Smoothness of Projections, Bernoulli Convolutions, and Dimensions Of

SMOOTHNESS OF PROJECTIONS, BERNOULLI CONVOLUTIONS, AND THE DIMENSION OF EXCEPTIONS

YUVAL PERES AND WILHELM SCHLAG

n Abstract. Erd˝os (1939, 1940) studied the distribution ν of the random series ∞ λ ,andshowedthat λ 0 ± ν is singular for infinitely many λ (1/2, 1), and absolutely continuous for a.e. λ in a small interval λ ∈ P (1 δ, 1). Solomyak (1995) proved a conjecture made by Garsia (1962) that ν is absolutely continuous − λ for a.e. λ (1/2, 1). In order to sharpen this result, we have developed a general method that can be used ∈ to estimate the Hausdorffdimension of exceptional parameters in several contexts. In particular, we prove: For any λ0 > 1/2, the set of λ [λ0, 1) such that ν is singular has Hausdorffdimension strictly less than 1. • ∈ λ For any Borel set A d with Hausdorffdimension dim A>(d +1)/2, there are points x A such that • ⊂ R ∈ the pinned distance set x y : y A has positive Lebesgue measure. Moreover, the set of x where this {| − | ∈ } fails has Hausdorffdimension at most d +1 dim A. − Let denote the middle-α Cantor set for α =1 2λ and let K be any compact set. Peres and • Kλ − ⊂ R Solomyak (1998) showed that for a.e. λ (λ0, 1/2) such that dim K +dim > 1, the sum K + has ∈ Kλ Kλ positive length; we show that the set of exceptional λ in this statement has Hausdorffdimension at most 2 dim K dim . − − Kλ0 For any Borel set E d with dim E>2, almost all orthogonal projections of E onto lines through • ⊂ R the origin have nonempty interior, and the exceptional set of lines where this fails has dimension at most d +1 dim E. − d If µ is a Borel probability measure on R with correlation dimension greater than m +2γ,thenfora • 1 d m “prevalent” set of C maps f : R R (in the sense described by Hunt, Sauer and Yorke (1992)), the → 2 m image of µ under f has a density with at least γ fractional derivatives in L (R ).

1. Introduction n The distribution νλ of the random series 0∞ λ , where the signs are chosen independently with proba- 1 ± bility 2 , has been studied by many authors since the two seminal papers by Erd˝osin 1939 and 1940. It is 1 1 immediate that νλ is singular for λ< 2 . In [6], Erd˝osshowed that νλ is singular for infinitely many λ ( 2 , 1), 1 ∈ namely those λ such that λ− is a Pisot number, and in [7] he proved that νλ is absolutely continuous for a.e. λ (1 δ, 1) where δ<0.01. It had been observed before by Jessen and Wintner [21] that νλ is either absolutely∈ continuous− or singular with respect to Lebesgue measure for any choice of λ. Works of Alexander and Yorke [2], Przytycki and Urbański [37], and Ledrappier [27] showed the importance of these distributions in several problems in dynamical systems. Solomyak [39] proved a conjecture made by Garsia in 1962, that νλ 2 is absolutely continuous for a.e. λ (1/2, 1); in fact, he showed that νλ has density in L for a.e. λ (1/2, 1). Peres and Solomyak [34] then gave∈ a simpler proof of this. A survey of the results obtained on∈ Bernoulli convolutions from the 1930’s to 1999 is presented in [33]. 2 In the present paper, we show that the density of νλ has a fractional derivative in L for a.e. λ (1/2, 1), and obtain a bound on the integrated Sobolev norm. We then use this bound to establish that∈ for any closed interval I (1/2, 1], the set of λ I such that νλ is singular, has Hausdorffdimension less than 1 (Theorem 5.4 and⊂ Corollary 5.6). In order∈ to prove these results, we were led to develop a general method, which can be applied to improve previous results on Hausdorffdimensions of sums of Cantor sets, distance sets,

1991 Mathematics Subject Classiﬁcation. Primary 42B25; Secondary 28A78. Y. Peres was partly supported by NSF grant DMS–9404391. W. Schlag was supported by a NSF postdoctoral fellowship at the Mathematical Sciences Research Institute, Berkeley. 1 2 YUVALPERESANDWILHELMSCHLAG

and self-similar sets with deleted digits. (An exposition of our method for the case of Bernoulli convolutions is in [33]). Heuristically, the measures νλ may be viewed as nonlinear projections of the uniform measure on sequence space. Indeed, the analysis of νλ in [39] and [34] employed techniques developed by Kaufman [23] and Mattilla [30] to study orthogonal projections of sets and measures in Euclidean space. The following (informally stated) principles were first discovered by Marstrand [28], Kaufman [23, 24] and Mattila [29, 30] in the study of orthogonal projections, and then extended to a variety of other settings by Falconer [12], Hu and Taylor [16], Pollicott and Simon [36], Solomyak [39, 40, 41], Peres and Solomyak [34, 35] and Sauer and Yorke [38]. If an m dimensional measure is “projected” to m dimensional space, then “typically” its di- • mension− is preserved. If an m + dimensional measure is “projected” to m dimensional space, then “typically” the • projected measure is absolutely continuous, with a density in L2. In making these principles precise, the appropriate notion of dimension of measures must be used (Correlation or information dimension, but not Minkowski or packing dimension; see Jarvenpää[20] and [38]), and the parametrized family of generalized projections considered must be sufficiently rich (at least m dimensional). A more delicate requirement is a certain “transversality condition” (e.g., Lemma 3.11 in [31]). For Bernoulli convolutions and other self-similar measures, this condition involves bounds on the double zeros of certain power series, and it was first made explicit by Pollicott and Simon [36]. Transversality is crucial in the works of Solomyak [39, 40, 41], Peres and Solomyak [34, 35], and in the present paper. The general principles that underly our work can be informally stated as follows: If an m dimensional measure µ is “projected” to an m dimensional space, then the set of • parameters− where the projection of µ has lower dimension than µ itself, has “codimension” at least . If an m + dimensional measure is “projected” to m dimensional space, then “typically” the • projected measure has a density with a fractional derivative of order /2inL2, and the set of parameters where the projected measure is singular has “codimension” at least . The notion of Sobolev dimension of a measure, defined in (2.3) below, yields a unification of these two principles. For orthogonal projections, these strengthened principles have been stated precisely, and established as theorems, by Kaufman [24], Falconer [9] and Mattila [29], except that they did not consider fractional derivatives. Their proofs are based on averaging with respect to an appropriate Frostman measure on parameter space. In the setting of Bernoulli convolutions, this averaging approach is not powerful enough to establish the last principle, so we were forced to develop a different technique. Our methods apply to several concrete problems, which we now describe. Throughout the paper, dim (without subscripts) will mean Hausdorffdimension. Orthogonal projections. To motivate the general formulations, we start by clarifying the impor- • tant results of Falconer [9] on exceptional directions for projections in Rd (see Sections 2 and 6). As a corollary, we find that for any Borel set E Rd with dim E>2 and for a.e. direction θ in the d 1 ⊂ sphere S − , the orthogonal projection projθ(E) of E to the line through 0 and θ, has nonempty interior. More precisely (see corollary 6.2), d 1 dim θ S − :proj(E) has empty interior d +1 dim E. { ∈ θ }≤ − n Symmetric Bernoulli Convolutions. Let νλ be the distribution of the random series n∞=0 λ , • n ± so νλ has the Fourier transform νλ(ξ)= ∞ cos(λ ξ). Here we sharpen the result of Solomyak [39] n=0 that νλ is absolutely continuous for a.e. λ (1/2, 1), by showing that the 2,γ–Sobolev norm satisfies ∈ ∞ ν 2 = ν (ξ) 2 ξ 2γ dξ< λ2,γ | λ | | | ∞ −∞ SMOOTHNESS OF PROJECTIONS 3

