∅I+
ei∩Bj+6=∅.
I 1
5sk> bk−1 o all k, hen
#Lb
k.bk−1
bk
1
sk#Jb
k2.
FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 9
Wi h hese wo lemmas we a e now eady o p o e he main esul o his
sec ion. We ha e he ollowing heo em. Recall ha hα(x) = xα.
Theo em 4.3 (hg →h√g,h≻hα).Le g∈H,h∈Hdbe wo dimension
unc ions such ha h(x).xα o some 0< α < 1and le Ebe an Fhg-se .
Le h∈Hwi h h≺h√g. I X
kh(2−k)√g(2−k)
h(2−k)2α
2α+1 <∞, hen Hh(E)>0.
P oo . Le E∈Fhg and le {Bj}j∈Nbe a co e ing o Eby balls wi h Bj=
B(xj; j). De ine ∆ = h√g
hand conside he sequence a=n∆2α
2α+1 (2−k)ok.
Also de ine, as in he p e ious sec ion, o each k∈N,Jk={j∈N: 2−k<
j≤2−k+1}and Ek=E∩ ∪j∈JkBj. Since by hypo hesis a∈ℓ1, we can
apply Lemma 2.6 o ob ain he decomposi ion o he se o di ec ions as
L=SkLkassocia ed o his choice o a, whe e Lkis de ined as
Lk:= e∈S:Hh
δ(ℓe∩Ek)≥ak
2kak1.
We can apply Lemma 4.1 wi h η=ak
2kak1 o ℓe∩Ek. The e o e we ob ain
wo in e als I−
eand I+
e, con ained in ℓewi h
Hh
δ(I±
e∩Ek)&ak
ha a e h−1( ak)-sepa a ed o =1
16kak1.
Now, le {ek
j}Nk
j=1 be a 2−k-sepa a ed subse o Lk. Taking in o accoun
he es ima e o he en opy gi en in Lemma 2.4. We ob ain hen ha
(15) Nk&Hg
∞(Lk)
g(2−k).
De ine Πk:= Jk×Jk×{1, .., Nk}and
(16) Tk:= (j−, j+, i)∈Πk:I−
ei∩Ek∩Bj−6=∅I+
ei∩Ek∩Bj+6=∅.
The idea is o coun he elemen s o Tkin wo ways. I we ix a pai j−and
j+and coun o how many alues o i he iple (j−, j+, i) belongs o Tk,
we ob ain, by using Lemma 4.2 o he choice b={2−k}, ha
(17) #Tk.1
h−1( ak)(#Jk)2.
Second, ix i. In his case, we ha e by hypo hesis ha Hh
δ(I+
ei∩Ek)&ak,
so Pj+h( j+)&ak. The e o e,
ak.X
(j−,j+,i)∈Tk
h( j+)≤Kh(2−k),
whe e Kis he numbe o elemen s o he sum. The e o e K&ak
h(2−k).
The same holds o j−, so
(18) #Tk&Nkak
h(2−k)2
.