Improving dimension estimates for Furstenberg-type sets

a Xi :0904.2710 2 [ma h.CA] 18 No 2009 IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS URSULA MOLTER AND EZEQUIEL RELA Abs ac . In his pape we s udy he p oblem o es ima ing he gen- e alized Hausdo ff dimension o Fu s enbe g se s in he plane. Fo α∈(0,1], a se Fin he plane is said o be an α-Fu s enbe g se i o each di ec ion e he e is a line segmen ℓein he di ec ion o e o which dimH(ℓe∩F)≥α. I is well known ha dimH(F)≥max{2α, α +1 2}- and i is also known ha hese se s can ha e ze o measu e a hei c i - ical dimension. By looking a gene al Hausdo ff measu es Hhdefined o doubling unc ions, ha need no be powe laws, we ob ain fine es ima es o he size o he mo e gene al h-Fu s enbe g se s. Fu he , his app oach allow us o sha pen he known bounds on he dimension o classical Fu s enbe g se s. The main difficul y we had o o e come, was ha i Hh(F) = 0, he e always exis s g≺hsuch ha Hg(F) = 0 (he e ≺ e e s o he na u al o de ing on gene al Hausdo ff dimension unc ions). Hence, in o de o es ima e he measu e o gene al Fu s enbe g se s, we ha e o conside dimension unc ions ha a e a ue s ep down om he c i ical one. We p o ide a he p ecise es ima es on he size o his s ep and by doing so, we can include a amily o ze o dimensional Fu s enbe g se s associa ed o dimension unc ions ha g ow as e han any powe unc ion a ze o. Wi h some addi ional g ow h condi ions on hese ze o dimensional unc ions, we ex end he known inequali ies o include he endpoin α= 0. 1. In oduc ion In his pape we s udy dimension p ope ies o se s o Fu s enbe g ype. We a e able o sha pen he known bounds abou he Hausdo dimension o hese se s using gene al doubling dimension unc ions o he es ima es. Le us ecall he no ion o Fu s enbe g se s. Fo αin (0,1], a subse E o R2is called Fu s enbe g se o Fα-se i o each di ec ion ein he uni ci cle 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. We will also say ha such se Ebelongs o he class Fα. I is known ([18], see also [17], [19], [20], [7], [16] o ela ed opics and [8], [15] o a disc e ized e sion o his p oblem) ha o any Fα-se E⊆R2 he Hausdo dimension (dim(E)) mus sa is y he inequali y dim(E)≥max{2α, α +1 2}and he e a e examples o Fα-se s Ewi h dim(E)≤1 2+3 2α. I we deno e by γ(α) = in {dim(E) : E∈Fα}, 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. This esea ch is pa ially suppo ed by G an s: PICT2006-00177 and UBACyT X149. 1 2 URSULA MOLTER AND EZEQUIEL RELA hen (1) max{α+1 2; 2α} ≤ γ(α)≤1 2+3 2α, α ∈(0,1]. In his pape we s udy a mo e gene al no ion o Fu s enbe g se s. To ha end we will use a ine no ion o dimension al eady de ined by Hausdo [6]. De ini ion 1.1. 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 1.2. 