SIXTY YEARS OF BERNOULLI CONVOLUTIONS
YUVAL PERES, WILHELM SCHLAG, AND BORIS SOLOMYAK
n Abstract. The distribution νλ of the random series ±λ is the infinite convolution product 1 of (δ λn + δλn ). These measures have been studied since! the 1930’s, revealing connections with 2 − harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorffdimension estimation. In this survey we describe some of these connections, and the progress that has been ∈ 1 made so far on the fundamental open problem: For which λ ( 2 , 1) is νλ absolutely continuous? Our main goal is to present an exposition of results obtained by Erd˝os, Kahane and the authors on this problem. Several related unsolved problems are collected at the end of the paper.
Table of Contents
1. Introduction 2. Historical notes 3. Laws of pure type 4. Dimensions of Bernoulli convolutions 5. Bernoulli convolutions and Salem numbers 6. Close to one; the Erd˝os-Kahane argument 7. Smoothness and the dimension of exceptions 8. Applications, generalizations, and problems
1. Introduction
n 1 Let ν be the distribution of ∞ λ where the signs are chosen independently with probability . λ 0 ± 2 ! 1 It is the infinite convolution product of (δ λn +δλn ), hence the term “infinite Bernoulli convolution”. 2 − These measures have been studied since the 1930’s, revealingsurprisingconnectionswithharmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorffdimension estimation.
There are several ways to think about νλ which hint at these connections.
Research of Peres was partially supported by NSF grant #DMS-9803597. Research of Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright foundation, and the Institute of Mathematics at the Hebrew University of Jerusalem. 1 2YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK
itξ (i) The Fourier transform νλ(ξ)= ∞ e dνλ(t)iseasilycomputed: #−∞ " ∞ n νλ(ξ)= cos(λ ξ), (1.1) n$=0 " and this formula has been crucial for number-theoretic considerations.
(ii) νλ can be characterized by the functional equation for its cumulative distribution function
Fλ(x)=νλ( ,x]: −∞ 1 x 1 x +1 Fλ(x)= Fλ − + Fλ . 2% & λ ' & λ '( In other words, ν is the self-similar measure for the iterated function system λx 1,λx+1 with λ { − } 1 1 probabilities ( 2 , 2 )(see[17]).Thispointofviewisusefulinapplicationstodynamical systems and dimension estimation. (iii) ν can be viewed as a “non-linear projection”: let Ω= 1, 1 N be the sequence space with λ {− } 1 1 N the Bernoulli measure µ =(2 , 2 ) .Then
1 ∞ n νλ = µ Πλ− where Πλ(ω)= ωnλ . ◦ n)=0
This representation has been most useful in the recent work on νλ which used ideas of geometric measure theory. The fundamental question about ν is to decide for which λ ( 1 , 1) this measure is absolutely λ ∈ 2 continuous and for which λ it is singular. If the density exists, it is natural to inquireaboutits smoothness. If the measure is singular, one would like to compute, or estimate, its dimension. 1 Denote by S the set of λ ( , 1) such that νλ is singular. The only elements of S that are ⊥ ∈ 2 ⊥ known were found in [9] by Erd˝os (1939): they are the reciprocals of Pisot numbers in (1, 2). It is an open problem whether they constitute all of S .Thefirstimportantresultintheoppositedirection ⊥ is also due to Erd˝os (1940): he proved in [10] that S (a, 1) has zero Lebesgue measure for some ⊥ ∩ a<1. Later, Kahane [19] indicated that the argument of [10] actually implies that the Hausdorff dimension of S (a, 1) tends to 0 as a 1. We included the Erd˝os-Kahane argument with explicit ⊥ ∩ ↑ numerical bounds since they have never appeared in the literature (see section 6). In [47] Solomyak (1995) proved that ν is absolutely continuous for a.e. λ ( 1 , 1). A simpler proof was found by λ ∈ 2 Peres and Solomyak [39] and we do not reproduce it here. In [38]PeresandSchlagestablished,as acorollaryofamoregeneralresult,thattheHausdorffdimension of S (a, 1) is less than one for ⊥ ∩ 1 any a> 2 .Ourmaingoalistogiveaself-containedproofofthis(seesection 7). The rest of the paper is organized as follows. In section 2 we give some additional historical background. Sections 3 and 4 contain several results which hold for all parameters λ.Insection3 we discuss two “laws of pure type” for self similar measures, one classical and one recently proved BERNOULLI CONVOLUTIONS 3 by Mauldin and Simon [36]. Together, they imply that any self similar measure ν with support K is either singular to Lebesgue measure ,orequivalenttotherestrictionof to K. L L In section 4 we discuss the Hausdorffand correlation dimensions of Bernoulli convolutions. We prove that the upper and lower correlation dimensions of νλ coincide for all λ and use this to show that the correlation dimension of ν equals one on a residual set of λ ( 1 , 1); these appear to be new λ ∈ 2 results. In section 5 we make some comments on Bernoulli convolutions for Salem numbers. The contents of sections 6 and 7 were described above. We concludeinsection8,withabriefdiscussion of some applications, generalizations and problems.
2. Historical notes
This section is by no means comprehensive; we just add a few remarks to what was said in the introduction. For instance, much of the early work on Bernoulli convolutions covers more general random variables r for an arbitrary sequence r with r2 < ,buthereweonlydiscussthe ± n n n ∞ case r = λn for λ! (0, 1). ! n ∈ In the 1930’s Bernoulli convolutions were studied by Wintnerandhisco-authors.Jessenand
Wintner (1935) showed that νλ is either absolutely continuous, or purely singular, depending on λ (see 3). Kershner and Wintner (1935) observed that ν is singular for λ (0, 1 )sinceitissupported § λ ∈ 2 on a Cantor set of zero Lebesgue measure (in fact, νλ is the standard Cantor-Lebesgue measure on this Cantor set). Wintner (1935) noted that ν is uniform on [ 2, 2] for λ = 1 ,andforλ =2 1/k λ − 2 − with k 2itisabsolutelycontinuous,withadensityinCk 2(R). For λ ( 1 , 1) the support of ν ≥ − ∈ 2 λ is the interval [ (1 λ) 1, (1 λ) 1], so one might surmise that ν is absolutely continuous for all − − − − − λ such λ.However,in[9]Erd˝os(1939)showedthatνλ is singular when λ is the reciprocal of a Pisot number. Recall that a Pisot number is an algebraic integer allofwhoseconjugatesarelessthan one in modulus. This gives a closed countable set of λ ( 1 , 1) with ν singular. Curiously, there ∈ 2 λ are only two Pisot numbers in (1, 21/2): the positive root θ 1.324718 of x3 x 1=0andthe 1 ∼ − − positive root θ 1.3802777 of x4 x3 1. The golden ratio 1 (1 + √5) is the only quadratic Pisot 2 ∼ − − 2 number in (1, 2), and it is also the smallest limit point of Pisot numbers, see [3]. No λ S ,other ∈ ⊥ than reciprocals of Pisot numbers, have been found.
The proof of Erd˝os [9] proceeds by showing that the Fourier Transform νλ(ξ)doesnottendto 1 zero at infinity if θ = λ− is Pisot. Salem [43], using Pisot’s Theorem (see [44," p.11]),showedthat for all λ (0, 1) such that λ 1 is not a Pisot number, ν (ξ)doestendtozerowhenξ .(I.e.,ν ∈ − λ →∞ λ is a Rajchman measure,see[31].)Forλ (0, 1 ), this result is related to the fact that a Cantor set ∈ 2 " 1 with dissection ratio λ is a set of uniqueness for Fourier series if and only if λ− is a Pisot number, see Salem [44]. 4YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK
Kahane and Salem (1958) obtained criteria for Bernoulli convolutions to have a density in L2.
