The power law for the Buffon needle probability of the four-corner Cantor set

THE PO WER LA W F OR THE BUFF ON NEEDLE PR OBABILITY OF THE F OUR-CORNER CANTOR SET FEDOR NAZAROV, YUV AL PERES, AND ALEXANDER V OLBER G Abstra ct. Let C n b e the n -th generation in the construction of the middle- half Can tor set. The Cartesian square K n of C n consists of 4 n squares of side- length 4 − n . The c hance that a long n eedle thrown at random in t he un it square will meet K n is es sentia lly the a v erage length of the pro jections of K n , also known as the F av a rd length of K n . A classical theorem of Besico vitc h implies that the F a v ard length of K n tends to zero. It is still an op en problem to determine its exact rate of decay . Until recen t ly , the only exp licit upp er b ound w as exp( − c log ∗ n ), d ue to Peres and Solomy ak . (log ∗ n is the num b er of times one needs to take log to obtain a num b er less than 1 starting from n ). W e obtain a p o w er la w b ound by com bining analytic and combinatorial ideas. 1. In tro duction The four-corner Can tor set K is constructed b y replaci ng t he unit square b y four su b-squares of side length 1 / 4 at its co rners, and iterating th is op eration in a self-similar mann er in eac h su b-square. Mo re formally , consider the set C n that is the union of 2 n segmen ts: C n = [ a j ∈{ 0 , 3 } ,j =1 , . .,n h n X j =1 a j 4 − j , n X j =1 a j 4 − j + 4 − n i , and let the middle h alf Can tor set b e C := ∞ \ n =1 C n . It can also b e written as C = { P ∞ n =1 a n 4 − n : a n ∈ { 0 , 3 } } . The four corner Cantor set K is the Cartesian square C × C . Since th e one-dimensional Hausdorff measure of K satisfies 0 < H 1 ( K ) < ∞ and the pro jections of K in t w o distinct dir ections ha ve zero length, a theorem of 1991 Mathematics Subje ct C l assific ation. Primary: 28A80; Secondary: 28A75, 60D05, 28A78. Researc h of the authors w as supp orted in part by NS F grants DMS- 050106 7 (Nazaro v and V olb erg) and DMS-0605166 (Peres). 1 2 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER V OLBERG Figure 1. K 3 , the third sta ge of th e construction of K . Besico vitc h (see[3, Theorem 6.1 3]) yields that the pr o j ection of K to almost ev ery line through t he origi n has ze ro length. This is e quiv alen t to sa ying that th e F a v ard length of K equals zero. Recall (see [1, p.357]) th at the F av ard length of a planar set E is d efined by F av( E ) = 1 π Z π 0 | Pro j R θ E | dθ , (1.1) where Pro j denotes the orthogonal pr o jection fr om R 2 to the h orizon tal axis, R θ is the counte rclo c k w ise rotation by angle θ , and | A | denotes the L eb esgue m easure of a measurable set A ⊂ R . The F a v ard length of a set E in the un it square has a probabilistic inte rpretation: up to a constan t f actor, it is the p robabilit y that the “Buffon’s needle,” a long line segment dropp ed at random, hits E (more precisely , supp ose the needle’s length is infi nite, pick its dir ection uniformly at random, and then lo cate the n eedle in a uniformly chose n p osition in that direction, at distance at m ost √ 2 from the ce n ter of the unit square). The set K n = C 2 n is a union of 4 n squares with s ide length 4 − n (see Figure 1 for a picture of K 3 ). By the dominated con vergence theorem, F a v ( K ) = 0 im p lies lim n →∞ F av( K n ) = 0. W e are int erested in goo d estimat es for F av( K n ) as n → ∞ . A low er b ound F a v ( K n ) ≥ c n for some c > 0 follo ws from Mattila [8, 1.4]. P er es and Solom y ak [10] p ro ved that BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 