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. As usual, he h-dimensional (ou e ) Hausdo measu e Hhwill be de ined as ollows. Fo a se E⊆R2and δ > 0, w i e Hh δ(E) = in (X i h(diam(Ei)) : E⊂∞ [ i Ei,diam(Ei)< δ). Then he h-dimensional Hausdo measu e Hho Eis de ined by Hh(E) = sup δ>0Hh δ(E). This no ion gene alizes he classical α-Hausdo measu e o unc ions h ha a e di e en o xα. I is well known ha a se o Hausdo dimension α can ha e ze o, posi i e o in ini e α-dimensional measu e. The desi able si ua ion, in gene al, is o wo k wi h a se which is uly α-dimensional, ha is, i has posi i e and ini e α-dimensional measu e. In his case we e e o his se as an α-se . Now, gi en an α-dimensional se Ewi hou his las p ope y, one could expec o ind in he class Han app op ia e unc ion h o de ec he p ecise “size” o i . By ha we mean ha 0 <Hh(E)<∞, and in his case Eis e e ed o as an h-se . We men ion one example: A Kakeya se is a compac se con aining a uni segmen in e e y possible di ec ion. I is known ha he e a e Kakeya se s o ze o measu e and i is conjec u ed ha hey mus ha e ull Hausdo dimension. The conjec u e was p o en by Da ies [2] in R2and emains open o highe dimensions. Since in he class o plana Kakeya se s he e a e se e al dis inc ypes o wo dimensional se s (i.e. wi h posi i e o null Lebesgue measu e), one would like o associa e a dimension unc ion o he whole class. A dimension unc ion h∈Hwill be called he exac Hausdo dimension unc ion o he class o se s Ci •Fo e e y se Ein he class C,Hh(E)>0. •The e a e se s E∈ C wi h Hh(E)<∞. IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 3 In he di ec ion o inding he exac dimension o he class o Kakeya se s in R2, Keich [9] has p o en ha in he case o he Minkowsky dimension he exac dimension unc ion is h(x) = x2log(1 x). Fo he case o he Hausdo dimension, he p o ided some pa ial esul s. Speci ically, he shows ha in his case he exac dimension unc ion hmus dec ease o ze o a he o igin as e han x2log(1 x) log log(1 x)2+ε o any gi en ε > 0, bu slowe han x2log(1 x). This no ion o speed o con e gence o ze o will allow us o de ine a pa ial o de be ween dimension unc ions ha ex ends he usual o de on he powe laws (see 1.3). In his pape we a e in e es ed in he p oblem o s udying he exac Hausdo dimension o he class o Fu s enbe g- ype se s ( o he p ecise de ini ion, see 1.5). We a e able o ind lowe bounds o he dimension unc ion, i.e. o a gi en class o Fu s enbe g- ype se s, we ind a dimension unc ion hwi h he p ope y ha any se in he class has posi i e Hh- measu e. Fo he cons uc ion o h-se s associa ed o ce ain sequences see he wo k o Cab elli e al [1] (see also [5]). We e e o he wo k o Olsen and Ren- o [13], [12], [11] o a de ailed s udy o he exac Hausdo dimension o he Liou ille numbe s L, which is a known example o a ze o dimensional se . Mo eo e , he au ho s p o e ha his is also a dimensionless se , i.e. he e is no h∈Hsuch ha 0 <Hh(L)<∞(equi alen ly, o any di- mension unc ion h, one has Hh(L)∈ {0,∞}). In ha di ec ion, u he imp o emen s a e due o Elekes and Kele i [3]. The e he au ho s p o e much mo e han ha he e is no exac Hausdo -dimension unc ion o he se Lo Liou ille numbe s: hey p o e ha o any ansla ion in a ian Bo el measu e Lis ei he o measu e ze o o has non-sigma- ini e measu e. So in pa icula hey answe he mo e in e es ing ques ion ha he e is no exac Hausdo -dimension unc ion o Le en in he s onge sense when equi ing only sigma- ini eness ins ead o ini eness. 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. 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 1.3. 