Although they could not apply these criteria to νλ,theyanalyzedthedistributionsofcertainseries
rn where the ratios rn/rn 1 are random. ± − Garsia (1962) found the largest explicitly given set of λ known to date, for which νλ is absolutely continuous (and even has bounded density). This set consistsofreciprocalsofalgebraicintegersin (1, 2) whose minimal polynomial has other roots outside the unit circle and the constant coefficient 2. (Such are for instance the polynomials xn+p xn 2wherep, n 1andmax p, n 2.) Garsia ± − − ≥ { }≥ n n 1 k n showed that for these λ,all2 sums 0− λ are distinct and at least C2− apart for some C>0. ! ± This implies that νλ has a bounded density.
Starting with Garsia (1963), many authors studied the measure νλ in the Pisot case, computing or estimating various dimensions and giving alternative proofs of singularity. This line of research is not the focus of our paper, and we only discuss it briefly in section 8. The interest in Bernoulli convolutions was renewed in the 1980’s when their importance in various problems of dynamics and dimension was discovered by Alexander and Yorke (1982), Przytycki and Urba´nski (1989), and Ledrappier (1992) (see section 8 for details).. The latest stage in the study of Bernoulli convolutions started with a seemingly unrelated de- velopment: the formulation of the “ 0, 1, 3 -problem” by Keane and Smorodinsky in the early 90’s { } (see [22]). Motivated by questions of Palis and Takens on sumsofCantorsets,theyaskedhowthe n dimension and morphology of the set ∞ a λ : a =0, 1, or 3 depends on the parameter { n=0 n n } λ.PollicottandSimon(1995)provedthattheHausdor! ffdimension equals the similarity dimension for a.e. λ ( 1 , 1 )usingself-similarmeasuresobtainedbytakingthedigitsa 0, 1, 3 indepen- ∈ 4 3 n ∈{ } dently with equal probabilities. A crucial tool in their paper was the notion of transversality for power series. This notion turned out to be crucial in all the recent work on Bernoulli convolutions [47, 39, 40, 38]; we discuss (a version of) transversality in section 7. Pollicott and Simon were influ- enced by the important paper of Falconer [12], where methods originating from geometric measure theory were used to obtain “almost sure” results on the dimension of self-affine sets.
3. Laws of pure type
Jessen and Wintner (1935) showed that any convergent infiniteconvolutionofdiscretemeasures is of pure type: it is either singular or absolutely continuous with respect to Lebesgue measure. In particular, this applies to νλ for any λ<1. The purity of νλ can also be obtained from its self similarity, as the following proposition demonstrates. BERNOULLI CONVOLUTIONS 5
Proposition 3.1. Suppose that ν is a self similar probability measure on Rd corresponding to the contracting similitudes S : j =1,...' and the (positive) probabilities p : j =1,...' , i.e., { j } { j } # 1 ν = pj ν Sj− . (3.1) j)=1 & ◦ '
Let K denote the closed support of ν.Ifν is not singular to Lebesgue measure on Rd,then L (i) ν is absolutely continuous with respect to . L (ii) The restriction of to K is absolutely continuous with respect to ν. LK L
Part (i) of this proposition is folklore; part (ii) was provedbyMauldinandSimon[36]forνλ,and we extend their proof to general self similar measures below.
Proof: (i) Apply the similitude Sj to the Lebesgue decomposition ν = νac +νs of ν into an absolutely continuous part and a singular part. Averaging the result over j with weights pj,weinferthatνac and νs also satisfy (3.1). Since ν is the only probability measure satisfying (3.1) (see [17]),itfollows that νac and νs are proportional, hence one of them must vanish. (ii) Since ν is not singular to ,necessarily (K) > 0andforsomeβ<1, L L sup (A) A Borel,A K,ν(A)=0 = β (K)(3.2) *L + ⊂ , L + + Let A0 be a Borel subset of K such that ν(A0)=0.Denotebyc the minimal contraction ratio for ∗ the maps S # .Fixx K and r (0, diamK). Since K = # S (K), there exist similitudes S { j}j=1 ∈ ∈ ∪j=1 j of the form S = S S ... S such that S(K)iscontainedintheopenballB(x, r). Choose i1 ◦ i2 ◦ ◦ im such an S with m minimal; clearly diamS(K) c r,andtherefore [S(K)] η [B(x, r)], where ≥ ∗ L ≥ L η>0doesnotdependonx and r (we can take η = cd [K]/ [B(0, diamK)]). ∗L L 1 The preimage S− A0 S(K) is a Borel subset of K,andν assigns it measure zero, since self & ∩ ' similarity of ν implies that ν S 1 is dominated by a constant multiple of ν.By(3.2)andscaling, ◦ − [A S(K)] β [S(K)] and consequently L 0 ∩ ≤ L
B(x, r) A0 (1 β) [S(K)] (1 β)η [B(x, r)] . L% \ ( ≥ − L ≥ − L Thus A cannot have a Lebesgue density point, whence (A )=0. ! 0 L 0 Remark: Suppose that ν satisfies (3.1) and let K be the support of ν.Considerthelowerderivative
d D(ν, x):=liminf(2r)− ν[B(x, r)] , r 0 ↓ and denote by A the set of x K where D(ν, x)=0.IfK A has nonempty interior then a 0 ∈ \ 0 variant of the above argument shows that dim(A0) < 1. 6YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK
4. Dimensions of Bernoulli convolutions
Recall that the Hausdorffdimension of a Borel measure ν on R is defined by
dim ν =inf dim A : A Borel,ν(R A)=0 . { \ } The correlation dimension of ν is defined by log(ν ν) (x, y): x y r C2(ν)=lim × { | − |≤ } , (4.1) r 0 log r → if the limit exists; otherwise, one considers the upper and lower correlation dimensions, obtained by taking limsup and liminf.
Proposition 4.1. (i) For any λ (0, 1) the limit in (4.1) with ν = ν exists. ∈ λ (ii) The set λ ( 1 , 1) : C (ν )=1 is a dense G set in ( 1 , 1). { ∈ 2 2 λ } δ 2 (iii) The set λ ( 1 , 1) : dim ν =1 is residual in ( 1 , 1) (it contains a dense G set). { ∈ 2 λ } 2 δ Remarks. 1. Part (i) of the proposition appears to be new for λ ( 1 , 1); in the literature that we are ∈ 2 aware of, this limit is only proved to exist under some separation conditions. We can prove a similar result in much greater generality (for arbitrary self-similar measures and quite general self-conformal measures, with or without overlaps) but we do not describe this here. 2. Part (iii) is immediate from part (ii) since dim ν is at least the lower correlation dimension of ν for any measure ν (see, e.g., [45]).
3. Of course, to prove (ii) we only have to show that the set under consideration is a Gδ set, since we already know that ν has a density in L2(R)fora.e.λ ( 1 , 1). λ ∈ 2 4. Ledrappier [29] indicated a proof of the statement that the set in (iii) is a Gδ set, based on the fact that every νλ is “exact-dimensional”, which in turn relies on the Ledrappier-Young theory.
Proof of Proposition 4.1. We begin with the proof of the implication (i) (ii) since it is easier. ⇒ Observe that for any positive r and t the following set is open:
λ :(νλ νλ) (x, y): x y r
an = φλn (y x) dνλ(x)dνλ(y)= φλn (Πλ(ω τ)) dµ(ω)dµ(τ) . (4.2) 11 − 1Ω1Ω − BERNOULLI CONVOLUTIONS 7
1 1 Since 2 1[ r/2,r/2] φr 1[ r,r],itisenoughtoprovethatthelimitlimn log an exists. Thus, it · − ≤ ≤ − suffices to show that for some C and k we have
am+n Caman k for m 1,n k +1 ≤ − ≥ ≥ n n (then bn =log(Can k)issub-additivehencelim(bn/n)exists).Denoteω' = σ ω and τ ' = σ τ − where σ is the left shift on Ω. We have n 1 2n − j n a = 2 φ m+n (ω τ )λ + λ Π (ω τ ) dµ(ω )dµ(τ ) . (4.3) m+n − 1 1 λ j j λ ' ' ' ' ω0,...) ,ωn 1 Ω Ω %j)=0 − − ( τ0,... ,τn −1 − The integral above equals
icξ 2 φ m (y x + c) dν (x)dν (y)= φ n (ξ)e− ν (ξ) dξ (4.4) 11 λ − λ λ 1 λ | λ | 2 n n 1 j 2 where c = λ− j=0− (ωj τj)λ ,andwehaveinvokedPlancherel’stheorem.Sinceφr(ξ) > 0, ! − the integral (4.4) is maximized for c =0,andthenitequalsam,see(4.2).Ontheotherhand,2 3 m φλ (y x + c) 0onthesupportofνλ νλ if c >C1 = 1 λ .Combiningthelasttwoobservations − ≡ × | | − with (4.3) we obtain
n 1 − j n am+n am Prob (ωj τj)λ C1λ ≤ · *| j)=0 − |≤ , n am Prob Πλ(ω τ) 2C1λ ≤ · *| − |≤ ,
n am 2 φ4C1λ (Πλ(ω τ)) dµ(ω)dµ(τ) ≤ · 1Ω1Ω − 2aman k ≤ − k where k is such that λ− > 4C1.Theproofiscomplete. !
