Buffon needle lands in ε -neighborhood of a 1-Dimensional Sierpinski Gasket with probability at most |logε | −c
aa r X i v : . [ m a t h . C A ] D ec BUFFON NEEDLE LANDS IN ǫ -NEIGHBORHOOD OF A -DIMENSIONAL SIERPINSKI GASKET WITH PROBABILITYAT MOST | log ǫ | − c MATT BOND AND ALEXANDER VOLBERG
Abstract.
In recent years, relatively sharp quantitative results in the spiritof the Besicovitch projection theorem have been obtained for self-similar setsby studying the L p norms of the “projection multiplicity” functions, f θ , where f θ ( x ) is the number of connected components of the partial fractal set thatorthogonally project in the θ direction to cover x . In [4], it was shown that n -thpartial 4-corner Cantor set with self-similar scaling factor 1 / Cn p , for p < /
6. In [1], this same estimate was provedfor the 1-dimensional Sierpinski gasket for some p >
0. A few observations wereneeded to adapt the approach of [4] to the gasket: we sketch them here. We alsoformulate a result about all self-similar sets of dimension 1. Definitions and result
Let E ⊂ C , and let proj θ denote orthogonal projection onto the line having angle θ with the real axis. The average projected length or Favard length of E ,Fav( E ), is given by Fav( E ) = 1 π Z π | proj θ ( E ) | d θ. For bounded sets, Favard length is also called
Buffon needle probability , sinceup to a normalization constant, it is the likelihood that a long needle dropped withindependent, uniformly distributed orientation and distance from the origin willintersect the set somewhere. B ( z , r ) := { z ∈ C : | z − z | < r } . For α ∈ {− , , } n let z α := n X k =1 ( 13 ) k e iπ [ + α k ] , G n := [ α ∈{− , , } n B ( z α , − n ) . This set is our approximation of a partial Sierpinski gasket; it is strictly larger. Wemay still speak of the approximating discs as “Sierpinski triangles.”The main result:
Theorem 1.
Fav ( G n ) ≤ Cn c , c > . Set G n is 3 − n approximation to Besicovitch irregular set (see [2] for definition)called Sierpinski gasket. Recently one detects a considerable interest in estimatingthe Favard length of such ǫ -neighborhoods of Besicovitch irregular sets, see [5], [6],[4], [3]. In [5] a random model of such Cantor set is considered and estimate ≍ n isproved. But for non-random self-similar sets the estimates of [5] are more in termsof ··· log n (number of logarithms depending on n ) and more suitable for generalclass of “quantitatively Besicovitch irregular sets” treated in [6].Let f n,θ := ν n ∗ n χ [ − − n , − n ] , where ν n := ∗ nk =1 e ν k and e ν k := 13 [ δ − k cos ( π/ − θ ) + δ − k cos ( − π/ − θ ) + δ − k cos (7 π/ − θ ) ] . For
K >
0, let A K := A K,n,θ := { x : f n,θ ≥ K } . Let L θ,n := proj θ ( G n ) = A ,n,θ .For our result, some maximal versions of these are needed: f ∗ N,θ := max n ≤ N f n,θ , A ∗ K := A ∗ K,n,θ := { x : f ∗ n,θ ≥ K } . Also, let E := E N := { θ : | A ∗ K | ≤ K − } for K = N ǫ , ǫ .Later, we will jump to the Fourier side, where the function ϕ θ ( x ) := 13 [ e − i cos( π/ − θ ) + e − i cos( − π/ − θ ) + e − i cos(7 π/ − θ ) ]plays the central role: c ν n ( x ) = Q nk =1 ϕ θ (3 − k x ).2. General philosophy
Fix θ . If the mass of f n,θ is concentrated on a small set, then || f n,θ || p should belarge for p > R f ≤ || f n,θ || p || χ L θ,n || q , so m ( L θ,n ) ≥ || f || − qp ,a decent estimate. The other basic estimate is not so sharp: m ( L θ,N ) ≤ − ( K − m ( A K,N,θ ) (2.1)However, a combinatorial self-similarity argument of [4] and revisited in [1] showsthat for the Favard length problem, it bootstraps well under further iterations ofthe similarity maps:
Theorem 2. If θ / ∈ E N , then |L θ,NK | ≤ CK . Note that the maximal version f ∗ N is used here. A stack of K triangles at stage n generally accounts for more stacking per step the smaller n is. For fixed x ∈ A ∗ K,N,θ , UFFON NEEDLE LANDS IN ǫ -NEIGHBORHOOD OF A 1-DIMENSIONAL SIERPINSKI GASKET WITH PROBABILITY AT MOST | log ǫ | − c the above theorem considers the smallest n such that x ∈ A K,n,θ , and uses self-similarity and the Hardy-Littlewood theorem to prove its claim by successivelyrefining an estimate in the spirit of (2.1). Of course, now Theorem 1 follows fromthe following:
Theorem 3.