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. Rema k 1.4. No e ha his pa ial o de , es ic ed o he class o powe unc ions, eco e s he na u al o de men ioned abo e. Tha is, xα≺xβ⇐⇒ α < β. Now we can make a p ecise s a emen o he p oblem. We begin wi h he de ini ion o he Fu s enbe g- ype se s. De ini ion 1.5. Le hbe a dimension unc ion. A se E⊆R2is a Fu s en- be g se o ype h, o an Fh-se , i o each di ec ion e∈S 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 4 URSULA MOLTER AND EZEQUIEL RELA 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 Hhmeasu e o any h≺himplies ha i has also posi i e Hhmeasu e (Theo em 42, [14]). Due o he exis ence o Fα-se s wi h Hα(E∩ℓe) = 0 o each e, i will be use ul o in oduce he ollowing subclass o Fα: De ini ion 1.6. A se E⊆R2is an F+ α-se i o each e∈S he e is a line segmen ℓesuch ha Hα(ℓe∩E)>0. Rema k 1.7. Gi en an Fh-se E o some h∈H, i is always possible o ind wo cons an s mE, δE>0 and a se ΩE⊆So posi i e σ-measu e such ha Hh δ(ℓe∩E)> mE>0∀δ < δE,∀e∈ΩE. Fo each e∈S, he e is a posi i e cons an mesuch ha Hh(ℓe∩E)> me. Now conside he ollowing pigeonholing a gumen . Le Λn={e∈S: 1 n+1 ≤me<1 n}. A leas one o he se s mus ha e posi i e measu e, since S=∪nΛn. Le Λn0be such se and ake 0 <2mE<1 n0+1 . Hence Hh(ℓe∩E)>2mE>0 o all e∈Λn0Finally, again by pigeonholing, we can ind ΩE⊆Λn0o posi i e measu e and δE>0 such (2) Hh δ(ℓe∩E)> mE>0∀e∈ΩE∀δ < δE. To simpli y no a ion h oughou he pape , since inequali y (2) holds o any Fu s enbe g se and we will only use he ac ha mE,δEand σ(ΩE) a e posi i e, i will be enough o conside he ollowing de ini ion o Fh-se s: De ini ion 1.8. Le hbe a dimension unc ion. A se E⊆R2is Fu s enbe g se o ype h, o an Fh-se , i o each e∈S he e is a line segmen ℓein he di ec ion o esuch ha Hh δ(ℓe∩E)>1 o all δ < δE o some δE>0. The pu pose o his pape is o ob ain an es ima e o he dimension o an Fh-se . By analogy o he classical es ima e (1), we i s no e ha i his a gene al dimension unc ion (no xα), α+1 2 ansla es o h√·and 2α o h2. Hence, when aiming o ob ain an es ima e o he Hausdo measu e o ou se E, he nai e app oach would be o p o e ha i a dimension unc ion h sa is ies (3) h≺h2o h≺h√·, hen Hh(E)>0. Howe e , he e is no hope o ob ain such a gene al esul , since o he special case o he iden i y unc ion h(x) = x, his equi emen would con adic (again by Theo em 42, [14]) he exis ence o ze o measu e plana Kakeya se s. The e o e, i is clea ha one needs o ake a s ep down om he con- jec u ed dimension unc ion. The main esul o his pape is o show ha his s ep does no need o be as big as a powe . I can be, o example, jus he powe o a log. P ecisely, we ind condi ions on he s ep ha gua an ee IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 5 lowe bounds on he dimension o Fh-se s. Fu he , ou echniques allow us o analyze Fu s enbe g- ype se s o Hausdo dimension ze o. This can be done conside ing dimension unc ions h ha a e smalle han xα o any α > 0. To keep he analogy wi h he classical Fu s enbe g se s, we will in oduce he ollowing no a ion: De ini ion 1.9. Gi en wo dimension unc ions g, h ∈H, we de ine he ollowing quo ien s which a e ela ed o he s ep-size be ween wo unc ions: ∆0(g, h)(x) := ∆0(x) = g(x) h(x)∆1(g, h)(x) := ∆1(x) = g(x) h2(x). When p o ing he i s case o he inequali ies in (3), he ele an quo ien is ∆1, which gi es he (be e ) bound dim ≥2αin he classical case a he endpoin α= 1. A he o he endpoin , α= 0, he bes bound is dim ≥α+1 2 and he quo ien o analyze he e in ou gene alized p oblem is ∆0. This pape is o ganized as ollows. In Sec ion 2, we in oduce some u he no a ion and p o e a p elimina y lemma o be used in he emainde o he pape . In Sec ion 3 we p o e he h2bound, in Sec ion 4 he h√·bound unde some posi i i y assump ions on he unc ion hand in Sec ion 5 we d op his las condi ion o ob ain a pa ial esul on he ze o dimensional Fu s enbe g se s. In addi ion we discuss ou me hods and s udy he Fu s enbe g p oblem in he ex eme case o he coun ing measu e. This is, oughly speaking, he case h≡1. 2. Rema ks, no a ion and mo e de ini ions 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 Aand B. By A∼Bwe mean ha bo h A.Band B.Ahold. On he ci cle Swe conside he a cleng h measu e σ. By L2(S) we mean L2(S, dσ). Fo each e∈S,ℓewill be a uni line segmen in he di ec ion e. As usual, by a δ-co e ing o a se Ewe mean a co e ing o Eby se s Uiwi h diame e s no exceeding δ. We in oduce he ollowing no a ion: De ini ion 2.1. 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 (4) Jb k:= {j∈N:bk< j≤bk−1}, and 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 (5) Jk:= {j∈N: 2−k< j≤2−k+1} The nex lemma in oduces a echnique we bo ow om [18] o decompose he se o all di ec ions. 6 URSULA MOLTER AND EZEQUIEL RELA Lemma 2.2. Le Ebe an Fh-se o some h∈Hand a={ak}k∈N∈ℓ1a 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 Ωk:= e∈S:Hh δ(ℓe∩Ek)≥ak 2kak1. Then S=∪kΩk. P oo . Clea ly Ωk⊂S. To see why S=∪kΩk, assume ha he e is a di ec ion e∈S ha is no in any o he Ωk. Then o ha di ec ion we would ha e ha 1<Hh δ(ℓe∩E)≤X kHh δ(ℓe∩E∩[ j∈Jk Bj)≤X k 1ak 2kak1 =1 2, which is a con adic ion.  As a inal ema k we no e ha in he ollowing sec ions ou aim will be o p o e essen ially (6) X j h( j)&1, p o ided ha his a small enough dimension unc ion. The idea will be o use he dyadic pa i ion o he co e ing o ob ain ha X j h( j)&∞ X k=0 h(2−k)#Jk. The lowe bounds we need will be ob ained i we can p o e lowe bounds on he quan i y Jkin e ms o he unc ion hbu independen o he co e ing. 3. The h→h2bound In his sec ion we gene alize he i s inequali y o (1), ha is, dim(E)≥ 2α o any Fα-se . Fo his, gi en a dimension unc ion h≺h2, we impose some su icien g ow h condi ions on he gap ∆1(x) := h(x) h2(x) o ensu e ha Hh(E)>0. We ha e he ollowing heo em: Theo em 3.1. Le h∈Hdbe a dimension unc ion and le Ebe an Fh-se . Le h∈Hsuch ha h≺h2. I X k h(2−k)qk h(2−k)<∞, hen Hh(E)>0. The main ool o he p oo o his heo em will be an L2bound o he Kakeya maximal unc ion on R2. Fo an in eg able unc ion on Rn, he Kakeya maximal unc ion a scale δwill be ∗ δ:Sn−1→R, ∗ δ(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. I is well known ha in R2 he Kakeya maximal unc ion sa is ies he bound (see [18]) IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 7 (7)   ∗ δ 2 2.log(1 δ)k k2 2. I is also known ha he log g ow h is necessa