The investigation of dimension(s) of νλ for particular λ essentially goes back to Garsia (1963). In [15] he considered H (λ), the entropy of the distribution of the random sum N λn,andthe N n=0 ± HN (λ) ! limit Gλ =limN which he showed exists. If there are no coincidences among thefinite →∞ N+1 sums, then Gλ =log2.If,ontheotherhand,therearecoincidences,thenGλ < log 2. Garsia proved 1 1 that if Gλ < log( λ )thenνλ is singular, and this inequality holds for Pisot λ− .AlexanderandYorke 1 [1] proved that Gλ/ log( λ )isalwaysanupperboundfortheR´enyi(information)dimension of νλ, and equality holds in the Pisot case. In fact,
1 λ− is Pisot dim ν = G / log(1/λ). (4.5) ⇒ λ λ
As observed by Ledrappier and Porzio [30], this follows from νλ being “exact-dimensional” and [53]. Adirectproofof(4.5)wasgivenbyLalley[25]. 8YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK
In several papers numerical estimates of dim νλ were pursued, especially for the golden ratio √5 1 case λg = 2− .Althoughtheformula(4.5)lookssimple,itisinefficienttouse it directly for such estimates. Alexander and Zagier [2] found a formula for dim νλg by analyzing the “Fibonacci graph”, and used it to show that 0.99557 < dim νλg < 0.99574. Recently Sidorov and Vershik [46] gave another proof of the Alexander-Zagier formula relatingittotheentropyoftherandomwalk on the Fibonacci graph. (They also gave a nice ergodic-theoretic proof of singularity of νλg .) On the other hand, Ledrappier and Porzio [30] and independently, Lalley [25], expressed dim νλg as the top Lyapunov exponent of certain random matrix products; Lalley covered the general case of Pisot numbers and biased Bernoulli convolutions. We should also mention the paper by Bovier [5] who gave yet another proof of singularity in the golden mean case using automata theory. Lau [26] and Lau and Ngai [27, 28] computed the Lq-spectrum of Bernoulli convolutions for the golden ratio andotherPisotnumbers.Thespectrum of local dimensions was investigated by Hu [16].
5. Bernoulli convolutions and Salem numbers
Recall that an algebraic integer θ>1isaSalem number if its Galois conjugates satisfy θ 1 | j|≤ and at least one of the conjugates has modulus equal to one (i.e. θ is not Pisot). The set of Salem numbers is rather poorly understood. In particular, the following is open: Problem: is there b>1suchthateverySalemnumberisgreaterthanb? This is related to the well-known Lehmer problem on the range of Mahler measure for integer polynomials, see [7]. Below we show that obtaining a topological analog of “almost sure” results for Bernoulli convo- lutions (such as Corollary 6.2(ii) below) would settle the problem on Salem numbers. Throughout, 2 fractional derivatives will be expressed in terms of the standard 2,γ–Sobolev space Lγ which is 2 2 2γ defined by the norm νλ 2,γ = ∞ νλ(ξ) ξ dξ. 1 1 #−∞ | | | | " Proposition 5.1. If there exist γ>0 and a<1 such that the set λ (a, 1) : ν L2 is residual { ∈ λ ∈ γ} in (a, 1),thenSalemnumbersdonotaccumulatetoone.
The proof is based on several easy lemmas.
1 Lemma 5.2. Let θ be a Salem number and λ = θ− .Then
ε lim sup νλ(ξ) ξ = for all ε>0. (5.1) ξ | || | ∞ →∞ " Proof. Let x denote the distance of x R to the nearest integer. It has been observed by several || || ∈ authors (see [6], [37, 6.9], [3, 5.5.1]) that for each Salem number θ and any δ>0onecanfindt 1 ≥ BERNOULLI CONVOLUTIONS 9 such that
tθn δ for all n 1. (5.2) || || ≤ ≥ 1 We have by (1.1) and (5.2) for λ = θ− : n n k νλ(πtθ ) νλ(πt) cos(π tθ ) | |≥| | k$=1 || || " " 2 n n ε ν (πt) (1 cδ ) c' ν (πt) (πtθ )− , ≥|λ | − ≥ | λ | log(1 cδ2) " " where ε = − .Sinceε 0asδ 0, it remains to show that ν (πt) =0. − log θ → → λ 2 2k+1 l n Suppose νλ(πt)=0,thent = θ for some l 0. But θ n 1 is" dense mod 1 (see [44]), hence 2 ≥ { } ≥ n 1 1 ! tθ is dense" mod 2 which contradicts (5.2) for δ<2 .
2 R 2 ˆ 2 2γ Lemma 5.3. Suppose that f L ( ) has compact support and f 2,γ = ∞ f(ξ) ξ dξ< . ∈ 1 1 −∞ | | | | ∞ Then #
γ fˆ(ξ) = o( ξ − ), ξ . | | | | | |→∞ Proof. This is a standard fact from harmonic analysis but we provide aproofforthereader’s convenience. Let f˜(x)=f( x)andletφ be a smooth compactly supported function on R equal to − one on the support of f f˜.Thenf f˜ =(f f˜)φ hence ∗ ∗ ∗ fˆ(ξ) 2 =(fˆ 2 φˆ)(ξ)= fˆ(ξ η) 2φˆ(η) dη (5.3) | | | | ∗ 1R | − | C fˆ(ξ η) 2 dη + C φˆ(η) dη. ≤ 1 η < ξ/2 | − | 1 η ξ/2 | | | | | | | |≥| | Since φˆ(η) is rapidly decreasing, the second integral is O( ξ q)foranyq>0. The first integral | | | |− can be estimated above by
2γ 2 2γ 2γ 2 2γ 2γ ξ/2 − fˆ(ξ η) ξ η dη ξ/2 − fˆ(η) η dη = o( ξ − ) , | | 1 η < ξ/2 | − | | − | ≤| | 1 η ξ/2 | | | | | | | | | | | |≥| | and the claim follows. !
Lemma 5.4. Let Γ be the set of λ (0, 1) for which there exists ε>0 such that ν (ξ)=O( ξ ε), ∈ λ | |− as ξ .Then(0, 1) Γ is a G set. | |→∞ \ δ " Proof. Let ε 0forj N.Itisenoughtoobservethat j → ∈
εj Γ= λ : ν (ξ) ξ − { | λ |≤| | } 0j k01 ξ/k ≥ | |≥ " is an Fσ set. ! 10 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK
Proof of Proposition 5.1. By Lemma 5.2, reciprocals of Salem numbers belong to (0, 1) ΓwhereΓ \ is defined in Lemma 5.4. Any integer power of a Salem number is a Salem number, by definition. It is easy to see that for any sequence x 1theunionofits(integer)powersisdensein[1, + ). n ↓ ∞ Thus, if Salem numbers accumulate to 1, the set (0, 1) ΓisdenseG in (0, 1). If ν is generically \ δ λ in L2 for λ (a, 1) for some a<1, then using Lemma 5.3 we see that Γis residual in (a, 1), a γ ∈ contradiction. ! Remark. The statement (5.1) is contained in [19] but with a typo, claiming that it holds with lim rather than lim sup. As a consequence of this typo, the statements in [8] concerning Salem numbers are unjustified. In fact, for all λ (0, 1) and any ε>0wehave ∈ log 2 log λ ε lim inf νλ(ξ) ξ − − =0. (5.4) ξ | |·| | | |→∞ " 1 n Indeed, let θ = λ− .ByKoksma’stheorem,fora.e.t>0thesequence θ t n 1 is uniformly { } ≥ distributed mod 1 (see [24, Cor. 1.4.3]). Fix such a t.Wehaveby(1.1) 1 1 n 1 log ν (πtθn) log cos(πtθk) log cos(πu) du = log 2, n | λ |≤n | |→1 | | − k$=1 0 " by the definition of the uniform distribution. This clearly implies (5.4).