for a.e. λ<0.649 such that λ1+2γ > 1/2. Moreover, for some constant C>0, and all small >0, we have dim λ (1/2+, 1) : ν is singular < 1 C. ∈ λ − p n Asymmetric Bernoulli Convolutions. Let νλ be the distribution of the sum n∞=0 λ • where the signs are chosen randomly and independently with probabilities (p, 1 p). Peres± and p p 1 p − p Solomyak [35] showed that ν is absolutely continuous for a.e. λ (p (1 p) − , 1), but ν has a λ ∈ − λ density in L2 for a.e. λ (p2 +(1 p)2, 1), and not for any smaller λ. The Pisot numbers still ∈ − p provide inﬁnitely many examples of singular νλ. In the present paper, we show that

∞ νp(ξ) 2 ξ 2γ dξ< | λ | | | ∞ −∞ for a.e. λ<0.649 such that λ1+2γ >p2 +(1 p)2. Moreover, if p (1/3, 2/3), then for some constant C>0, and all >0, − ∈ dim λ (p2 +(1 p)2 + , 1) : νp L2 < 1 C, ∈ − λ ∈ − and p 1 p p dim λ (p (1 p) − + , 1) : ν is singular < 1 C. ∈ − λ − d d Intersections of sets with spheres Suppose E R is a Borel set with d 2. Let S0 R be • ⊂ ≥ ⊂ a strictly convex, closed C∞–hypersurface surrounding the origin. Then d dim x R :(x + rS0) E = for a.e. r>0 1+d dim E. ∈ ∩ ∅ ≤ − This statement fails if S0 is the boundary of a cube or more generally, if S0 contains a piece of a d 1 hyperplane. If S0 is the sphere S − , then the estimate above reduces to a strengthened form of Falconer’s result in [10] about distance sets: For any Borel set A Rd with Hausdorffdimension dim A>(d+1)/2, there are points x A such that the pinned distance⊂ set x y : y A has ∈ {| − | ∈ } positive Lebesgue measure. Moreover, the set of x Rd where this fails has Hausdorffdimension at most d +1 dim A (see Corollary 8.4). Variants∈ involving the Hausdorffdimension of the radii, as well as estimates− where x is restricted to a hyperplane, can be found in section 8.2. We also 2,δ obtain results if S0 is assumed to be only in C for some 1 >δ>0. Sums of Cantor sets. Let • ∞ = (1 λ) a λn : a 0, 1 Kλ − n n ∈{ } n=0 be the middle-α Cantor set for α =1 2λ. Let K R be any compact set. Peres and Solomyak [35] showed that 1(K + ) > 0 for almost− every λ ⊂ (0, 1 ) such that dim K +dim > 1, and that H Kλ ∈ 2 Kλ dim(K+ λ)=dimK+dim λ for a.e. λ (0, 1/2) such that dim K+dim λ < 1. In Theorem 5.12 we proveK that K ∈ K dim λ (λ , 1/2) : 1(K + )=0 2 (dim K +dim ), ∈ 0 H Kλ ≤ − Kλ0 and dim λ (0,λ ):dim(K + ) < dim K +dim dim K +dim . ∈ 1 Kλ Kλ ≤ Kλ1 We also establish similar estimates for Cantor sets whose symbols belong to the Hölder space C1,δ for some δ>0. Keane–Smorodinsky 0, 1, 3 –problem. The Hausdorffdimension of the set • { } ∞ Λ(λ)= a λi : a 0, 1, 3 i i ∈{ } i=0 4 YUVALPERESANDWILHELMSCHLAG

has been studied by several authors, since it is perhaps the simplest example of a parametrized 1 family of self-similar sets with overlap. For λ 4 , it is easy to see that Λ(λ) is self–similar log 3 ≤ with Hausdorffdimension log λ . Keane, Smorodinsky, and Solomyak [26] proved that Λ(λ)= − 1 2 [0, 3/(1 λ)] if λ>2/5 and that dim Λ(λ) < 1 for infinitely many λ ( 3 , 5 ). Pollicott and − 1 1 log∈ 3 Simon [36] showed that for a.e. λ ( 4 , 3 ) one still has dim Λ(λ)= log λ and they found a 1 1 ∈ − dense subset of ( 4 , 3 ) where the dimension is strictly less than that fraction. Solomyak [39] finally 1 2 established that for a.e. λ ( 3 , 5 ), the set Λ(λ) has positive Lebesgue measure. In Theorem 5.10, we establish that ∈ log 3 dim λ (λ , 2/5) : 1(Λ(λ)) = 0 2 for any λ > 1 , ∈ 0 H ≤ − log λ 0 3 − 0 log 3 log 3 1 dim λ (1/2,λ1):dimΛ(λ) < for any λ1 > 4 . ∈ log λ ≤ log λ1 − − The latter inequality improves an estimate from [36]. Self–similar sets in the plane. Consider the self–similar sets • ∞ CS = a λn : a S λ n n ∈ n=0 2 S where S = s1,...,s C is a fixed set of symbols. In [41] Solomyak showed that (Cλ ) > 0 { 1/2 }⊂ H for a.e. λ> l − in a region of transversality. In Theorem 8.2 below we prove that the dimension of the exceptional| | set for this property has to be strictly less than 2. Typical C1 maps trade offdimension for smoothness. Hunt, Sauer and Yorke [18] defined a • Borel set A in a Banach space X to be prevalent if some Borel probability measure ν on X satisfies ν(A + x) = 1 for all x X. Sauer and Yorke [38] and Hunt and Kaloshin [17] showed that if µ is ∈ d a Borel probability measure on R with correlation dimension at most m,thenthesetAµ(d, m) of C1 maps f : Rd Rm that map µ to a measure with the same correlation dimension is prevalent. To prove this they→ showed that the definition of prevalence holds with ν equal to Lebesgue measure on the set of m d real matrices with entries bounded by 1 in absolute value (these matrices × are considered as linear maps from Rd to Rm). This result can be obtained as a special case of Theorem 7.3, which also yields the following complement (see Remark 7.4): Let µ be a Borel probability measure on Rd with correlation dimension greater than m+2γ. Then for a prevalent set of C1 maps f : Rd Rm the image of µ under f has a density with at least γ fractional derivatives → in L2(Rm). Consequently, if E Rd has dim E>m(respectively, dim E>2m), then for a ⊂ prevalent set of maps f : Rd Rm, the image f(E) has positive Lebesgue measure (respectively, → nonempty interior) in Rm. All these examples are special cases of our main result that is formulated in the following general setting. m Given a measure µ on a compact metric space (Ω,d) and a family of maps Πλ :Ω R parameterized n → smoothly by λ R with n m, we show that for a.e. λ the projection of µ under Πλ has as much “smoothness” as∈µ and that the≥ Hausdorffdimension of the exceptional parameters decreases with the loss of smoothness. This requires the family Πλ to satisfy some nondegeneracy assumption. In this paper we impose n the aforementioned transversality condition. In the case of Bernoulli convolutions, Πλ(ω)= n∞=0 ωnλ where ω Ω= 1, +1 N (N denotes the nonnegative integers). Here transversality means that the power ∈ {− } series Πλ(ω) Πλ(τ) do not have double zeros. Solomyak showed in [39] that this is the case if λ<0.649. Extending our− smoothness results to the entire unit interval then requires arguments that are special to Bernoulli convolutions, namely breaking the series up into various subseries. The paper is organized as follows. In section 2, which is still mainly expository, we discuss transversality and state the general projection theorem in one dimension. Our estimate on the Hausdorffdimension of the set of λ for which νλ is a singular measure does not rely on Frostman’s lemma. Instead we use the SMOOTHNESS OF PROJECTIONS 5