y (see [9]), because o he exis ence o Kakeya se s o ze o measu e in R2. See also [10] o es ima es on he Kakeya maximal unc ion wi h mo e gene al measu es on he ci cle. We now p o e Theo em 3.1. We ema k ha since his heo em says, oughly speaking, ha he dimension o an Fh-se should be abou h2, he s ep down mus be aken om his dimension unc ion. This is he ole played by ∆1(h, h2)(x) = h(x) h2(x)in his sec ion. P oo . By 1.8, since E∈Fh, we ha e Hh δ(ℓe∩E)>1 o all e∈Sand o any δ < δE. 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 (8) X j h( j)&1 o any h∈Hsa is ying he hypo hesis o he heo em. Clea ly we can es ic ou sel es o δ-co e ings wi h δ < δE 5. De ine a={ak}wi h ak=qk ∆1(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= E∩∪j∈JkBj. Since by hypo hesis a∈ℓ1, we can apply Lemma 2.2 o ob ain he decomposi ion S=SkΩkassocia 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. The L2 no m o can be easily es ima ed as ollows: 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. Hence, (9) k k2 2.#Jkh2(2−k). Now ix kand conside he Kakeya maximal unc ion ∗ δo le el δ= 2−k+1 associa ed o he unc ion de ined o his alue o k. 8 URSULA MOLTER AND EZEQUIEL RELA In Ωkwe ha e he ollowing poin wise lowe es ima e o he maximal unc ion. Le ℓebe he line segmen such ha Hh δ(ℓe∩E)>1, and le Tebe he ec angle o wid h 2−k+2 a ound his segmen . De ine, o each e∈Ωk, (10) Jk(e) := {j∈Jk:ℓe∩E∩Bj6=∅}. Wi h he aid o he Vi ali co e ing lemma, we can selec a subse o disjoin balls e Jk(e)⊆Jk(e) such ha [ j∈Jk(e) Bj⊆[ j∈ e Jk(e) B(xj; 5 j). No e ha e e y ball Bj,j∈Jk(e), in e sec s ℓeand he e o e a leas hal o Bjis con ained in he ec angle Te, yielding |Te∩Bj| ≥ 1 2π 2 j. Hence, by de ini ion o he maximal unc ion, using ha j≥2−k+1 o j∈Jk(e), | ∗ 2−k+1 (e)| ≥ 1 |Te|ZTe dx =h(2−k)2k |Te|Te∩∪Jk(e)Bj &h(2−k)22kTe∩∪e Jk(e)Bj &h(2−k)22kX j∈ e Jk(e) 2 j &h(2−k)# e Jk(e) &X e Jk(e) h( j). Now, since (11) ℓe∩Ek⊆[ j∈Jk(e) Bj⊆[ j∈ e Jk(e) B(xj; 5 j) and o e∈Ωkwe ha e Hh δ(ℓe∩Ek)&ak, we ob ain | ∗ 2−k+1 (e)|&X e Jk(e) h( j)&X j∈ e Jk(e) h(5 j)&ak. The e o e we ha e he es ima e (12) k ∗ 2−k+1 k2 2&ZΩk| ∗ 2−k+1 (e)|2dσ &a2 kσ(Ωk) = σ(Ωk)k ∆1(2−k). Combining (9), (12) and using he maximal inequali y (7), we ob ain σ(Ωk)k ∆1(2−k).k ∗ 2−k+1 k2 2.log(2k)k k2 2.k#Jkh2(2−k), and he e o e σ(Ωk) h(2−k).#Jk. IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 9 Now we a e able o es ima e he sum in (8). 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 σ(Ωk)≥σ(S)>0.  Applying his heo em o he class F+ α, we ob ain a sha pe lowe bound on he gene alized Hausdo dimension: Co olla y 3.2. Le Ean F+ α-se . I his any dimension unc ion sa is ying h(x)≥Cx2αlog1+θ(1 x) o θ > 2 hen Hh(E)>0. Rema k 3.3. A he endpoin α= 1, his es ima e is wo se han he one due o Keich. He ob ained, using s ongly he ull dimension o a ball in R2, ha i Eis an F+ 1-se and his a dimension unc ion sa is ying he bound h(x)≥Cx2log(1 x)log log(1 x)θ o θ > 2, hen Hh(E)>0. Rema k 3.4. No e ha he p oo abo e elies essen ially on he L1and L2size o he ball in R2, no on he dimension unc ion h. Mo eo e , we only use he “gap” be ween hand h2(measu ed by he unc ion ∆1). This las obse a