6. Close to one; the Erd˝os-Kahane argument
In [10] Erd˝os proved that νλ is absolutely continuous for a.e. λ sufficiently close to one. However, explicit bounds for the neighborhood of one were not given. Kahane [19] gave a brief outline of the argument and indicated that it actually yields that the dimension of λ (λ , 1) : ν is singular { ∈ 0 λ } tends to zero as λ 1. Below we give an exposition of this argument since it remains the only way 0 ↑ to prove the statement, while Kahane’s paper [19] is not widely known and is tersely written. We also give explicit numerical bounds for the neighborhoods where the statements hold.
Proposition 6.1. Let 1 Corollary 6.2. (i) For any s>0 there exists α(s) < 1 such that dim λ (α(s), 1) : ν is singular s. { ∈ λ }≤ (ii) For any k N and any s>0 there exists α(k, s) < 1 such that ∈ dν dim λ (α(k, s), 1) : λ Ck(R) s. { ∈ dx 2∈ }≤ The corollary is immediate from the proposition by the formula νλ(u)=νλ2 (u)νλ2 (λu)which implies " 3 3 3 2 m 2 m 2mB/k log[eA k] dim λ [b− − ,a− − ]: ν (u) = O(u− ) . { ∈ λ 2 }≤ k log a " 1 log[eA3k] 2 To get a concrete numerical estimate, take a =2 and b =2.Then k log a < 1fork =34and log[cos(πr)] − k log b > 0.0006, so Proposition 6.1 implies 1 1/2 0.0006 dim λ [2− , 2− ]: ν (u) = O(u− ) < 1, { ∈ λ 2 } hence " 2 10 2 11 0.6 dim λ [2− − , 2− − ]: ν (u) = O(u− ) < 1. { ∈ λ 2 } 10 Therefore, by this argument ν has a density in L2(R")forallλ [2 2− , 1) [0.99933, 1) outside a λ ∈ − ⊃ set of dimension less than one. This can be improved somewhat by optimizing the choice of a and b but not very significantly. 1 Proof of Proposition 6.1. Denote θ = λ− .From(1.1)wehave N N n νλ(πθ t)= cos(πθ t)νλ(πt) . (6.2) n$=1 " " Let θnt = c +ε where c N and ε [ 1 , 1 )(thedependenceonθ and t is not written explicitly n n n ∈ n ∈ − 2 2 but should be kept in mind). By assumption, we can fix δ>0sothat 1 log[cos(πρ)] ρ := [2(b +1)(b +1+δ)]− satisfies B − . (6.3) ≤ log b Fix also 1 cn+1 θεn εn+1 b εn + εn+1 const θ = − | | | | n . (6.4) + − cn + + cn + ≤ cn ≤ a + + + + We are going to cover E by intervals of size a N centered at cN ,sowewanttoestimatethe N − cN 1 ∼ − number of possible pairs (cN 1,cN )correspondingtoθ EN (and some t). In fact, we will estimate − ∈ the number of sequences c1,... ,cN with the help of the following lemma. Lemma 6.3. The following holds for n sufficiently large (n n (a, b, δ)). ≥ 0 (i) Given cn,cn+1 there are at most A' := 1 + (b +1)(b +1+δ) possibilities for cn+2,independent of θ [a, b] and t [1,θ). ∈ ∈ (ii) If 1 max ε , ε , ε <ρ= , {| n| | n+1| | n+2|} 2(b +1)(b +1+δ) then c is uniquely determined by c and c ,independentofθ [a, b] and t [1,θ). n+2 n n+1 ∈ ∈ Proof of the lemma. It is easy to see that that cn+1 θ +δ b+δ for n sufficiently large (depending cn ≤ ≤ on a and δ). Using this together with (6.4) we obtain 2 cn+1 cn+2 cn+1 cn+2 cn+1 θ + θ + − cn + ≤ &+ − cn+1 + + − cn +' + + + + + + b ε + ε +(c /c )(b ε + ε ) ≤ | n+1| | n+2| n+1 n | n| | n+1| (b +1)(b +1+δ)max ε , ε , ε . ≤ {| n| | n+1| | n+2|} Now both (i) and (ii) are immediate since c N. ! n+2 ∈ Proof of Proposition 6.1 concluded. For Γ [1,N] N consider those θ [a, b]forwhichthere ⊂ ∩ ∈ exists t [1,θ)suchthat ε <ρfor n [1,N] Γ. It follows from Lemma 6.3 that the number ∈ | n| ∈ \ 3card(Γ) of sequences c1,... ,cN corresponding to such θ is bounded above by Ca,b,δ(A') .Thus,the number of sequences c1,... ,cN corresponding to EN does not exceed N 3N/k const (A') N/k · BERNOULLI CONVOLUTIONS 13 (dealing with the possible non-integrality of N/k is left to the reader). By (6.4), this is the number of intervals of size const a N needed to cover E .Therefore, · − N N 3N/k log (A') N/k log[e(A )3k] dim E lim ' , ≤ N N log a ≤ k log a →∞ as one can easily deduce using Stirling’s formula. To complete the proof it remains to notice that A = A (δ) A as δ 0. ! ' ' → → 7. Smoothness and the dimension of exceptions 2 Following [38], we show in this section that the density of νλ has fractional derivatives in L for almost all λ ( 1 , 1) and we estimate the dimension of those λ so that ν is singular with respect ∈ 2 λ to Lebesgue measure. Throughout, fractional derivatives will be expressed in terms of the standard 2 2 2 2γ 2,γ–Sobolev space Lγ which is defined by the norm νλ 2,γ = ∞ νλ(ξ) ξ dξ.Firstwerecall 1 1 #−∞ | | | | the definition of δ–transversality from [39]. " Definition 7.1. Let δ>0. We say that J R is an interval of δ–transversality for the class of ⊂ power series ∞ n g(x)=1+ bnx , with bn 1, 0, 1 (7.1) n)=1 ∈{− } if g(x) <δimplies g (x) < δ for any x J. ' − ∈ Ausefulcriterionforcheckingδ–transversality was found in [39]. A power series h(x)iscalleda ( )–function if for some k 1anda [ 1, 1], ∗ ≥ k ∈ − k 1 − ∞ h(x)=1 xi + a xk + xi. − k )i=1 i=)k+1 In [47] Solomyak showed that among all convex combinations ofseriesoftheform(7.1),thepower series with the smallest double zero must be a ( )–function. The following lemma from [39] bypasses ∗ this fact and reduces the search for intervals of transversality to finding a suitable ( )–function. ∗ Lemma 7.2. Suppose that a ( )–function h satisfies ∗ h(x ) >δ and h'(x ) < δ 0 0 − for some x (0, 1) and δ (0, 1).ThenDefinition7.1issatisfiedon[0,x ]. 