3 F av( K n ) ≤ C exp[ − a log ∗ n ] for all n ∈ N , where log ∗ n = min    k ≥ 0 : log lo g . . . log | {z } k n ≤ 1    . This result can b e view ed as an attempt to make a qu an titativ e statemen t out of a qualitativ e Besico vitc h pr o j ection th eorem [1], [11], using this canonical example of the Besico vitc h irregular set. It is v ery inte resting to see what are quan titativ e analogs of Besico vitc h theo rem in general. The r eader can find more of that in [11]. W e no w state our m ain result, whic h impr o ves this upp er b ound to a p ow er la w. Theorem 1. F or every δ > 0 , ther e exists C > 0 such that F av( K n ) ≤ C n δ − 1 / 6 for al l n ∈ N . Remarks. • The 1 / 6 in the exp onent is certainly not optimal, and, ind eed, can b e impro v ed sligh tly w ith the metho ds of this pap er. Ho w ever, a b oun d deca ying faster than O  n − 1 / 4  w ould r equire n ew id eas. • In [10], Th eorem 2.2, a random analog of the Can tor set K is analyzed, and it is sh o wn that, w ith high p robabilit y , the F a v ard length of th e n -th stage in the co nstruction has u pp er and lo wer b ounds that are constant m ultiples of n − 1 . Ho w ev er, it is not clear to u s whether F a v ( K n ) also deca ys at this rate. • It follo ws from the results of Keny on [5] and Lagarias and W ang [6] that | Pr o j R θ K| = 0 for all θ such that tan θ is irrational. As n oted in [10], this information do es not seem to h elp obtain an upp er b ound for F av( K n ). • The set K wa s one of the first examples of sets of p ositiv e length and zero analytic capacit y , see [2] f or a survey . T he asymptotic b ehavior of the analytic capacit y of K n w as determined in 2003 by Mateu, T olsa and V erd er a [7], it is equiv alent to 1 √ n . It will b e con venien t to translate K n so that its con ve x hull is the unit square cen tered at the origi n. Due to the symmetries of the square, one can a verag e ov er 4 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G θ ∈ (0 , π 4 ) in the defi n ition (1 .1) of F a v ( K n ). After translation, the pro jection of R θ  K n − ( 1 2 , 1 2 )  to the horizont al axis is the union of 4 n in terv als of length 4 − n (cos θ + sin θ ) cen tered at the p oin ts P n − 1 k =0 4 − k ξ k , wh ere ξ k ∈ n ± 3 √ 2 8 cos( π 4 − θ ) , ± 3 √ 2 8 sin( π 4 − θ ) o . Let now t = tan( π 4 − θ ) ∈ [0 , 1 ]. Since √ 2 2 ≤ cos( π 4 − θ ) ≤ 1 on (0 , π 4 ), the length of the p ro jection P ro j R θ ( K n ) is comparable to the length of the un ion of 4 n in terv als of length 4 − n ρ cen tered at the p oin ts P n − 1 k =0 4 − k ξ k with ξ k ∈ {± 1 , ± t } , where ρ = ρ ( θ ) = 8 3 √ 2 (1 + tan( π 4 − θ )). The exact v alue of ρ ( θ ) is of no imp ortance, the only thin g that matters is that it is separated from b oth 0 and + ∞ . W e shall also need the function f n that is th e p ro duct of 1 ρ and the su m of the charac teristic functions of these in terv als. In other words, f n = ν ( n ) ∗ 4 n ρ χ [ − ρ 2 4 − n , ρ 2 4 − n ] , where ν ( n ) = ∗ n − 1 k =0 ν k , and ν k = 1 4 [ δ − 4 − k + δ − 4 − k t + δ 4 − k t + δ 4 − k ] . Geometrical ly , f n is (up to minor r escaling) the n umber of squares wh ose pr o- jections con tain a giv en p oin t. Finally , since | dt dθ | = 1 cos 2 ( π 4 − θ ) is b etw een 1 and 2 for all θ ∈ [0 , π 4 ), w e can replace a verag ing o v er θ with that o v er t . 