ion leads o conjec u e ha his p oo can no be used o p o e ha an Fh-se has posi i e h2measu e, since in he case o h(x) = x, as we ema ked in he in oduc ion, his would con adic he exis ence o Kakeya se s o ze o measu e in R2. Also no e ha he absence o condi ions on he unc ion hallows us o conside he “ze o dimensional” Fu s enbe g p oblem. Howe e , his bound does no p o ide any subs an ial imp o emen , since he ze o dimensionali y p ope y o he unc ion his sha ed by he unc ion h2. This is because he p oo abo e, in he case o he Fα-se s, gi es he wo se bound (dim(E)≥2α) when he pa ame e αis in (0,1 2). 4. The h→h√·bound In his sec ion we will u n ou a en ion o hose unc ions h ha sa is y he bound h(x).xα o α≤1 2. Fo hese unc ions we a e able o imp o e on he p e iously ob ained bounds. We need o impose some g ow h con- di ions on he dimension unc ion h. This condi ions can be hough o as imposing a lowe bound on he dimensionali y o h o keep i away om he ze o dimensional case. Rema k 4.1. Th oughou his sec ion, he expec ed dimension unc ion should be abou h√·. We he e o e need a s ep down om his unc ion. Fo his, we will look a he gap ∆0(x) = h(x) h(x). The nex lemma says ha we can spli he h-dimensional mass o a se E con ained in an in e al Iin o wo se s ha a e posi i ely sepa a ed. Lemma 4.2. 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. 16 URSULA MOLTER AND EZEQUIEL RELA wi h some sui able modi ica ions o he cons uc ion made in [18], Rema k 1.5, (p. 10). The e, o each 0 < α ≤1, an Fα-se is cons uc ed whose dimension is no g ea e han 1 2+3 2α. I is s aigh o wa d o modi y ha cons uc ion o i o hold e en a he endpoin α= 0. We also include he ollowing example o an F2 0-se Go dimension ze o. I will be cons uc ed using he nex esul , which is Example 7.8 (p. 104) in [4]. In ha example, Falcone cons uc s se s E, F ⊆[0,1] wi h dim(E) = dim(F) = 0 and such ha [0,1] ⊆E+F. E×{1} −F×{0} x −y θ Figu e 3. Conside G=E×{1}∪−F×{0}. This se Ghas clea ly dimension 0, and con ains wo poin s in e e y di ec ion θ∈[0; π 4]. Fo , i θ∈[0; π 4], le c= an(θ), so c∈[0,1]. By he choice o Eand F, we can ind x∈Eand y∈Fwi h c=x+y. The poin s (−y, 0) and (x, 1) belong o Gand de e mine a segmen in he di ec ion θ(Figu e 3 on page 16). 6. Acknowledgmen s We would like o hank o Michael T. Lacey o ui ul con e sa ions du ing his isi o he Depa men o Ma hema ics a he Uni e si y o Buenos Ai es. We also hank he anonymous e e ee o ex emely ca e ul eading o he manusc ip and poin ing ou many sub le imp o emen s which made his pape mo e eadable. Re e ences [1] Ca los Cab elli, F anklin Mendi il, U sula M. Mol e , and Ronald Shonkwile . On he Hausdo ff h-measu e o Can o se s. Paci ic J. Ma h., 217(1):45–59, 2004. [2] Roy O. Da ies. Some ema ks on he Kakeya p oblem. P oc. Camb idge Philos. Soc., 69:417–421, 1971. [3] M´a on Elekes and Tam´as Kele i. Bo el se s which a e null o non-σ-fini e o e e y ansla ion in a ian measu e. Ad . Ma h., 201(1):102–115, 2006. [4] Kenne h Falcone . F ac al geome y. John Wiley & Sons Inc., Hoboken, NJ, second edi ion, 2003. Ma hema ical ounda ions and applica ions. 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