Let ǫ < / . Then for N >> , | E | < N − ǫ . It turns out that L theory on the Fourier side is of great use here. It is provedin [4], [1]: Theorem 4.
For all θ ∈ E N and for all n ≤ N , || f n,θ || L ≤ CK . One can then take small sample integrals on the Fourier side and look for lowerbounds as well. Let K = N ǫ , and let m = 2 ǫ log N . Theorem 4 easily impliesthe existence of ˜ E ⊂ E such that | ˜ E | > | E/ | and number n , N/ < n < N/ θ ∈ ˜ E , Z n n − m n Y k =0 | ϕ θ (3 − k x ) | dx ≤ CKmN ≤ ǫ N ǫ − log N. Number n does not depend on θ ; n can be chosen to satisfy the estimate in theaverage over θ ∈ E , and then one chooses ˜ E . Let I := [3 n − m , n ] . Now the main result amounts to this (with absolute constant A large enough): Theorem 5. θ ∈ ˜ E : Z I n Y k =0 | ϕ θ (3 − k x ) | dx ≥ c m − · Am = cN − ǫ (2 A − . The result: 2 ǫ log N ≥ N − ǫ (4 A − , i.e., N ≤ N ∗ . Now we sketch the proofof Theorem 5. We split up the product into two parts: high and low-frequency: P ,θ ( z ) = Q n − m − k =0 ϕ θ (3 − k z ), P ,θ ( z ) = Q nk = n − m ϕ θ (3 − k z ). Theorem 6.
For all θ ∈ E , R I | P ,θ | dx ≥ C m . Low frequency terms do not have as much regularity, so we must control thedamage caused by the set of small values , SSV ( θ ) := { x ∈ I : | P ( x ) | ≤ − ℓ } , ℓ = α m with sufficiently large constant α . In the next result we claim the existenceof E ⊂ ˜ E , |E| > | ˜ E/ | with the following property: Theorem 7. Z ˜ E Z SSV ( θ ) | P ,θ ( x ) | dx dθ ≤ m − ℓ/ ⇒ ∀ θ ∈ E Z SSV ( θ ) | P ,θ ( x ) | dx dθ ≤ c K m − ℓ/ . Then Theorems 6 and 7 give Theorem 5.