following fact: Let h ∞ be a family of nonnegative C∞–functions on [0, 1] whose derivatives grow at most { j}j=0 1 j ∞ exponentially in j.If 0 0 R hj(x) dx < , then one can bound the dimension of the set of x [0, 1] for j ∞ ∈ which ∞ r h (x)= for any 1 r

2. Ageneralschemeforfamiliesofprojections To motivate this section we review some simple facts about orthogonal projections in Euclidean space.

n n 1 Definition 2.1. Let µ be a finite measure on R , n 2, with compact support. For any θ S − its ≥ ∈ projection νθ onto the line tθ : t R is given by fdνθ = f((x θ)θ) dµ(x) for any continuous f. For any α (0,n)theα–energy{ of µ is defined∈ } to be · ∈ dµ(x)dµ(y) α(µ)= α . E n n x y R R | − | We measure smoothness of measures ν in Rn in terms of the homogeneous Sobolev norm

2 2 2γ ν 2,γ = νˆ(ξ) ξ dξ. n | | | | R Finiteness of ν for some γ>0, means that ν has γ (fractional) derivatives in L2. 2,γ The following proposition relates the smoothness of the projected measures to the energy of the original 1 measure. If a, b > 0, then the notation a b means that C− a

Proof. The projected measures ν clearly satisfy ν (ξ)=ˆµ((ξ θ)θ). Hence θ θ · 2 ∞ 2 2γ ∞ 2 2γ νθ 2,γ dθ = νθ(tθ) t dt dθ = µˆ(tθ) t dt dθ Sn 1 Sn 1 | | | | Sn 1 | | | | − − −∞ − −∞ 2 1+2γ n =2 µˆ(ξ) ξ − dξ = cγ,n 1+2γ (µ). n | | | | E R The ﬁnal equality follows from Plancherel’s Theorem and the deﬁnition of energy; see Lemma 12.12 in [31]. The main purpose of this section is to study regularity properties of projected measures in a more general setting. Moreover, we bound the dimension of the set of those parameters for which the projected measure has less regularity than generically. In case of projections in the plane the latter question was considered by Kaufman [24] and Falconer [9], see also section 6. For the exceptional set in Proposition 2.2 with γ = 0, Falconer found the estimate 1 2 dim θ S : νθ L (R) 2 α { ∈ ∈ }≤ − if α(µ) < . However, his method does not apply to Bernoulli convolutions or other examples considered inE this paper.∞ Our approach is based on the Littlewood–Paley decomposition from harmonic analysis, which provides a simple characterization of various degrees of smoothness, see section 4. Moreover, the analogue of Falconer’s bound is obtained without Frostman measures.

Definition 2.3. For any finite measure ν on Rn,letitsSobolev dimension be defined as

2 α n (2.1) dims(ν)=sup α R : νˆ(ξ) (1 + ξ ) − dξ< . { ∈ n | | | | ∞} R n Clearly, if 0 < dims(ν) n,thenν is absolutely continuous and its density has fractional derivatives of order (σ n)/2inL2(Rn). Throughout this paper dim (without sub– or superscripts) will mean Hausdorﬀdimension.− The following deﬁnition introduces the general framework we will be working with.

Definition 2.4. Let (Ω,d) be a compact metric space, J R an open interval, and let Π: J Ω R be ⊂ × → a continuous map. We assume that for any compact I J and =0, 1,... there exists a constant C,I such that ⊂ d (2.2) Π(λ,ω) C dλ ≤ ,I 1 for all λ I and ω Ω. Given any finite measure µ on Ωlet ν = µ Π− ,whereΠ( )=Π(λ, ). The λ λ λ ∈ ∈ dµ(ω1)dµ(ω2) ◦ · · α–energy of µ is defined as α(µ)= α . E Ω Ω d(ω1,ω2) Informally, one can think of Πλ( )= Π(λ, ) as a family of projections parameterized by λ. We state two results on the smoothness of a typical· ν in terms· of (µ) assuming certain transversality conditions. λ Eα Definition 2.5. Let Ω, J, and Πbe as in Definition 2.4. For any distinct ω ,ω Ωand λ J let 1 2 ∈ ∈ Π(λ,ω1) Π(λ,ω2) Φλ(ω1,ω2)= − . d(ω1,ω2) J is an interval of strong transversality for Πif there exist positive constants C, C so that for all λ J ∈ and ω1,ω2 Ω ∈ ((i)) d Φ (ω ,ω ) C d Φ (ω ,ω ) for =2, 3,... | dλ λ 1 2 |≤ | dλ λ 1 2 | ((ii)) d Φ C. | dλ λ|≥ SMOOTHNESS OF PROJECTIONS 7

Theorem 2.6. Let J and Π be as in Definition 2.4 and suppose that J is an interval of strong transversality for Π. Let µ be a finite positive measure on Ω with finite α–energy for some α>0. Then for any compact I J ⊂ 2 α (2.3) ν (ξ) dλ C ξ − (µ). | λ | ≤ α| | Eα I Moreover, dim (ν ) α for a.e. λ J. More precisely, for any σ (0,α], s λ ≥ ∈ ∈ (2.4) dim λ J :dim(ν ) <σ σ +min(1 α, 0). { ∈ s λ }≤ −

The condition of strong transversality turns out to be too restrictive for most applications. Since the cosine function has vanishing derivative at 0,π it fails for projections onto lines. More importantly, (2.3) fails for Bernoulli convolutions because of Pisot numbers, see Lemma 5.7. The correct notion of transversality in the context of Bernoulli convolutions turns out to be the following one.