2. F ourier-ana l y tic p ar t. In what follo ws, we will use ≍ and . , & to denote, resp ectiv ely , equalit y or the corresp onding inequalit y up to some p ositiv e multiplic ativ e constant. Let K, S b e large p ositive num b ers. Our fi rst aim is to sho w th at there exists a p o wer p > 0 (w e’ll see that any p > 4 fits) such that the measure of the set E =  t ∈ [0 , 1] : max 1 ≤ n ≤ ( K S ) p Z R f 2 n ≤ K  is at most 1 S . S upp ose not. Let N b e th e least ev en intege r exceeding 1 2 ( K S ) p . F or ev ery t ∈ E , w e m ust ha ve K ≥ Z R | f N ( x ) | 2 dx ≍ Z R | b f N ( y ) | 2 dy & Z 4 N/ 2 1 | b ν ( N ) ( y ) | 2 dy , BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 5 b ecause ψ = 4 N ρ χ [ − ρ 2 4 − N , ρ 2 4 − N ] satisfies b ψ ( y ) & 1 for all | y | < 4 N/ 2 if N is suffi- cien tly large. Th us 1 | E | Z E h N/ 2 X n =1 Z 4 n 4 n − 1 | b ν ( N ) ( y ) | 2 dy i dt ≤ K and for eac h m ≤ N / 2, there exists n ≤ N / 2 satisfying 1 | E | Z E h Z 4 n 4 n − m | b ν ( N ) ( y ) | 2 dy i dt ≤ 4 K m N . Th us E ∗ = n t ∈ E : Z 4 n 4 n − m | b ν ( N ) ( y ) | 2 dy ≤ 8 K m N o satisfies | E ∗ | ≥ | E | / 2. Our assu m ption on E implies th at | E ∗ | ≥ 1 2 S . No w for y ∈ [4 n − m , 4 n ], w e ha ve | b ν ( N ) ( y ) | 2 ≍ n Y k =0    cos 4 − k y + cos 4 − k ty 2    2 , since the remaining terms (that corresp ond to k ∈ [ n + 1 , N ]) in the pro duct con verge geo metrically to 1. Making th e c hange of v ariable y 7→ 4 n y , w e get Z 4 n 4 n − m | b ν ( N ) ( y ) | 2 dy ≍ 4 n Z 1 4 − m    n Y k =0 cos 4 k y + cos 4 k ty 2    2 dy . No w split the last pro duct in to P 1 ( y ) = m Y k =0 cos 4 k y + cos 4 k ty 2 and P 2 ( y ) = n Y k = m +1 cos 4 k y + cos 4 k ty 2 . Consider the int egral Z 1 4 − m | P 2 ( y ) | 2 dy first. W riting the cosines as sums of exp onent ials, w e ha ve P 2 ( y ) = 4 m − n 4 n − m X j =1 e iλ j y , where { λ j } 4 n − m j =1 are the sums of all subs ets of {± 4 k , ± 4 k t : k ∈ [ m + 1 , n ] } . F or t ∈ E ∗ ⊂ E , the definition of E yields that 6 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G 1 −1/L 1/L 0 2/L 1 −1 −2/L Figure 2. T riangle k ernel function. Z R  X j χ [ λ j − ρ 2 4 m ,λ j + ρ 2 4 m ]  2 ≤ K · 4 n (this is equ iv alen t to R R f 2 n − m ≤ K ). The last in equalit y can b e viewed as a separation co ndition on the sp ectrum, so one can h op e th at a v ariation of Salem’s tric k sh ould allo w u s to conclude that Z 1 4 − m | P 2 ( y ) | 2 dy & 4 m − n , pro vided that L = 4 m is c h osen appropriately . W e sh all choose m su c h that 4 m = L is a large constan t m ultiple of K . S ince | P 2 ( − y ) | = | P 2 ( y ) | , we can in tegrate o ver [ − 1 , 1] \ [ − L − 1 , L − 1 ]. Consider the function g giv en by g ( y ) = (1 − | y | ) + − 2(1 − L − 1 )(1 − L 2 | y | ) + + (1 − 2 L − 1 )(1 − L | y | ) + . Note th at g is ev en , 0 ≤ g ≤ 1, s upp g ⊂ [ − 1 , 1] \ [ − L − 1 , L − 1 ] and R 1 − 1 g ≥ 1 2 if L is n ot to o small. No w, let h denote “the triangle fu n ction” that is 1 at 0, v anishes on R \ ( − 1 , 1) and is linear on [ − 1 , 0] and on [0 , 1]. Then g ( y ) = h ( y ) − 2(1 − L − 1 ) h ( L 2 y ) + (1 − 2 L − 1 ) h ( Ly ) , . As b h ( λ ) = 2 1 − cos λ λ 2 ∈ [0 , C 1+ λ 2 ], w e get b g ( λ ) ≥ − C L · 1 1 + ( λ/L ) 2 . BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 7 So w e got the estimate b g ( λ ) ≥ − C L λ 2 + L 2 with some numerical constan t C . Denote M = 4 n − m . Let us call k ∈ { 1 , . . . , M } go o d if C X j L L 2 + ( λ j − λ k ) 2 ≤ 1 8 . Then Z [ − 1 , 1] \ [ − L − 1 ,L − 1 ]    X k e iλ k y    2 dy ≥ Z R g ( y )    X k e iλ k y    2 dy ≥ X { k : k is goo d } 1 2 + Z R g ( y )    X { k : k is bad } e iλ k y    2 dy − 2 X { k : k is go o d } C X j L L 2 + ( λ j − λ k ) 2 ≥ 1 4 # { k : k is goo d } . No w we need only to sho w th at the num b er of go o d in dices is comparab le to M . T o this