0 ∈ ∈ 0 14 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK In [39] a particular ( )–function was found that satisfies h(2 2/3) > 0.07 and h(2 2/3) < 0.09, ∗ − − − 2/3 so transversality in the sense of 7.1 holds on [0, 2− ]bythislemma.Ontheotherhand,in[47] Solomyak proved that there is a power series of the form (7.1) with a double zero at roughly 0.68, whereas 2 2/3 0.63. We will return to this issue below. − 6 Following [39] and [38], we consider Bernoulli convolutionsfromthepointofviewof“projections” —inanappropriatesense.Moreprecisely,letΩ= 1, +1 N be equipped with the product {− } 1 1 measure µ = 0∞( 2 δ 1 + 2 δ1). For any distinct ω,τ Ωwedefine < − ∈ ω τ =min i 0:ω = τ . | ∧ | { ≥ i 2 i} R n Fix some interval J =[λ0,λ1] (0, 1) and define Π: J Ω via Πλ(ω)= n∞=0 ωnλ .The ⊂ ω×τ → ! metric on Ω(depending on J)isgivenbyd(ω,τ)=λ1| ∧ |.Bydefinitionthedistributionνλ is 1 dµ(ω1)dµ(ω2) equal to νλ = µ Π− .Theα–energy of µ is defined as α(µ)= α .Onechecksthat ◦ λ E Ω Ω d(ω1,ω2) (µ) < if and only if λα > 1 .Hereweaddressthefollowingquestion:Howmuchregularit# # y Eα ∞ 1 2 2 does νλ inherit from µ for a typical value of λ?In[39]itwasshownthatνλ has an L –density for a.e. λ> 1 .Thisisbasedonthefactthat (µ) < for any compact J ( 1 , 1). In [38], Peres and 2 E1 ∞ ⊂ 2 Schlag improved this statement in two ways. Firstly, they showed that ν L2 for a.e. λ1+2γ > 1 λ ∈ γ 2 using that (µ) < on intervals J =[λ ,λ ]withλ1+2γ > 1 .Infact,theyprovedthat E1+2γ ∞ 0 1 0 2 ν 2 dλ< .Secondly,theyusedthis“meanderivativebound”onthedensities to show that J 1 λ12,γ ∞ the# Hausdorffdimension of the set of parameters λ J for which ν L2 is strictly less than one. ∈ λ 2∈ Arigorousformulationoftheseprinciplesisgivenbythefollowing" theorem, which is a special case of Theorem 2.8 in [38] (see also section 5.1 in that paper). Theorem 7.3. Suppose J =[λ ,λ ] ( 1 , 1) is an interval of δ–transversality for the power series 0 0' ⊂ 2 (7.1). Then 2 1+2γ 1 νλ 2,γ dλ< if λ0 > . (7.2) 1J 1 1 ∞ 2 Furthermore, 2 log 2 dim λ J : dνλ/dx L (R) 2 . (7.3) { ∈ 2∈ }≤ − log 1 λ0 The relation between the dimension of a set in Euclidean spaceandthatofagenericprojectionhas been studied by many authors, see [21], [11], [34]. In these works the dimension of the exceptional parameters is typically estimated by averaging a suitable functional (e.g. energy) against a Frostman measure on the set of exceptional parameters, see [11]. This depends crucially on a simple relation between the Fourier transform of a measure and the Fourier transform of its projections. Such a BERNOULLI CONVOLUTIONS 15 relation is not available in the case of Bernoulli convolutions, and a new idea is required. In [38] this is accomplished by means of Lemma 7.4 below, which allowsonetoderive(7.3)from(7.2). The idea behind that lemma is as follows: Let h be a family of nonnegative smooth functions { j }j∞=0 1 j on [0, 1] whose derivatives grow at most exponentially in j.If ∞ R h (x) dx < ,thenone 0 0 j ∞ # j ! can bound the dimension of the set of x [0, 1] for which ∞ r h (x)= for any 1 r 2 3 log 2 dim λ J : dνλ/dx L (R) (7.4) { ∈ 2∈ }≤2 − 2log 1 λ0 and then sketch briefly how (7.3) can be obtained. For more details concerning the full statement of Theorem 7.3 above, we refer the reader to [38], section 3. The following lemma (essentially Lemma 3.1 in [38]) is the basic tool for bounding the Hausdorff dimension of the exceptional parameters. Lemma 7.4. Let I R be a nonempty open interval and N N.Suppose h CN (I) satisfy ⊂ ∈ { j}0∞ ∈ sup A jn h(n) C for all n N and sup Rj h (λ) dλ C < , (7.5) − j n 1 j j 0 = =∞ ≤ ≤ j 0 I | | ≤ ∗ ∞ ≥ = = ≥ where A>1.SupposethatR>r 1 satisfy Aαrα/N = R Ar1/N with 0 <α 1.Then ≥ r ≤ ≤ ∞ j dim λ I : r hj(λ) = 1 α. (7.6) * ∈ j)=0 | | ∞, ≤ − 2 j Proof for N =1. Define Ej = λ I : hj(λ) >j− r− .Then - ∈ | | . ∞ j λ I : r hj (λ) = lim sup Ej. (7.7) * ∈ ) | | ∞, ⊂ j j=0 →∞ Let s>1 α.Wewillestimatethes–Hausdorffmeasure of lim sup E by covering each E with − j j intervals of side length (rA) j.TheideaisthatanypointinE has a neighborhood of size 6 − j 6 (rA) j on which h is at least Cj 2r j.Moreprecisely,fixsomej and let I Mj be a covering − | j| − − { ij}j=1 of E by intervals of size j 2(rA) j(2C ) 1.Wecanassumethatall I Mj are contained in I and j − − 1 − { ij}j=1 16 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK no point of I is covered more than twice. Since sup h (λ + y) h (λ) C Aj y ,itfollowsthat λ | j − j |≤ 1 | | Mj 1 j Iij λ I : hj (λ) > 2 r− . (7.8) i0=1 ⊂ - ∈ | | 2j . By Markov’s inequality and assumption (7.5), we conclude from (7.8) that r M 8C j4(rA)jrj h (λ) dλ 4C C j4(rA)j j . (7.9) j 1 1 j 1 ≤ I | | ≤ ∗ >R? Let s>1 α.Inviewof(7.9), − Mj s ∞ s ∞ 4 j r j 2 j s (lim sup Ej) lim Iij lim Cj (rA) j− (rA)− =0, H ≤ k | | ≤ k R →∞ j)=k )i=1 →∞ j)=k > ? > ? using that (rA)α = R when N =1.Thus(7.6)followsfrom(7.7)bylettings (1 α). ! r ↑ − Next we sketch briefly how the cases N>1arehandled.ItturnsoutthatanypointinEj (defined in the same way as for N =1)hasaneighborhoodofsize r j/NA j on which the average of h 6 − − | j| 1 j is at least C j2 r− .ThisfollowsbyconsideringNth order differences. Using a covering of Ej by intervals of this size leads to the desired estimate. Let us make this more precise for N =2.We have 2j 2 sup hj(λ +2y) 2hj (λ + y)+hj(λ) C2A y , λ | − |≤ | | hence for L>0andλ E 0 ∈ j 2C2 2j 3 A L hj(λ0 +2y) 2hj (λ0 + y)+hj(λ0) dy 3 ≥ 1[ L,L] | − | − 2L hj (λ0) hj(λ0 + y) dy 2 hj (λ0 +2y) dy ≥ ·| |−1[ L,L] | | − 1[ L,L] | | − − 2L j 2 r− 2 hj(λ0 + y) dy. ≥ j − 1[ 2L,2L] | | − Therefore, 1 1 j C2 2j 2 hj(λ0 + y) dy 2 r− A L , 4L 1[ 2L,2L] | | ≥ 4j − 12 − 1/2 so taking L = C− j 1r j/2A j yields that the average of h on [λ 2L,λ +2L]isatleast 2 − − − | j | 0 − 0 1 j 6j2 r− . The following proposition shows how to obtain a dimension bound from a suitable Sobolev estimate by means of the previous lemma. 