Definition 2.7. Let Φλ be as in Definition 2.5. For any β [0, 1) we say that J is an interval of transversality of order β for Πif there exists a constant C so∈ that for all λ J and ω ,ω Ωthe condition β ∈ 1 2 ∈ Φ (ω ,ω ) C d(ω ,ω )β implies | λ 1 2 |≤ β 1 2 d (2.5) Φ C d(ω ,ω )β. dλ λ ≥ β 1 2 In addition, we say that Πλ is L–regular on J for some positive integer L or L = , if under the same ∞ condition and for some constants Cβ,, d β (2.6) Φ (ω ,ω ) C d(ω ,ω )− for =1, 2,...,L. dλ λ 1 2 ≤ β, 1 2 Our main result in this section is the following theorem. Theorem 2.8. Let Ω, J, Π be as in Definition 2.4 and suppose that J is an interval of transversality of order β for Π for some β [0, 1) and that Πλ is –regular on J. Let µ be a finite positive measure on Ω with finite α–energy for some α∈ >0. Then for any compact∞ I J ⊂ (2.7) ν 2 dλ C (µ) if 0 < (1 + 2γ)(1 + a β) α, λ2,γ ≤ γ Eα 0 ≤ I where a0 is some absolute constant. Moreover, for any σ (0,α], ∈ α (2.8) dim λ J :dims(νλ) σ 1+σ , { ∈ ≤ }≤ − 1+a0β and for any σ (0,α 3β], ∈ − (2.9) dim λ J :dim(ν ) <σ σ. { ∈ s λ }≤

In all our applications of this theorem we will be able to take β arbitrarily small. Moreover, the geometric problems we consider in this paper satisfy transversality with β = 0, see sections 6 and 8.2. Some of these geometric estimates are known to be optimal. Since they are covered by our general theory, it follows that the corresponding cases of Theorem 2.8 are sharp, see section 6 for more discussion. However, we do not know whether Theorem 2.8 is optimal in all cases. The basic idea behind the proof of (2.7) is that for any finite measure ν on R and γ ( 1/2, 1) ∈ − 1 ∞ 2j(γ+1) j j 2 2 (2.10) ν 2,γ 2 ν(x 2− ,x) ν(x, x +2− ) | − − | 2 j= L (dx) −∞ 8 YUVALPERESANDWILHELMSCHLAG as one can easily check by applying Plancherel’s theorem to the right–hand side. However, it is difficult to work with the square–function in (2.10) because of the singularities of the kernel χ[ 1,0] χ[0,1]. As is standard in harmonic analysis, one circumvents this difficulty by considering a smoother kernel.− − The most convenient form is given in terms of the Littlewood–Paley decomposition, see Lemma 4.1. Theorem 2.8 has a simple corollary concerning the dimension of the exceptional set with respect to absolute continuity without making any assumptions on the energy of µ. The hypothesis will be in terms of the upper and lower information dimension of µ. Recall that the lower pointwise dimension is given by log µ(B(ω,r)) πµ−(ω)=liminf r 0+ → log r and set dimI (µ)= π− and dim (µ)=essinfdµ(ω) π−(ω) . µ L∞(dµ) I µ Notice that α(µ) < implies dim(µ) α, but the converse is clearly false. E ∞ I ≥ Corollary 2.9. Suppose J is an interval of transversality of order β for Π and that Πλ is –regular on J. Then ∞

dimI (µ) (2.11) dim λ J : νλ is singular 2 { ∈ }≤ − 1+a0β

dimI (µ) (2.12) dim λ J : νλ is not absolutely continuous 2 , { ∈ }≤ − 1+a0β where a0 is the constant from Theorem 2.8.

Proof. Let α = dimI (µ) and fix someα ˜ <αand >0. Then log µ(B(ω,r)) Ω˜ = ω :liminf > α˜ r 0+ { → log r } satisfies µ(Ω˜) > 0. By Egoroff’s theorem there exists Ω Ω˜ so that µ(Ω˜ Ω) <and ⊂ \ log µ(B(ω,r)) lim inf α˜ uniformly on Ω. r 0+ → log r ≥ () 1 Let µ = µ Ω and ν = µ Π− . By definition, α˜ (µ ) < , and thus by Theorem 2.8, λ ◦ λ E − ∞ () 1 dim λ J :dim(ν ) σ 1+σ (˜α )(1 + a β)− . { ∈ s λ ≤ }≤ − − 0 In particular, () 1 (2.13) dim λ J : ν is singular 2 (˜α )(1 + a β)− . { ∈ λ }≤ − − 0 Since j (2− ) λ J : νλ is singular lim sup λ J : νλ is singular { ∈ }⊂ j { ∈ } →∞ (2.11) follows by lettingα ˜ α and 0+ in (2.13). The proof of (2.12) is very similar and is omitted. → → 3. Exceptional parameters for convergence of power series

3.1. The C∞ case. The dimension bounds (2.4) in case α 1 and (2.9) are analogous to Kaufman’s result for Orthogonal projections, and involve Frostman measures as≤ in his original proof. However, if α>1, estimates (2.4) and (2.8) are derived from the smoothness bounds by means of the following lemma, which we state for n n parameters in R for an arbitrary n 1. Recall that the length of a multi-index η =(η1,η2,...,ηn) N is ≥ η ∈ η ∂| | deﬁned to be η = η1 + ...+ ηn and that ∂ = η η where λ =(λ1,...,λn). For applications to | | (∂λ1) 1 ... (∂λn) n Bernoulli convolutions it suﬃces to read the following· lemma· with n = 1. SMOOTHNESS OF PROJECTIONS 9

n Lemma 3.1. Let Q R be a nonempty open set. Suppose hj ∞ C∞(Q) satisfy ⊂ { }0 ∈ j η η n j (3.1) sup A− | | ∂ hj Cη for all multi–indices η N and sup R hj(λ) dλ C < , j 0 ∞ ≤ ∈ j 0 Q | | ≤ ∗ ∞ ≥ ≥ where A>1.Supposethat R>r 1. n R ≥j ((i)) If A < , then ∞ r h (λ) < for all λ Q. r j=0 | j | ∞ ∈ ((ii)) If Aα R An, then ≤ r ≤ ∞ (3.2) dim λ Q : rj h (λ) = n α. ∈ | j | ∞ ≤ − j=0 n R Proof. Fix some 0 2 r− .Then j ∈ | j | j