end, note that we hav e the condition Z R  X j χ [ λ j − ρ 2 L , λ j + ρ 2 L ] ( λ )  2 dλ ≤ M LK . Con v olving with the P oisson ke rnel P L ( λ ) = 1 π L L 2 + λ 2 and taking into acco unt that χ [ λ j − ρ 2 L,λ j + ρ 2 L ] ∗ P L ≥ cL P L ( · − λ j ) with c > 0 (here we u se that ρ sta ys b ounded aw a y fr om 0 and + ∞ ), w e get L 2 Z R h X j P L ( λ − λ j ) i 2 dλ ≤ C ′ M LK , but Z R P L ( λ − λ j ) P L ( λ − λ k ) dλ ≥ c ′ P L ( λ j − λ k ) . Th us c ′ X j,k P L ( λ j − λ k ) ≤ C ′ M K L − 1 and # { k : k is bad } ≤ 8 C π C ′ c ′ ( K L − 1 ) M ≤ M 2 , pro vided that L ≥ 16 C π C ′ c ′ K . Therefore, ind eed, R 1 4 − m | P 2 ( y ) | 2 dy & 4 m − n . 8 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G The danger is that this large in tegral can b e completely killed when the integrand is m ultiplied b y | P 1 | 2 . Note that cos 4 k y + cos 4 k ty 2 = cos 2 − 1 4 k ( y + ty ) cos 2 − 1 4 k ( y − ty ) so P 1 ( y ) = m Y k =0 cos 2 − 1 4 k ( y + ty ) cos 2 − 1 4 k ( y − ty ) . Using the formula 2 · 4 m sin( u 2 ) 2 m Y ℓ =0 cos 2 ℓ − 1 u = sin 4 m u . w e co nclude that | P 1 ( y ) | & 4 − 2 m | sin 4 m ( y + ty ) | · | sin 4 m ( y − ty ) | . This can b e small only if sin 4 m ( y + ty ) or sin 4 m ( y − ty ) is small. F or δ ∈ (0 , 1), denote by I δ the union of in terv als of length 4 − m δ centered at the p oin ts π ℓ 4 m , ℓ ∈ Z . Define ω ( t ; δ ) b y ω ( t ; δ ) = { y ∈ (4 − m , 1) : y + ty ∈ I δ or y − ty ∈ I δ } . W e would lik e to estimate R ω ( t ; δ ) | P 2 ( y ) | 2 dy f rom ab ov e. This ma y b e a hard task for an ind ividual t ∈ E ∗ , bu t we can boun d th e a v erage fairly ea sily . W e ha ve 1 | E ∗ | Z E ∗  Z ω ( t ; δ, ) | P 2 ( y ) | 2 dy  dt ≤ 2 S Z 1 0  Z ω ( t ; δ, ) | P 2 ( y ) | 2 dy  dt . 2 S Z [4 − m , 1] ∩I δ  n Y k = m +1 cos 2 2 − 1 4 k u  du u + v Z [0 , 1]  n Y k = m +1 cos 2 2 − 1 4 k v  dv + 2 S Z [4 − m , 1]  n Y k = m +1 cos 2 2 − 1 4 k u  du u + v Z [0 , 1] ∩I δ  n Y k = m +1 cos 2 2 − 1 4 k v  dv where u = y + ty and v = y − ty . Using the formula cos 2 α = 1 2 (1 + cos 2 α ) and the inequalit y 1 u + v ≤ L du dv , we can estima te the la st expression by E := C S L · 4 m − n h Z I n Y k = m +1 (1 + cos 4 k u ) du i · h Z [0 , 1] n Y k = m +1 (1 + cos 4 k v ) dv i . BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 9 Ab o v e we observ ed that n Y k = m +1 (cos 2 2 − 1 4 k u ) = 2 m − n n Y k = m +1 (1 + cos 4 k u ) = : 2 m − n R ( u ) . W e wa n t to see no w that E ≤ c S L · 4 m − n √ δ . T o this end we notice that the Riesz p ro duct R ( u ) is π 4 m -p erio dic. Note also that, for an y in terv al J of length 4 − m π 4 j ( j ∈ Z + ), w e ha ve Z J R ( u ) du = Z J R 1 ( u ) R 2 ( u ) du, where R 1 ( u ) = Q m + j k = m +1 (1 + co s 4 k u ) and R 2 ( u ) = Q n k = m + j +1 (1 + co s 4 k u ). Ob serv e that R 1 ( u ) ≤ 2 j for all u and R 2 ( u ) is π 4 m + j -p erio dic, so Z J R 2 ( u ) du = 1 4 m + j Z π 0 R 2 ( u ) du = π 4 m + j . Th us R J R ( u ) du ≤ π 2 j 4 − m . Ch o ose j in s uc h a w a y that δ ≍ 4 − j . It follo ws that, for eac h constituting in terv al J of I δ , w e ha v e R J R ( u ) du . 4 − m √ δ . Z [4 − m , 1] ∩I δ R ( u ) du . 4 m · 4 − m √ δ . √ δ . In co njun ction with the estimate R [0 , 1] ∩I η R ( v ) du . 1, w e finally get E ≤ c S L · 4 m − n √ δ . The resu lting estimate is muc h less than 4 m − n if δ is m uc h less than S − 2 L − 2 . Th us, for at least one t ∈ E , w e m u st h a ve (recall that L = 4 m ) Z [ L − 1 , 1] \ Ω( t ) | P 2 ( y ) | 2 dy ≥ c 4 m − n and, thereb y , (if we rememb er that K w as a small constant ti mes 4 m ) Z [ L − 1 , 1] | P 1 ( y ) | 2 | P 2 ( y ) | 2 dy ≥ 4 − 4 m ( S − 2 L − 2 ) 4 · 4 m − n ≥ cS − 8 K − 11 4 − n . 