2 Proposition 7.5. If I νλ 2,γ dλ< with some I (0, 1) and 0 <γ<1/2,then # 1 1 ∞ ⊂ dim λ I : ν L2(R) 1 2γ. { ∈ λ 2∈ }≤ − " BERNOULLI CONVOLUTIONS 17 Proof. For any j =1, 2,... define 2j 2j j 2 j hj(λ)=2− νλ(ξ) dξ =2− exp iξ Πλ(ω) Πλ(τ) dµ(ω)dµ(τ)dξ. 1 j 1 1 j 1 1 1 2 − | | 2 − Ω Ω & @ − A' It follows that " n k (n) jn d jn hj (λ) Cn2 k Πλ(ω) L (I Ω) Cn' 2 . = = ≤ )=dλ = ∞ × ≤ = = k=1= = Thus, the first condition of (7.5) is satisfied with A =2,forallN N.Thesecondconditionof ∈ 1+2γ (7.5) holds with R =2 by the main assumption and the definition of the L2,γ-norm. Letting r =2andN yields by Lemma 7.4 that →∞ 2 ∞ j dim λ J : νλ L (R) =dim λ J : 2 hj(λ)= 1 2γ. - ∈ 2∈ . - ∈ j)=0 ∞. ≤ − " Observe that the case N =1ofLemma7.4(forwhichcompletedetailsweregiven)yields(7.4). ! Next we turn to the proof of (7.2). As a preliminary step we present the standard construction of aLittlewood–Paleydecomposition,seeStein[50]orFrazier, Jawerth, Weiss [13]. Recall that (R) S is the Schwartz space of smooth functions all of whose derivatives decay faster than any power. It is a basic property of the Fourier transform that it preserves . S Lemma 7.6. There exists ψ (R) so that ψˆ 0, ∈S ≥ ∞ j supp(ψˆ) ξ R :1 ξ 4 , and ψˆ(2− ξ)=1 if ξ =0. (7.10) ⊂{ ∈ ≤| |≤ } j=) 2 −∞ Moreover, given any finite measure ν on R and any γ R ∈ 2 ∞ 2jγ ν 2,γ 2 (ψ2 j ν)(x) dν(x)(7.11) 1R − 1 1 8 j=) ∗ −∞ j j where ψ j (x)=2ψ(2 x). 2− Proof. Choose φ (R)withφˆ 0, φˆ(ξ)=1for ξ 1andφˆ(ξ)=0for ξ > 2. Define ψ via ∈S ≥ | |≤ | | ψˆ(ξ)=φˆ(ξ/2) φˆ(ξ). It is clear that ψˆ(ξ) 0andthatψˆ(ξ)=0if ξ < 1or ξ > 4. (7.10) − ≥ | | | | holds since the sum telescopes. Moreover, it is clear from (7.10) that there exists some constant Cγ depending only on γ so that for any ξ =0 2 1 2γ ∞ 2jγ ˆ j 2γ Cγ− ξ 2 ψ(2− ξ) Cγ ξ . | | ≤ j=) ≤ | | −∞ j Since ψ j (ξ)=ψˆ(2 ξ), Plancherel’s theorem implies 2− − 3 ˆ j 2 (ψ2 j ν)(x) dν(x)= ψ(2− ξ) ν(ξ) dξ 1R − ∗ 1 | | 2 18 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK and (7.11) follows. ! Now we come to the main technical statement needed to prove theSobolevestimate(7.2). log λ0 Lemma 7.7. Let J =[λ0,λ1] be an interval of δ-transversality for some δ>0 and let β = 1. log λ1 − Suppose that ρ C∞(R) is supported in the interior of J and ρ 1.Letφ be any function in ∈ 1 1∞ ≤ the Schwarz space and ψ(x)=2φ(2x) φ(x).Fixs (1, 2).Thenforanydistinctω,τ Ω,with − ∈ ∈ ω τ = k,andanyR>0, | ∧ | s := ρ(λ)ψ R[Π (ω) Π (τ)] dλ C Rλk(1+3β) − , (7.12) +1 λ λ + 1 |J | + R > − ? + ≤ > ? + + where C depends only on ρ,β+ and s. + Proof. Fix ω,τ Ωsuchthat ω τ = k,andR>0. We may assume that ∈ | ∧ | Rλk(1+3β) 1 (7.13) 1 ≥ since otherwise the estimate is obvious. We can write Π (ω) Π (τ)=2λkf(λ)wheref(λ)isa λ − λ power series of the form (7.1). Recall that δ-transversality says that f (λ) < δ for λ J whenever ' − ∈ f(λ) <δ.Letη be the distance between the support of ρ and the boundary of J. δ-transversality implies that if f(λ) <δηfor some λ supp(ρ), then f has a zero λ J which is the only zero of | | ∈ ∈ f on J.Denoteu = λ λ.Wearegoingtolinearizeeverythingaroundλ.Clearly, − k k k 2 k 1 2 k 2 λ f(λ) λ f '(λ)u f '' λ1u + f ' kλ1− u C1kλ1u , (7.14) | − |≤1 1∞ 1 1∞ ≤ with the constant C depending only on J.Letχ C be non-negative with χ =1on[ 1 , 1 ]and 1 ∈ ∞ − 2 2 supp(χ) ( 1, 1). Then ⊂ − 2C k ρ(λ)ψ 2Rλkf(λ) dλ = ρ(λ)ψ 2Rλkf(λ) χ 1 f(λ) dλ 1R 1 2 kβ > ? > ? &δ ηλ1 ' 2C k + ρ(λ)ψ 2Rλkf(λ) 1 χ 1 f(λ) dλ. (7.15) 1 − 2 kβ > ?% &δ ηλ1 '( The integrand of the second integral is non-zero only if 2 kβ δ ηλ1 2kβ f(λ) > Cβλ1 . | | 4C1k ≥ k k(1+β) k k(1+3β) Then, using that λ λ1 for λ [λ0,λ1]weconcludethat 2Rλ f(λ) 2CβRλ1 .By ≥ ∈ | |≥ s k(1+3β) − the rapid decay of ψ,thesecondintegralin(7.15)isthereforelessthanCβ' (Rλ1 ) . Thus it suffices to estimate the first integral in (7.15), which we denote by 1.Noticethatits 2 kβ J integrand is nonzero only if f(λ) δ ηλ1 ,hence | |≤ 2C1k δηλkβ δλkβ u = λ λ 1 < 1 (7.16) | | | − |≤ 2C1k 2C1k BERNOULLI CONVOLUTIONS 19 by δ-transversality. This implies that k 2 1 k 1 k C kλ u < λ δ u λ f '(λ) u, 1 1| | 2 | |≤ 2 | | hence by (7.14) 1 k k 3 k λ f '(λ) u λ f(λ) λ f '(λ) u. (7.17) 2 | | ≤ | |≤ 2 | | 2C1k Let g(λ)=χ 2 kβ f(λ) ρ(λ). Since g(λ)=g(λ + u)=g(λ)+O( g' u)wehave > δ ηλ1 ? 1 1∞ k = g(λ + u) ψ(2Rλkf(λ)) ψ(2Rλ f(λ)u) du 1 1 J @ − A k + g(λ)ψ(2Rλ f(λ)u) du (7.18) 1 k + O( g' u) ψ(2Rλ f(λ)u) du = I1 + I2 + I3. 1 1 1∞ The integral I2 is the easiest one: since ψ(t)=2φ(2t) φ(t)wehave ψ(t) dt =0,soI2 =0. − # To estimate I1 we use that g 1, the mean value theorem, (7.14), (7.17), the rapid decay of 1 1∞ ≤ ψ ,andthat f (λ) >δ,toobtain ' | ' | k k k 2 k 4 ψ(2Rλ f(λ)) ψ(2Rλ f(λ)u) CkRλ1u min 1, (Rλ u)− . (7.19) + − + ≤ - . + + (The exponent 4abovecanbereplacedbyanynegativeintegerbychangingtheconstantbutis − sufficient for our purposes.) Since s<2wecanfindε so that 2 s 0 <ε< − . (7.20) 3 We write k 2 k 4 I1 CkRλ1u min 1, (Rλ u)− du = + = I11 + I12. ≤ 1R 1 u (Rλk)ε 1 1 u (Rλk)ε 1 - . | |≤ 1 − | |≥ 1 − We have k 2 k 2+3ε I11 CkRλ1u du = C'k(Rλ1)− =: C'S11. ≤ 1 u (Rλk)ε 1 | |≤ 1 − 1 (1+β) Estimating I we use that λ− λ− to get 12 ≤ 1 k 4 4k(1+β) 2 2 ε k(2+4β+ε) I12 CkRλ1 R− λ1− u− du = C''kR− − λ−1 =: C''S12. ≤ 1 u (Rλk)ε 1 · | |≥ 1 − s k(1+3β) − Astraightforwardcomputationshowsthatmax S11,S12 Cβ Rλ1 .Letusdemonstrate kβ { }≤ > ? this for S .Sincek C λ− it is enough to establish that 11 ≤ β 1 s 2+3ε k(2+β 3ε) k(1+3β) − R− λ−1 − Rλ1 ≤ > ? 20 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK which is equivalent to s 2+3ε k[(1+3β)s (2+β 3ε)] R − λ − − 1. 1 ≤ Applying (7.20) and (7.13) reduces the last inequality to 3(2 3ε) 1whichiscertainlytrue.The − ≥ expression S is handled similarly, eventually reducing the estimate to 3(2 + ε) 5. This concludes 12 ≥ the estimation of the integral I1 from (7.18). It remains to estimate I3.Firstrecallthedefinitionofg(λ)toget kβ g' Ckλ1− . 