∞ j (3.3) λ Q : r hj(λ) = lim sup Ej. ∈ | | ∞ ⊂ j j=0 →∞ We will estimate the (n α)–Hausdorﬀmeasure of lim sup Ej by covering each Ej with cubes of side length j − j A− . The idea is that any point in Ej has a neighborhood of size A− on which the average of hj is at 1 j | | least C j2 r− . More precisely, for any positive integer N, N N i N η j N (3.4) ( 1) hj(λ + iy) y sup ∂ hj CN y A . i − ≤| | η =N ∞ ≤ | | i=0 | | For any j =0, 1,... and λ0 Q let ∈ Ij,N (λ0)= hj(λ0 + λ) dλ [ NL ,NL ]n | | − j,N j,N where Lj,N > 0 will be determined below. In view of (3.4), with some dimensional constant bn, N bn CN N+n jN N i (3.5) Lj,N A ( 1) hj(λ0 + iy) dy N + n ≥ [ L ,L ]n i − − j,N j,N i=0 N N 1 (2L )n h (λ ) I (λ ). ≥ j,N | j 0 |− i in j,N 0 i=1 j 2 j 1 In particular, setting Lj,N = A− (CN j r )− N with a suitable constant CN , one has

1 2 j 1 n n N (j r )− (2L ) (2L ) h (λ ) 2 I (λ ) 2 j,N ≥ j,N | j 0 |− j,N 0 and therefore, for any λ E , 0 ∈ j N 2 j n (3.6) I (λ ) 2− j− r− (2L ) , j,N 0 ≥ j,N if j is suﬃciently large (depending on dist(Q,∂Q)). Fix some positive integer N and let Uij : i = 1,...,M be a covering of E with disjoint cubes of side length 2NL . Pick any λ { U E . j,N } j j,N ij ∈ ij ∩ j Setting λ0 = λij in (3.6) and summing over i =1,...,Mj,N yields

N 2 j n j Mj,N 2− j− r− (2Lj,N ) 2n hj(λ) dλ 2nC R− . ≤ | | ≤ ∗ Q Therefore, by the deﬁnition of Lj,N , 2(1+ n ) n 1+ n j (3.7) M C˜ j N A r N /R . j,N ≤ N 10 YUVAL PERES AND WILHELM SCHLAG

α α R Let α (0, 1) satisfy A r N < . In view of (3.7), ∈ r n α ∞ n α − (lim sup Ej) CÑ lim Mj,N (2NLj,N ) − H ≤ k →∞ j=k α 1+ j ∞ r N α 2(1+ α ) = CÑ lim A j N =0. k R →∞ j=k Thus (3.2) follows from (3.3) by letting N . →∞ n n R N n R To prove assertion (i) assume that A < r and let N be a fixed positive integer such that r A < r . Let λ E for some sufficiently large j. It follows from (3.6) that 0 ∈ j n R j j ˜ 2(1+ N ) C R Ij,N (λ0) CN j− 1+ n , ∗ ≥ ≥ Ar N which is a contradiction for large j.Thuslimsupj Ej = and (i) follows from (3.3). →∞ ∅ n L 3.2. The case of limited regularity. Suppose Q R and hj ∞ C (Q) for some positive integer L. ⊂ { }0 ⊂ Here CL denotes the space of functions which are L times continuously differentiable. Under the assump- α n α L R n L tion (3.1) for η L the previous proof with N = L yields (3.2) provided A r r A r .Belowwe show how to weaken| |≤ this condition on α under a slightly different assumption. For≤ a positive≤ integer L and δ [0, 1) we let CL,δ denote the space of functions which are L times continuously differentiable and such that∈ the derivatives of order L satisfy a Hölder condition of order δ (we will write C0,δ = Cδ). n Lemma 3.2. Let Q R be an open set and suppose hj 0∞ are nonnegative and uniformly bounded functions j ⊂ { }j i j satisfying supj 0 R Q hj(λ) dλ 1. ≥ ∗ ∞ ≥ 1 ((i)) Let 1 rj− r− , r h (λ) T j ∈ j 1 0 i ≤ i=0 SMOOTHNESS OF PROJECTIONS 11

for j =0, 1,.... Clearly,

∞ i ∞ i ∞ (T ) (3.12) λ Q : r1 hi(λ)= , r0 hi(λ) < lim sup Ej . ∈ ∞ ∞ ⊂ j i=0 i=0 T=1 →∞ Mj (T ) j Fix some positive integer T and let U be a covering of E with disjoint cubes of side length W

j j j 1 (3.14) hj(λ) hj(λ0) = B− B [Hj(λ) Hj(λ0)] B − [Hj 1(λ) Hj 1(λ0)] | − | − − − − − 1 A j ∞ C 1 r /B − λ λ ri h (λ ) ≤ − 0 | − 0| r 0 i 0 0 i=0 j (T ) if λ λ

j 1 j 1 n j (3.15) C R− h (λ) dλ r− dλ = M W r− j M 2 1 2 j j 1 ∗ ≥ Q ≥ j U 2j 2j i=1 ij S and thus, by choice of Wj, j jn 2(1+n) r1 n r1 Mj 2C j A . ≤ ∗ R r0 Hence

n α (T ) ∞ n α − lim sup Ej lim MjWj − H j ≤ k →∞ →∞ j=k j jn j(n α) ∞ 2(1+n) r1 n r1 n α 2(n α) r0 − j(n α) lim 2C j A C˜ − j− − A− − ≤ k ∗ R r r →∞ 0 1 j=k ∞ r j r jα = C lim j2(1+α) 1 Aα 1 =0 k R r →∞ 0 j=k α provided Aα < R r0 . r1 r1 12 YUVAL PERES AND WILHELM SCHLAG

In view of (3.12) therefore

∞ ∞ (3.16) dim λ Q : ri h (λ)= , ri h (λ) < n α ∈ 1 i ∞ 0 i ∞ ≤ − i=0 i=0 α R r0 α if A and 0 ρm 1 >...>ρ2 > 1 >ρ1.Then ≤ r1 r1 ≤ − m ∞ i ∞ i ∞ i (3.17) λ Q : r hi(λ)= = λ Q : ρk hi(λ)= , ρk 1 hi(λ) < ∈ ∞ ∈ ∞ − ∞ i=0 i=0 i=0 k=2 and (3.16) imply that ∞ dim λ Q : ri h (λ)= n α ∈ i ∞ ≤ − i=0 α R ρi 1 α ρi if A min − . Letting max 1 finishes the proof. r i=2,...,m ρi ρi 1 ≤ n − → If An < R r0 , then (3.15) with M = 1 leads to a contradiction if j is sufficiently large. Subdividing as r1 r1 j R n in (3.17) therefore shows that the left–hand side of (3.17) is empty if r >A , and the proof of (i) is complete. For the proof of (ii) let r>r = B and T (0, ), and define E(T ) as in (3.11). The definition of H 0 ∈ ∞ j k k (T ) implies that supk 0 B Hk(λ0) T for all λ0 Ej and all j. By (3.13) therefore ≥ ≤ ∈ k k (3.18) H (λ) H (λ ) CT λ λ and thus H (λ) CT B− provided λ λ