10 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G Th us, if p > 12 then our c hoice of N at the b eginnin g of the pro of giv es N > ( K S ) 12+ ε / 2 > ( K S ) 12+ ε/ 2 , h ence 2 K log K N is m uc h less than S − 8 K − 11 , and w e get a con tradiction. Ho wev er, we promised to sh o w that N > ( K S ) 4+ ε already lea ds to a cont radic- tion. T o d o this, w e make our consid er ations more elab orate, b ut w e follo w the same lines. In fact, let us consider Ω( t ; δ, η ) = { y ∈ (4 − m , 1) : y + ty ∈ I δ and y − ty ∈ I η } . W e c hanged the w ord “or” in the definition of ω ( t ; δ ) by the wo rd “and” in th e definition of Ω( t ; δ, η ). This will allo w us to mak e a sub tler estimate. No tice that { y : | sin 4 m ( y + ty ) | · | sin 4 m ( y − ty ) | ≤ 2 − l } ⊂ ℓ [ k =0 Ω( t ; 2 − k , 2 − ℓ + k +1 ) . W e wo uld lik e to estimate R Ω( t ; δ,η ) | P 2 ( y ) | 2 dy from abov e. As b efore w e ha ve 1 | E ∗ | Z E ∗  Z ω ( t ; δ, ) | P 2 ( y ) | 2 dy  dt ≤ 2 S Z 1 0  Z Ω( t ; δ,η ) | P 2 ( y ) | 2 dy  dt . 2 S Z [4 − m , 1] ∩I δ  n Y k = m +1 cos 2 2 − 1 4 k u  du u Z [0 , 1] ∩I η  n Y k = m +1 cos 2 2 − 1 4 k v  dv where u = y + ty and v = y − ty as before. W e already int ro duced R ( u ) = Q n k = m +1 (1+cos 4 k u ) and established the follo wing estimate Z [0 , 1] ∩I η R ( v ) dv . √ η . No w w e can estimate Z [4 − m , 1] ∩I δ R ( u ) du u . X 1 ≤ j ≤ 1 π 4 m 4 m π j · 4 − m √ δ ≤ √ δ m . Therefore, we obtain 1 | E ∗ | Z E ∗  Z Ω( t ; δ,η ) | P 2 ( y ) | 2 dy  dt . S m p δ η 4 m − n . Let us denote Ω ℓ ( t ) := S ℓ k =0 Ω( t ; 2 − k , 2 − ℓ + k +1 ). W e kno w no w that 1 | E ∗ | Z E ∗  Z Ω ℓ ( t ) | P 2 ( y ) | 2 dy  dt . S mℓ · 2 − ℓ/ 2 4 m − n . BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 11 If S mℓ · 2 − ℓ/ 2 is a small constan t (m uc h less than 1), then it follo w s that there exists t ∈ E ∗ suc h that Z [4 − m , 1] \ Ω ℓ ( t ) | P 2 ( y ) | 2 dy ≥ c · 4 m − n . But Ω ℓ ( t ) con tains { y : | sin 4 m ( y + ty ) | · | sin 4 m ( y − ty ) | ≤ 2 − l } . This means that | P 1 | & 4 − 2 m 2 − ℓ on (4 − m , 1) \ Ω ℓ , so for this t , w e ha ve Z 1 4 − m | P 2 ( y ) | 2 dy ≥ 4 − 4 m 2 − 2 ℓ 4 m − n . If K m N is muc h less than 4 − 4 m 2 − 2 ℓ 4 m , we get a con tradiction. Sin ce 4 m ≍ K , we see that w e get a con tradiction if it is possib le to fi nd ℓ suc h th at S mℓ · 2 − ℓ/ 2 is muc h less than 1 and N is muc h greater than K 4 m 2 2 ℓ sim ultaneously . If N > ( K S ) 4+ γ with γ > 0, we can tak e 2 ℓ/ 2 ≍ ( S K ) γ 8 S , thus finish ing the p ro of of ou r cl aim with an y p > 4. 3. Combina torial p ar t. Fix the rotation angle θ and some large p ositive inte ger N . As b efore, let F n ( x ) b e the num b er of the squares in R θ K n whose pr o j ections to the h orizon tal axis con tain x . Define F ∗ ( x ) = max 0 ≤ n ≤ N F n ( x ) . Our k ey observ ation is the follo win g inequalit y: for any p ositiv e integ ers K, M , w e ha ve µ { F ∗ ≥ 4 K M } ≤ 108 K µ { F ∗ ≥ K } µ { F ∗ ≥ M } , where µ denotes the u sual Leb esgue measure on the real line. Pro of: F or eac h p oin t x ∈ R where F ∗ ( x ) ≥ 2 K , c ho ose the least n = n ( x ) f or whic h F n ( x ) ≥ 2 K and mark all the squares in R θ K n whose pro jections con tain x . Note that the n u m b er of suc h squ ares for a giv en p oin t x cannot exceed 4 K : otherwise we w ould h a ve F n − 1 ( x ) ≥ 2 K , w h ic h contradicts our choi ce of n . No w unmark all marked squares th at are con tained in larger mark ed squares and consider the f amily of the r emaining maximal marked squares. The desired inequalit y is immediately implied by the follo wing t w o claims: Claim 1. In or der to r e ach the level 4 K M at x , we have to r e ach level M in at le ast one maximal marke d squar e whose pr oje ction c ontains x . 