1 1∞ ≤ By the rapid decay of ψ and since f (λ) δ we obtain | ' |≥ kβ k 3 I3 Ckλ1− u min 1, (Rλ u)− du = + = I31 + I32. ≤ 1 1 u (Rλk)ε 1 1 u (Rλk)ε 1 - . | |≤ 1 − | |≥ 1 − where ε is again defined by (7.20). Now kβ 2+2ε k(2+β 2ε) I31 Ckλ1− udu= C'kR− λ1− − =: C'S31 ≤ 1 u (Rλk)ε 1 | |≤ 1 − and kβ 3 3k(1+β) 2 2 ε k(2+4β+ε) I32 Ckλ1− R− λ1− u− du = C''kR− − λ1− =: C''S32. ≤ 1 u (Rλk)ε 1 | |≥ 1 − s It remains to check that max S ,S Rλk(1+3β) − which is done similarly to the above (in { 31 32}≤ 1 fact, S = S and the estimate of S reduces> to s Proof of Theorem 7.3. If an interval of δ-transversality is enlarged slightly, it is obviously goingto δ be an interval of 2 -transversality. Thus, (7.2) will be established if we show that for an arbitrary interval of δ-transversality J =[λ ,λ ] ( 1 , 1) and any function ρ C (R)supportedinits 0 1 ∈ 2 ∈ ∞ 2 α 1 interior, R ν ρ(λ) dλ< . Let λ = .Wehaveα<2bythediscussionoftransversality 1 λ12,γ ∞ 0 2 above. Fix# some γ<1 with λ1+2γ > 1 and s (1 + 2γ,2). Assume first that J is so short that 2 0 2 ∈ 0 < (1 + 2γ)(1 + 3β) α where β = log λ0 1. Let ψ be the Littlewood-Paley function from ≤ log λ1 − BERNOULLI CONVOLUTIONS 21 Lemma 7.6. In view of (7.11), the definition of νλ,andLemma7.7 ∞ 2 ∞ 2jγ ∞ νλ ρ(λ) dλ 2 (ψ j νλ)(x) dνλ(x) ρ(λ) dλ 1 2,γ 1 1 2− 1 1 8 j=) ∗ −∞ −∞ −∞ ∞ 2j(1+2γ) ψ 2j[Π (ω) Π (τ)] ρ(λ) dλ dµ(ω)dµ(τ) 1 1 +1 λ λ + ≤ Ω Ω ) + > − ? + j= + + −∞ + + s ∞ j(1+2γ) j ω τ (1+3β) C 2 min 1, 2 λ| ∧ | − dµ(ω)dµ(τ) β,γ 1 1 1 ≤ Ω Ω ) * > ? , −∞ dµ(ω)dµ(τ) Cβ,γ (1+3β)(1+2γ) ω τ Cβ,γ α(µ) < , ≤ 1Ω 1Ω ≤ E ∞ λ1 | ∧ | as claimed. This argument depended on β being sufficiently small. In the general case fix any small 1+β β>0andpartitionJ into subintervals Ji =[λi,λi+1]fori =0,... ,m so that λi >λi+1 .Applying the previous calculation to each of the Ji and summing concludes the proof of (7.2). The dimension estimate (7.3) follows from (7.2) and Proposition 7.5, letting γ 1 ( log 2 1). ! 2 log λ0 → − − 1 log 2 It is well–known that for 0 <λ< 2 the support of νλ is a Cantor set of dimension log λ .In log 2 − 1 fact, νλ is a Frostman measure on that set, which implies that dim(νλ)= log λ for 0 <λ< 2 . − Solomyak [47] showed that the first double zero for a power series of the form (7.1) lies in the interval [0.649, 0.683]. In particular, the previous theorem will apply only up to some point in this interval. Nevertheless, one can show that ν has some smoothness for a.e. λ ( 1 , 1). This follows λ ∈ 2 from Theorem 7.3 by “thinning and convolving”, see [47] and [39]. As one expects, the number of derivatives tends to as λ 1. ∞ → Lemma 7.8. For any 4>0 there exists a γ = γ(4) > 0 so that 1 √2 2 νλ dλ< . (7.21) 1 1 2,γ 2 +4 1 1 ∞ Furthermore, there exists some ' (2 1/2, 2 1/4) and a γ > 0 so that 0 ∈ − − 0 #0 2 νλ 2,γ dλ< . (7.22) 1 2 0 #0 1 1 ∞ Proof. As mentioned above, [0,λ1]isanintervaloftransversalityforthepowerseries(7.1)for some λ > 2 2/3.Fixanyλ ( 1 , 2 2/3]. Partitioning the interval [λ ,λ ]asintheproofof 1 − 0 ∈ 2 − 0 1 Theorem 7.3, one obtains from (7.2) that λ1 2 1+2γ 1 νλ 2,γ dλ< provided λ0 > . (7.23) 1λ0 1 1 ∞ 2 22 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK 2/3 To go beyond 2− we remove every third term from the original series. More precisely, let Πλ(ω)= n 3 n ωnλ and denote the distribution of this series by νλ.Itwasshownin[47]and[39]thattheB -| !class of power series (7.1) that satisfy either b =0forallj 0orb =0forallj 0 3j+1 C ≥ 3j+2 ≥ have [0,λ3]asanintervalofδ–transversality for some λ3 > 1/√2. After some straightforward modifications the argument given above for the full series shows that λ3 2 1+2γ 2/3 νλ 2,γ dλ< provided λ2 > 2− . (7.24) 1λ2 1 1 ∞ C For more details we refer the reader to section 5.1 of [38]. Since ν ν and λ > 2 2/3,(7.21) | λ|≤| λ| 1 − follows from (7.23) and (7.24). Moreover, we have shown (7.22). " C" ! 1 Corollary 7.9. For any λ0 > 2 there exists 4(λ0) > 0 such that dim λ (λ , 1) : ν does not have L2–density < 1 4(λ ). { ∈ 0 λ } − 0 Proof. This follows from the previous lemma and Proposition 7.5 using the identity νλ(ξ)=νλ2 (ξ)νλ2 (λξ) . " 3 3 ! As observed by Kahane [19], Erd˝os’s argument yields that 4(λ ) 1asλ 1(seesection6). 0 → 0 ↑ 8. Applications, generalizations and problems 8.1. Applications to dimension and dynamics. Alexander and Yorke [1] considered the “fat baker’s transformation” (λx +(1 λ), 2y 1) if y 0 Tλ(x, y)= − − ≥ (λx (1 λ), 2y +1) if y<0 − − on the square [ 1, 1]2.TheyprovedthattheSinai-Bowen-Ruellemeasureη for T is the product − λ λ of ν (more precisely, its affine copy supported on [ 1, 1]) and the uniform measure in y-direction. λ − They showed further that absolute continuity of νλ implies the equality of the information (R´enyi) and Lyapunov dimension for ηλ but this breaks down in the Pisot case. Another application of Bernoulli convolutions has to do withfractalgraphs.Letφ be a Z-periodic function and λ ( 1 , 1). Define ∈ 2 ∞ n n Γλ,φ = (x, y): x [0, 1],y= λ φ(2 x) . * ∈ n)=0 , BERNOULLI CONVOLUTIONS 23 If φ(x)=cos(2πx)thisdefinesafamilyofWeierstrassnowheredifferentiablefunctions. It is an open problem to compute dim Γλ,cos(2πx),evenforatypicalλ.Thisproblemservedasamotivationfor studying Γλ,φ with an easier choice of φ. Przytycki and Urba´nski [42] considered the case φ(x)=r(x)=1ifx [0, 1 )mod1andr(x)= 1 ∈ 2 − otherwise. This is a discontinuous function with a self-affinegraph.Itisprovedin[42]thatif dim νλ =1then log(1/λ) dim Γ =dim Γ =2 . λ,r(x) M λ,r(x) − log 2 Here dimM is the Minkowski (box) dimension. The equality for Minkowskidimensionholdsforall λ ( 1 , 1) but the Hausdorffdimension drops for reciprocals of Pisot numbers. We note that the ∈ 2 methods of [42] readily extend to the case of more general self-affine sets invariant for the iterated function system (γx,λx 1), (γx+(1 γ),λx+1) for