2 j (T ) j B L+δ Mj if λ E and y Cj− L+δ A− = W where C = C(L,δ,T ). Now let U be a covering 0 ∈ j | |≤ r j { ij}i=1 of E(T ) with cubes of side length 2NW and ﬁx some λ U E(T ) for all i and j. Integrating (3.19) with j j ij ∈ ij ∩ j λ = λ over [λ W , λ + W ]n yields 0 ij ij − j ij j N 1 j n n N 1 r− W h (λ )W h (λ + λ) dλ 2j2 j ≥ j ij j − i in j ij i=1 [ NWj ,NWj ] − and thus 1 N 2 j − n (3.20) hj(λij + λ) dλ 2− 2j r Wj . [ NW ,NW ] ≥ − j j j Summing (3.20) over i =1,...,Mj implies in view of our assumption C R− hj(λ) dλ that ∗ ≥ Q j 1 j 1 n j (3.21) 2nC R− 2n h (λ) dλ r− dλ = M (2NW ) r− . j M N+1 2 N+1 2 j j ∗ ≥ Q ≥ j U 2 j 2 j i=1 ij By deﬁnition of Wj, therefore S j jn 2n r nj r L+δ M CjL+δ A . j ≤ R B SMOOTHNESS OF PROJECTIONS 13

Hence α n α (T ) α r L+δ R (3.22) − lim sup Ej =0 if A < H j B r →∞ i for all T . Since for any r>B,theset λ Q : ∞ r h (λ)= is contained in ∈ i=0 i ∞ ∞ ∞ ∞ (3.23) λ Q : ri h (λ)= , Bi h (λ) < λ Q : Bi h (λ)= , ∈ i ∞ i ∞ ∪ ∈ i ∞ i=0 i=0 i=0 α α r L+δ R (3.10) follows from (3.12), (3.22), and (3.8). To apply (3.8) to (3.23) one needs to check that A B < r α R implies that A B . However, this is an immediate consequence of B

4.1. The C∞ case. To estimate Sobolev norms it will be convenient to decompose frequency space dyadically. For the sake of completeness we first recall the construction of such a Littlewood–Paley decomposition, see Stein [43] and Frazier, Jawerth, Weiss [13]. (Rn) is the Schwartz space of smooth functions all of whose derivatives decay faster than any power. It is aS basic property of the Fourier transform that it preserves . S Lemma 4.1. There exists ψ (Rm) so that ψˆ 0, ∈S ≥ ˆ m ∞ ˆ j (4.1) supp(ψ) ξ R :1 ξ 4 , and ψ(2− ξ)=1 if ξ =0. ⊂{ ∈ ≤| |≤ } j= −∞ Moreover, given any finite measure ν on Rm and any γ R ∈ 2 ∞ 2jγ j (4.2) ν 2,γ 2 (ψ2− ν)(x) dν(x) m ∗ j= R −∞ jm j j where ψ2− (x)=2 ψ(2 x). Proof. Choose φ (Rm)withφˆ 0, φˆ(ξ) = 1 for ξ 1 and φˆ(ξ) = 0 for ξ > 2. Define ψ via ∈S ≥ | |≤ | | ψˆ(ξ)=φˆ(ξ/2) φˆ(ξ). It is clear that ψˆ(ξ) 0 and that ψˆ(ξ)=0if ξ < 1 or ξ > 4. (4.1) holds since the − ≥ | | | | sum telescopes. Moreover, it is clear from (4.1) that there exists some constant Cγ depending only on γ so that for any ξ =0 1 2γ ∞ 2jγ j 2γ C− ξ 2 ψˆ(2− ξ) C ξ . γ | | ≤ ≤ γ | | j= −∞ j j ˆ Since ψ2− (ξ)=ψ(2− ξ), Plancherel’s theorem implies

j 2 j ˆ (ψ2− ν)(x) dν(x)= ψ(2− ξ) νˆ(ξ) dξ m ∗ | | R and (4.2) follows. Notice that (4.2) is closely related to (2.10). The following proposition shows how to obtain a dimension bound from a suitable Sobolev estimate. This will depend on Lemma 3.1. Proposition 4.2. Let J and ν be as in Deﬁnition 2.4. If ν 2 dλ< with some I J and γ> 1/2 λ I λ2,γ ∞ ⊂ − then for all σ [0 2γ,1+2γ) one has dim λ I :dim(ν ) σ σ 2γ. ∈ ∧ { ∈ s λ ≤ }≤ − 14 YUVAL PERES AND WILHELM SCHLAG

Proof. Let ψ be the Littlewood–Paley function from Lemma 4.1. For any j =0, 1, 2,... deﬁne

j 2 j (4.3) h (λ)=2− ψj (ξ) ν (ξ) dξ = ψ 2 Π(λ,ω ) Π(λ,ω ) dµ(ω )dµ(ω ). j 2− | λ | 1 − 2 1 2 Ω Ω Lemma 4.1 implies that ∞ (4.4) C ν 2 2j(1+2γ)h (λ). λ2,γ ≥ j j=0 Moreover, for any >0,

∞ (4.5) λ I :dim(ν ) σ λ I : (2σ+)jh (λ)= { ∈ s λ ≤ }⊂ ∈ j ∞ j=0 by (2.1) and Lemma 4.1. Since (2.2) implies that for any j, =0, 1,...

l i () d j j d j hj (λ) = ψ 2 Π(λ,ω1) Π(λ,ω2) dµ(ω1)dµ(ω2) C 2 Π C 2 , | | dλ − ≤ dλi L∞(I Ω) ≤ Ω Ω i=0 × the proposition follows from Lemma 3.1 with n = 1, A = 2, r =2σ+ , and R =22γ+1 as 0+. → Proposition 4.4 below deals with the case α 1 in Theorems 2.6 and 2.8, see (2.4) and (2.8). In that case it is more eﬃcient to rely on Frostman measures≤ than on the previous proposition. The idea of using Frostman measures appears repeatedly in geometric measure theory, see [31]. First we prove a simple technical lemma that describes the location of the zeros of Φλ.