12 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G Claim 2. Th e sum of side lengths of al l maximal marke d squar es do es not exc e e d 108 K µ { F ∗ ≥ K } . Pro of of C laim 1: Ob viously , in ord er to reac h the lev el 4 K M at x , one has to reac h th e level M in at least one of the squares of generation n ( x ) whose p ro jection con tains x (recall that there are not more than 4 K suc h squ ares!) Eac h s u c h squ are is con tained in some maximal m ark ed square, whence the claim. Pro of of Claim 2 : Consider all 4-adic int erv als I ⊂ R s u c h that I inte rsects a pr o j ection of s ome maximal mark ed square Q whose side length is at least | I | . Clearly , the u nion of all s uc h in terv als con tains the pro jections of all maximal mark ed s quares. No w consider maximal interv als I with this pr op ert y . Clearly , eac h suc h maximal inte rv al I in tersects the pr o j ection of some maximal mark ed square Q with side length | I | , but no pr o jection of a maximal marke d square with a la rger side lengt h. No w let us estimate the sum of side le ngths of the maximal mark ed squares in tersecting one su c h maximal 4-adic int erv al I . Let σ = sin θ + co s θ . Note that ea c h maximal mark ed square wh ose pr o j ection intersects I is con tained in some square of generation log 4 1 | I | with side lengt h | I | , wh ose p ro jection in tersects I . Since th e pr o j ection of eac h su c h square is con tained in (2 σ + 1) I , ha ving more th an 2 σ +1 σ · 4 K su c h squ ares wo uld imply existence of a p oint x ∈ I that is con tained in more than 4 K p ro jections of squares of generation log 4 1 | I | . But this implies that there are at least 2 K squares of the previous generation ab ov e x , so n ( x ) ≤ log 4 1 | I | − 1 and there exists a marke d squ are of side length greater than | I | whose pr o jection in tersects I . The maximal mark ed sq u are contai ning it h as at least the same side lengt h and its pro jection still in tersects I . But this con tradicts maximalit y of I . Since the maximal mark ed squares are disjoin t, the sum of side lengths of maxi- mal mark ed squares con tained in one square of generation log 4 1 | I | do es not exceed | I | . He nce the su m of side lengths of all maximal m ark ed squares whose pro jections in tersect I is at most 2 σ +1 σ · 4 K | I | ≤ 12 K | I | . T h us, the total sum of side lengths of all maximal m ark ed squares is a t most 12 K P I | I | = 12 K | U I I | , b ecause the maximal in terv als are disjoin t. BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 13 No w le t I b e one of ou r maxima l inte rv als a nd let Q 1 b e a m aximal m ark ed square with side length | I | , whose pro jection inte rsects I . S ince Q 1 is a mark ed square, there exists a p oin t x and 2 K − 1 other squares Q 2 , . . . , Q 2 K of side length | I | such that Pr o j Q j ∋ x for all j = 1 , . . . , 2 K . Cho osing K su c h squares whose cen ters lie on one side of x , w e see that there exists an in terv al J of length σ / 2 con taining x . such that F ∗ ≥ F log 4 1 | I | ≥ K on J . S ince dist( x, I ) ≤ σ | I | , we ha v e I ⊂ 5 σ +4 σ J ⊂ 9 J . Hence, if J ′ is the constituting interv al of the s et { F ∗ ≥ K } , con taining J , we also ha v e I ⊂ 9 J ′ . Ther efore, | S I I | ≤ 9 µ { F ∗ ≥ K } and w e are done. No w fix θ , K , and N . L