MATT BOND AND ALEXANDER VOLBERG Locating zeros of P We can consider Φ( x, y ) = 1 + e ix + e iy . The key observations are | Φ( x, y ) | ≥ a ( | x − | + | y − | ) , sin 3 x sin x = 4 cos x − . Changing variable we can replace 3 ϕ θ ( x ) by φ t ( x ) = Φ( x, tx ). Consider P ,t ( x ) := Q nk = n − m φ t (3 − k x ), P ,t ( x ) := Q n − mk =0 13 φ t (3 − k x ). We need SSV ( t ) := { x ∈ I : | P ,t ( x ) | ≤ − ℓ } . One can easily imagine it if one considers Ω := { ( x, y ) ∈ [0 , π ] : |P ( x, y ) | := | Q mk =0 Φ(3 k x, k y ) | ≤ m − ℓ } . Moreover, (using that if x ∈ SSV ( t ) then3 − n x ≥ − m , and using xdxdt = dxdy ) we change variable in the next integral: Z ˜ E Z SSV ( t ) | P ,t ( x ) | dxdt = 3 − n +2 m · n Z ˜ E Z − n SSV ( t ) | n Y k = m Φ(3 k x, k tx ) | dxdt ≤ − n +3 m Z Ω | n Y k = m Φ(3 k x, k y ) | dxdy . Now notice that by our key observations Ω ⊂ { ( x, y ) ∈ [0 , π ] : | sin 3 m +1 x | + | sin 3 m +1 y | ≤ a − m m − ℓ ≤ − ℓ } . The latter set Q is the union of 4 · m +2 squares Q of size 3 − m − ℓ/ × − m − ℓ/ . Fix such a Q and estimate Z Q | n Y k = m Φ(3 k x, k y ) | dxdy ≤ ℓ Z Q | n Y k = m + ℓ/ Φ(3 k x, k y ) | dxdy ≤ ℓ · (3 − m − ℓ/ ) Z [0 , π ] | n − m − ℓ/ Y k =0 Φ(3 k x, k y ) | dxdy ≤ ℓ · (3 − m − ℓ/ ) · n − m − ℓ/ = 3 − m · n − m − ℓ/ . Therefore, taking into account the number of squares Q in Q and the previousestimates we get Z E Z SSV ( t ) | P ,t ( x ) | dxdt ≤ m − ℓ/ . Theorem 7 is proved.To prove Theorem 6 we need the following simple lemma.
Lemma 8.
Let C be large enough. Let j = 1 , , ...k , c j ∈ C , | c j | = 1 , and α j ∈ R .Let A := { α j } kj =1 . Suppose Z R ( X α ∈ A χ [ α − ,α +1] ( x )) dx ≤ S .
Then Z | X α ∈ A c α e iαy | dy ≤ C S . eferences 5
Some key facts useful for its proof: Z | X α ∈ A c α e iα y dy | ≤ e Z ∞ | X α ∈ A c α e i ( α + i ) y dy | = e Z R (cid:12)(cid:12)(cid:12)(cid:12) X α ∈ A c α α + i − x (cid:12)(cid:12)(cid:12)(cid:12) dx , and the fact that H ( C + ) is orthogonal to H ( C + ), so one can pass to the Poissonkernel. 4. The general case
Let us have k closed disjoint discs of radii 1 /k located in the unit disc. We build k n small discs of radii k − n by iterating k linear maps from small discs onto the unitdisc. Call the resulting union S k ( n ). We would like to show that exactly as in thecase of k = 3 considered above and in a very special case of k = 4 considered in [4]Fav( S k ( n )) ≤ C n − c , c >
0. However, presently we can prove only a weaker result.
Theorem 9.
Fav ( S k ( n )) ≤ C e − c (log n ) / , c > . References [1] M. Bond, A.Volberg,
The Power Law For Buffon’s Needle Landing Near the SierpinskiGasket , arXiv:math. 0911.0233v1, 2009, pp. 1-34.[2] K. J. Falconer, The geometry of fractal sets. Cambridge Tracts in Mathematics, 85.C.U.P., Cambridge–New York, (1986).[3] I. Laba, K. Zhai,
Favard length of product Cantor sets , arXiv:0902:0964v1, Feb. 52009.[4] F. Nazarov, Y. Peres, A. Volberg
The power law for the Buffon needle probability ofthe four-corner Cantor set , arXiv:0801.2942, 2008, pp. 1–15.[5] Y. Peres and B. Solomyak,
How likely is Buffon’s needle to fall near a planar Cantorset? Pacific J. Math. 204 , 2 (2002), 473–496.[6] T. Tao,
A quantitative version of the Besicovitch projection theorem via multiscaleanalysis, pp. 1–28, arXiv:0706.2446v1 [math.CA] 18 Jun 2007.
Matt Bond, Department of Mathematics, Michigan State University, [email protected]