Lemma 4.3. Let J and Π be as in Deﬁnition 2.4. Suppose J =(λ0,λ1) is an interval of transversality of order β for Π and that Πλ is 1–regular on J. Let Cβ be the constant from Deﬁnition 2.7, (2.5) and set r = d(ω1,ω2). Then

Nβ (4.6) λ J : Φ

β d β Proof. Let the intervals I be defined by (4.6). Since C r Φ C r− on each I by Definition 2.7, j β ≤|dλ λ|≤ β,1 j their lengths are as claimed. The other statements are now clear. Proposition 4.4. Let Ω, J,andΠ be as in Definition 2.4. Assume that J is an interval of transversality of order β for Π,thatΠλ is 1–regular on J,andthatthemeasureµ on Ω has finite α–energy for some α (0, 1]. 1 ∈ Then ν = µ Π− satisfies λ ◦ λ σ( λ J :dim(ν ) <σ ) = 0(4.7) H { ∈ s λ } for any σ (0,α 3β].IfJ is an interval of strong transversality then (4.7) holds for all σ (0,α]. ∈ − ∈ Proof. Suppose (4.7) fails for some σ. By outer regularity of Hausdorffmeasure there exists 0 > 0 so that σ( λ J :dim(ν ) σ ) > 0. H { ∈ s λ ≤ − 0} By Frostman’s lemma there exists a nonzero measure ρ on J so that ρ(I) I σ for all intervals I and ≤| | (4.8) ρ( λ J :dim(ν ) >σ )=0. { ∈ s λ − 0} SMOOTHNESS OF PROJECTIONS 15

Frostman’s lemma can be applied since λ J :dim(ν ) >κ is an –set for any κ>0. Indeed, { ∈ s λ } Fσ ∞ ∞ 2 κ+δ 1 (4.9) λ J :dim(ν ) >κ = λ J : ν (ξ) ξ − dξ M . { ∈ s λ } { ∈ | λ | | | ≤ } δ>0 M=1 −∞ Since

iξΠλ(ω) iξΠλ0 (ω) νλ(ξ) νλ0 (ξ) e e dµ(ω) ξ Πλ(ω) Πλ0 (ω) dµ(ω) C ξ λ λ0 , | − |≤ Ω − ≤| | Ω | − | ≤ | || − | the sets on the right–hand side of (4.9) are closed, as claimed. Thus for small >0 and with r = d(ω ,ω ) 1 2 ∞ ∞ dν (x)dν (y) dµ(ω )dµ(ω ) λ λ dρ(λ)= 1 2 dρ(λ) σ σ J x y − J Ω Ω Π(λ,ω1) Π(λ,ω2) − −∞ −∞ | − | | − | (σ ) dµ(ω1)dµ(ω2) = χ[ Φ C rβ ] Φλ(ω1,ω2) − − dρ(λ) | λ|≥ β | | d(ω ,ω )σ Ω Ω J 1 2 − (σ ) dµ(ω1)dµ(ω2) (4.10) + χ[ Φ C rβ ] Φλ(ω1,ω2) − − dρ(λ) | λ|≤ β | | d(ω ,ω )σ Ω Ω J 1 2 − N dµ(ω )dµ(ω ) β dρ(λ) dµ(ω )dµ(ω ) C 1 2 + C 1 2 ≤ d(ω ,ω )(σ )(1+β) σ d(ω ,ω )(σ )(1+β) Ω Ω 1 2 − Ω Ω j=1 λ λj − 1 2 − | − | dµ(ω )dµ(ω ) (4.11) C 1 2 , ≤ d(ω ,ω )α Ω Ω 1 2 since σ +3β α. The sum is obtained by applying Lemma 4.3 to (4.10). However, (4.11) contradicts (4.8) if ≤ <0. The easier case of strong transversality is implicit in the above and is omitted.

Proof of Theorem 2.6. Fix some compact I J and let ρ C∞(R) be nonnegative so that ρ = 1 on I and supp(ρ) J.Fixdistinctω ,ω Ωand let⊂r = d(ω ,ω ).∈ Then for any nonnegative integer N ⊂ 1 2 ∈ 1 2 ∞ iξ[Π(λ,ω ) Π(λ,ω )] ∞ iξrΦ (ω ,ω ) ∞ iξrΦ (ω ,ω ) N (4.12) e 1 − 2 ρ(λ) dλ = e λ 1 2 ρ(λ) dλ = e λ 1 2 L (ρ)(λ) dλ −∞ −∞ −∞ d 1 where L is the differential operator L( )= dλ (( iξrΦλ (ω1,ω2))− ). It follows from Definition 2.5 that k k k (i) · − · L (f) Ck ξr − f for all k. In particular, applying (4.12) with N = 0 and N α yields | |≤ | | i=0 ∞ ≥ ∞ iξ[Π(λ,ω ) Π(λ,ω )] α e 1 − 2 ρ(λ) dλ C ( ξ d(ω ,ω ))− . ≤ α | | 1 2 −∞ Therefore ∞ ∞ 2 iξ[Π(λ,ω1) Π(λ,ω2)] α dµ(ω1)dµ(ω2) νλ(ξ) ρ(λ) dλ = e − ρ(λ) dλdµ(ω1)dµ(ω2) C ξ − α . | | Ω Ω ≤ | | Ω Ω d(ω1,ω2) −∞ −∞ This proves (2.3) and also that ν 2 dλ< for any small >0. The dimension bound (2.4) in I λ2,(α 1)/2 ∞ case α 1 thus follows from Proposition 4.2,− whereas− (2.4) with α 1 is covered by Proposition 4.4. ≥ ≤ In view of Propositions 4.2 and 4.4, Theorem 2.8 will follow from the Sobolev estimate (2.7). This will be proved using the characterization (4.2). The main technical statement is Lemma 4.6 below. The following routine calculus lemma will be needed in the proof of that lemma. L Lemma 4.5. Let L be a positive integer and suppose h C (I) satisfies h =0on some interval I R. Let H denote the inverse of h. Then for any positive integer∈ L ⊂ ≤ 1 () − (+p +...+p ) ( ) p ( ) p (4.13) H = a (h H)− 1 t (h 1 H) 1 ... (h t H) t ,p ◦ ◦ · · ◦ t=0 ,p 16 YUVAL PERES AND WILHELM SCHLAG

with some integer coeﬃcients a,p . The inner sum runs over vectors =(1,...,t) and p =(p1,...,pt) with integer coordinates. Moreover, if 2 one has a =0unless 2 for all i, min p 1,and ≥ ,p ≥ i ≥ j ≥ p + ...+ p 2( 1). 1 1 t t ≤ − Proof. If = 1 then (4.13) holds with t = 0. The general case now follows easily by induction.