et ν = µ { F ∗ ≥ K } . By induction, we get µ { F ∗ ≥ (4 K ) j K } ≤ [108 K ν ] j ν, j = 1 , 2 , . . . .. Hence, for all n = 0 , 1 , . . . , N , we get Z R f 2 n = Z { f n ≤ K } f 2 n + Z { K ≤ f n ≤ 4 K 2 } f 2 n + X j ≥ 1 Z { (4 K ) j K ≤ f n ≤ (4 K ) j +1 K } f 2 n ≤ √ 2 K + 16 K 4 ν + X j ≥ 1 [108 K ν ] j (4 K ) 2 j 16 K 4 ν ≤ 2 K, pro vided that 108 · 16 K 3 ν ≤ 1 2 . T h e F ourier-analytic part implies that the m easure of all angles θ w ith this prop erty is less than some absolute constan t times K N 1 / 4 − γ with arbitrarily small γ > 0. Assume now that ν > 32 − 1 · 108 − 1 · K − 3 . F or eac h p oin t x ∈ R wh ere F ∗ ( x ) ≥ K , c ho ose some n ∈ { 0 , 1 , . . . , N } for which F n ( x ) ≥ K and mark all sq u ares of the n -th generation whose pro jections con tain x . No w, in the N -th generation, color green all squares con tained in th e marked squ ares. Let ϕ b e th e su m of the charact eristic functions of pro jections of green squares an d let Ξ b e he un ion of the pr o j ections of all marke d squares. W e w ant to sho w first that Ξ ⊂ { y ∈ R : M ϕ ( y ) ≥ K 4 } , where M ϕ ( y ) = sup r > 0 1 2 r R y + r y − r ϕ ( s ) ds is the cen tral Hardy -Littlewoo d maximal function. In deed, if y ∈ Ξ, then the vertica l line thr ough y in tersects at least one mark ed s q u are Q 1 . Th u s, there exists x ∈ R and K − 1 o ther marked squares Q 2 , . . . , Q K of the same size as Q 1 , su c h that x ∈ Pro j Q j for all j = 1 , . . . , K . No w, the in terv al J cen tered at y of length 4 | Pro j Q 1 | conta ins all the pr o j ections of the squares Q 1 , . . . , Q K . Integ ral R J ϕ is then n ot less than the sum of all 14 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G lengths of the pr o jections of the green squares con tained in Q 1 , . . . , Q k , which is K | P r o j Q 1 | . Hence, M ϕ ( y ) ≥ 1 | J | R J ϕ ≥ K 4 . Using the w eak type L 1 estimate for the Hardy-Littlew o o d maximal function, we conclude that µ (Ξ) . 1 K Z R ϕ. Since F ∗ ( x ) ≥ K implies x ∈ Ξ, we deduce that R R ϕ & K ν & K − 2 , i.e., there are at least cK − 2 4 N green squares. On the other han d , Ξ cont ains the pro jections of all green squ ares, and R R ϕ is (up to a constan t factor) th e sum of all sid e lengths of all green squares. T h us, the length of the pro jection of the union of all green squares is at most C K times the sum of their s id e lengths. The net outcome of the previous construction is that in the N -th generation r otated Can tor square R θ K N , we ha v e U & K − 2 4 N green squares, wh ose pro jections o v er lap a lot (more p recisely , their total p ro jection is only ab out 1 K times th eir total side length) and 4 N − U other (wh ite) squares ab out whic h we know n othin g. This just giv es the estimate C K U · 4 − N + √ 2 · 4 − N (4 N − U ) for the total length of the pr o jection of R θ K N , w hic h do esn’t lo ok very imp ressiv e. But here is where the self-similarit y comes in to p la y . Let u s rep eat the construction of green squares in eac h of the white squares (this will bring us to the consid eration of K 2 N instead of K N ). No w w e will h a ve U · 4 N squares con tained in the original green squares, w hic h still giv e us the pro j ections ≤ C K U · 4 − N , bu t we shall also ha v e (4 N − U ) U n ew small green squares and the total length of their pro jection w ill b e ≤ C K U (4 N − U ) · 4 − 2 N . Thus the total length of the pro jections of all these squ ares will b e at most C K U · 4 − N [1 + (1 − U 4 N )]. T o this w e should add √ 2 · 4 − 2 N (4 N − U ) 2 = √ 2(1 − U 4 N ) 2 , which is the