γ (0, 1 ). In particular, dim ν =1suffices { − − } ∈ 2 λ for the equality of the Hausdorffand Minkowski dimensions. Ledrappier [29] studied the family of continuous graphs Γλ,φ where φ(x)=dist(x, Z)(sometimes called Takagi graphs). Their analysis is quite a bit harder. Ledrappier proved that if dim ν 1 =1 (2λ)− then dim Γ =2 log(1/λ) . λ,φ − log 2 Observe that in all applications mentioned here it is the equality dim νλ =1thatgetsused,not the absolute continuity of νλ. 8.2. Generalizations. There are many natural generalizations of Bernoulli convolutions; many of them can be treated similarly to the classical case with some additional work. (i) Biased Bernoulli convolutions:asintheclassicalcasebutthesignsaretakenwithproba- bilities (p, 1 p). We will generalize a bit further: − (ii) Suppose that D R is an arbitrary finite set of digits, with card(D)=m,andp = ⊂ D,p (p1,... ,pm)isaprobabilityvector.Letνλ be the distribution of the random series n ∞ a λ where a D independently with probabilities p . n=0 n n ∈ i ! “Almost sure” results on the existence of a density in Lq(R)forνD,p,whenq [1, 2], λ ∈ were obtained in [40], and the dimension of exceptions was estimated in [38] (for q =1and 2). These results were proved on an interval of transversality which, in this case, means an interval free of double zeros for power series with coefficients in D D.Checking − transversality is not always easy, so some of these results are less complete than those for classical Bernoulli convolutions. For instance, it is proved in [40] that the (p, 1 p) − Bernoulli convolutions are absolutely continuous for a.e. λ (pp(1 p)1 p, 1), but only ∈ − − for p [1/3, 2/3]. ∈ 24 YUVALPERES,WILHELMSCHLAG,ANDBORISSOLOMYAK D,p It is easy to see that the Erd˝os-Kahane argument transfers tothecaseofνλ .The D,p question of convergence to zero at infinity of νλ was considered by Salem (Borwein and Girgensohn [4] were apparently unaware of this" when they discussed some special cases.) Making a linear change of variable we can assume that the first two digits in D are 0 and D,p 1 1. Then νλ tends to zero at infinity if and only θ = λ− is Pisot and and D lies in the field of θ,see[44,Ch.VII]." (iii) Consider the same set-up as in (ii) but with complex ai and λ complex of modulus less than one. Some results were obtained in [49] (and the dimension of exceptions was estimated in [38]) but checking transversality becomes more formidable.Notethathere,determining the support of the measure in the two-digit case is non-trivial. (iv) Convolutions of self-similar measures and arithmetic sums of Cantor sets: see [48, 40, 38] for some “almost sure” results; see also the references in [48] for other work on sums of Cantor sets and the connection with smooth dynamics and the Palis-Takens problem. BERNOULLI CONVOLUTIONS 25 8.3. Problems on the Bernoulli convolutions νλ. 1. Other properties of the density. Since ν has a density dνλ in L2(R)fora.e. λ ( 1 , 1), it λ dx ∈ 2 follows from the formula ν ( )=ν 2 ( ) ν 2 (λ )(8.1) λ · λ · ∗ λ · that dνλ is continuous for a.e. λ (2 1/2, 1). It is not known whether dνλ is continuous, dx ∈ − dx or even bounded, for a.e. λ ( 1 , 2 1/2). Using (8.1) again and the result of Mauldin and ∈ 2 − Simon [36], we may infer that for a.e. λ (2 1/2, 1), the density of ν is strictly positive in ∈ − λ the interior of its support. We do not know whether for a.e. λ ( 1 , 2 1/2), the essential ∈ 2 − dνλ infimum of dx on any compact subinterval of supp(νλ)ispositive. Numerical approximation of self-similar measures was studied in several papers, among them [51] which contains histograms of νλ for some λ. 2. Is absolute continuity generic? We saw in Proposition 4.1 that for λ ( 1 , 1), the Bernoulli ∈ 2 convolution generically has correlation and Hausdorffdimension equal to one. The anal- ogous question for absolute continuity is open. We are grateful to Elon Lindenstrauss for simplifying the original proof of the following proposition. 1 Proposition 8.1. The set S = λ ( , 1) : νλ is singular is Gδ. ⊥ { ∈ 2 } Proof. It is easy to see that the function λ ν (a, b)iscontinuousforanyinterval(a, b). 9→ λ Let be the collection of all finite unions of open intervals. Fix a sequence ε converging G n to 0. Now observe that S = λ (1/2, 1) : νλ(G) > 0.5 ⊥ { ∈ } /n (G0)<εn L where the union is over all G with (G) <εn.ThusS is a Gδ set. ∈G L ⊥ Aconsequenceofthispropositionisthatifabsolutecontinuity holds on a residual set in ( 1 , 1), then the exceptional set S is nowhere dense, and hence, by (8.1), there is a left 2 ⊥ neighborhood of 1 which is disjoint from S . ⊥ 3. Let 1 1/n Does the limit limn Jn(λ) exist for all λ (1/2, 1)? →∞ ∈ Apositiveanswerwouldimplythatsomeneigborhoodof1doesnot contain Salem num- bers. To see this, fix 1 <λ < 1. As proved in [38] (see 7), there exists γ>0suchthat 2 0 § ν L2 for a.e. λ [λ , 1). The set λ ∈ γ ∈ 0 ∞ ∞ W := λ [λ , 1) : J (λ) < (1 4)n 1 ∈ 0 n − k/=1 n0=k * , is a Gδ set in [λ0, 1) for any 4>0. 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Math. 57 (1935), 827–838. 53. L.-S. Young, Dimension, entropy and Lyapunov exponents. Ergodic Th. Dynam. Sys. 2 (1982), 109–124. Yuval Peres, Department of Mathematics, Hebrew University, Jerusalem and Department of Statistics, University of California at Berkeley. [email protected] Wilhelm Schlag, Department of Mathematics, Fine Hall, Princeton University Princeton, NJ 08544, USA [email protected] Boris Solomyak, Box 354350, Department of Mathematics, University of Washington, Seattle, WA 98195, USA, [email protected] Current address: Institute of Mathematics, The Hebrew UniversityofJerusalem,GivatRam, Jerusalem, Israel, [email protected]