trivial upp er b oun d for the total pro jection of the remaining (4 N − U ) 2 squares in R θ K 2 N . Pro ceeding to K 3 N in a similar manner, we shall get | Pro j R θ K 3 N | ≤ C K U · 4 − N [1 + (1 − U 4 N ) + (1 − U 4 N ) 2 ] + √ 2(1 − U 4 N ) 3 , and so on. By the time w e reac h R θ K X N with a large p ositiv e in teger X , we shall get | Pro j R θ K X N | ≤ C K U 4 N X − 1 X ℓ =0 (1 − U 4 N ) ℓ + √ 2(1 − U 4 N ) X . The first term d o es n ot exceed C K U 4 N P ∞ ℓ =0 (1 − U 4 N ) ℓ = C K , while the second is at most √ 2 e − 4 − N U X , whic h is less than √ 2 K if 4 − N U X > log K , i.e., if X is m uch greater than K 2 log K . The moral of the story is that, give n t w o p ositiv e int egers BUFFON NEEDLE PR OBABILITY OF THE FOUR-CORNER CANTOR SET 15 K and N , w e can find an exceptional set of m easur e . K N 1 / 4 − γ , suc h that for all θ outside this set, we hav e | Pro j R θ K X N | ≤ 1 K for all in tegers X that are muc h greater than K 2 log K . The last result can b e restated as follo ws: If K , S are large enough and N ≥ K p S q with p > 6, q > 4, then µ n θ ∈ (0 , π 4 ) : | Pro j R θ K N | ≥ C K o . 1 S . This giv es us the w eak typ e inequalit y µ n θ ∈ (0 , π 4 ) : | Pro j R θ K N | ≥ t o . ( N − 1 t − p ) 1 /q , pro vided that N − 1 t − p is m uch less than 1. Com bining it with the trivial estimate µ { θ ∈ (0 , π 4 ) : . . . } ≤ π 4 for all other t , we finally ge t: Z π 4 0 | Pro j R θ K N | dθ = Z ∞ 0 µ n θ ∈ (0 , π 4 ) : | Pr o j R θ K N | ≥ t o dt . Z C N − 1 /p 0 1 dt + Z ∞ C N − 1 /p N − 1 /q t − p/q dt . N − 1 p + N − 1 q N 1 p ( p q − 1) = 2 N − 1 p , finishing the pro of. 4. h -Ha usd orff m easures of the projections. If a f u nction h is increasing, con tin u ous and h (0) = 0, we can defin e Hausdorff measure H h on compact set by the usual p ro cedure. When h ( t ) = t , this is exactly the Hausd orff measur e H 1 of d im en sion 1. W e kn ow that H 1 measure of almost all pr o jections of 1 / 4 Can tor set is zero, and the Hausd orff dimension of almost ev ery pro jection is 1. W e ca n ge t more information ab out these pro j ections by measuring their H h using a more refined scale of gauge f unctions than just p o wers of t . Namely , the main result obtained in this pap er readily implies th e follo wing corollary . Consider the gauge fun ction h ( t ) = t (lo g 1 t ) c with small p ositiv e c . W e ha ve pro ve d Theorem 2. If c is sufficiently smal l ( c ∈ (0 , 1 / 6) ) then almost every pr oje ction of the four c orner Cantor set K has zer o H h me asur e. 16 FEDOR N AZARO V, YUV AL PERES, AND ALEXANDER VOLBER G Referen ces 1. A. S . Besicovitc h, T angential pr op erties of sets and ar cs of infinite line ar m e asur e , Bul l. Amer. Math. So c. 66 (1960), 353–359 . 2. G. David, Analytic c ap acity, Calder´ on-Zygmund op er ators, and r e ctifiability, Publ. Mat. 43 (1999),3–2 5. 3. K. J. F alconer, The geome try of fractal sets. Cambridge T racts in Mathematics, 85. C.U.P ., Cam bridge–New Y ork, (1986). 4. P . W. Jones and T. Murai, Positive analytic c ap acity but zer o Buffon ne e d l e pr ob abili ty , Pacific J. Math. 133 (1988), 99–114. 5. R. 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Fedor Nazaro v, Dep ar tment of Ma th ema tics, Un iversity of Wisconsin. nazarov@ma th.wisc.edu Yuv al Peres, Microsoft Research, Redm ond and Dep ar tme nts o f St a tistics and Ma thema tics, Unive rsity of California, B erkeley. peres@micr osoft.com Alexander V olberg, Dep ar tme nt of Ma the ma tics, Michigan St a te Un iversity and the University of Edinburgh volberg@math.msu. edu and a.volberg@ed.ac.u k