Lemma 4.6. Let J and Π be as in Deﬁnition 2.4. Assume that J is an interval of transversality of order β for Π and that Πλ is –regular on J.Supposeρ C∞(R) is supported on J. Let ψ be the Littlewood–Paley function from Lemma∞ 4.1. Then for any distinct ω∈ ,ω Ω, any integer j, and any positive integer q, 1 2 ∈

∞ j j 1+a β q (4.14) ρ(λ)ψ 2 [Π(λ,ω ) Π(λ,ω )] dλ C (1 + 2 d(ω ,ω ) 0 )− 1 − 2 ≤ q 1 2 −∞ where Cq depends only on q, ρ,andβ and a0 is some absolute constant. j Proof. Fix distinct ω1,ω2 Ωand j, q as above. We may assume that 2 r>1wherer = d(ω1,ω2). Let ∈ 1 1 φ C∞ be nonnegative with φ = 1 on [ , ] and supp(φ) [ 1, 1]. Then ∈ − 2 2 ⊂ −

∞ ρ(λ)ψ 2j[Π(λ,ω ) Π(λ,ω )] dλ 1 − 2 −∞ j 1 β j 1 β (4.15) = ρ(λ)ψ(2 rΦ (ω ,ω ))φ(C− r− Φ ) dλ + ρ(λ)ψ(2 rΦ (ω ,ω ))[1 φ(C− r− Φ )] dλ. λ 1 2 β λ λ 1 2 − β λ Here Cβ is the constant from Deﬁnition 2.7. By the rapid decay of ψ

j 1 β j 1+β q j 1+β q ρ(λ)ψ 2 rΦ (ω ,ω ) [1 φ(C− r− Φ )] dλ C ρ(λ) (1 + 2 r )− dλ C (1 + 2 r )− . λ 1 2 − β λ ≤ q,β | | ≤ q,β Thus it suﬃces to estimate the ﬁrst integral in (4.15). By Lemma 4.3 there exists χ C∞(R) depending only on β so that supp(χ) is compact, χ = 1 on a neighborhood of the origin, and such that∈

2β 2β 1 β χ r− (λ λ ) = χ r− (λ λ ) φ(C− r− Φ ), − j − j β λ where these functions have disjoint supports for distinct j. We can therefore write the ﬁrst integral in (4.15) as follows.

Nβ j 1 β 2β j ρ(λ)ψ(2 rΦ )φ(C− r− Φ ) dλ = ρ(λ)χ r− (λ λ ) ψ(2 rΦ ) dλ λ β λ − j λ i=1 Nβ Nβ 2β j 1 β (j) + ρ(λ) 1 χ r− (λ λ ) ψ(2 rΦ )φ(C− r− Φ ) dλ = A + B. − − j λ β λ i=1 i=1 3β To estimate the second term B notice that Φλ Cr on the support of the integrand. The rapid decay of ψ therefore implies | |≥ j 1+3β q B C (1 + 2 r )− . | |≤ q,β For simplicity, we assume henceforth that i = 1 and set λ = λ . By choice of χ one has d Φ C rβ on 1 | dλ λ|≥ β the support of the integrand of A(1). In order to change variables in A(1) we deﬁne H via

2β (4.16) Φ = u λ = λ + H(u)providedχ r− (λ λ) =0. λ ⇐⇒ − SMOOTHNESS OF PROJECTIONS 17

2β Let F (u)=ρ(λ + H(u))χ(r− H(u))H(u). Thus

A(1) = F (u)ψ(2jru) du 2(q 1) − F ()(0) (4.17) = ψ(2jru) u + O(F (2q 1)(u)u2q 1) du 1 − − u <(2j r)− 2  !  | |  =0  + O (2jr u ) 2q 1 F (u) du = A(1) + A(1). 1  − − 1 2  u >(2j r)− 2 | | | | | | β β (1) j q 1 β H(u) Cβ r− by the choice of χ and thus F (u) Cr− . In particular, A2 Cβ,q(2 r)− − r− .Since ψ| has vanishing|≤ moments of all orders | |≤ | |≤

2(q 1) − F ()(0) A(1) = ψ(2jru) u du + O F (2q 1) (2jr) q . 1 1 − − − u >(2j r)− 2 ! ∞ | | =0 It remains to estimate F ()(u). Suppose λ and u are related via (4.16). By Lemma 4.5 (with L = ) and (2.6) in Deﬁnition 2.7, ∞

1 () − (+p +...+p ) β( p +...+ p ) β(4 3) H (u) C a Φ − 1 t r− 1 1 t t C r− − . | |≤ β, | ,p || λ| ≤ β, t=0 ,p Therefore, by Leibnitz’s rule,

() 2β β(4 3) β(4l+1) 6β F Cβ, r− r− − + r− Cβ, r− . ∞ ≤ ≤ Thus

2(q 1) − ∞ A(1) C r 6β (2jru) 2q 1u du + O r 6(2q 1)β(2jr) q C r 6(2q 1)β(2jr) q. 1 β,q − 1 − − − − − − β,q − − − | |≤ j ≤ =0 (2 r)− 2 2β Since N C r− , one ﬁnally obtains (4.14) with a = 12. β ≤ β 0

Proof of (2.7). Fix some γ with 0 < (1 + 2γ)(1 + a0β) α. Let ρ be a smooth nonnegative function on the line so that supp(ρ) J. Fix any q>1+2γ. In view of≤ (4.2), the deﬁnition of ν , and Lemma 4.6 ⊂ λ

∞ 2 ∞ 2jγ ∞ ν ρ(λ) dλ 2 (ψ j ν )(x) dν (x) ρ(λ) dλ λ2,γ 2− ∗ λ λ j= −∞ −∞ −∞ ∞ 2j(1+2γ) ψ 2j[Π(λ,ω ) Π(λ,ω )] ρ(λ) dλ dµ(ω )dµ(ω ) ≤ 1 − 2 1 2 Ω Ω j= −∞ ∞ j(1+2γ) j 1+a0β q (4.18) Cβ,γ 2 (1 + 2 d(ω1,ω2) )− dµ(ω1)dµ(ω2) ≤ Ω Ω −∞ dµ(ω1)dµ(ω2) Cβ,γ Cβ,γ α(µ) < , ≤ d(ω ,ω )(1+a0β)(1+2γ) ≤ E ∞ Ω Ω 1 2 as claimed. 18 YUVAL PERES AND WILHELM SCHLAG

4.2. The case of limited regularity. We now discuss the case where Π: J Ω R only has a finite degree of smoothness as a mapping λ Π . × → → λ Definition 4.7. Let Ω, J be as in Definition 2.4. Suppose Π: J Ω R is continuous and let L be a L,δ × → positive integer and δ [0, 1). We write Πλ C (J) if the following conditions are satisfied: Given any compact I J we assume∈ that the bounds (2.2)∈ hold for all =0, 1,...,L and that ⊂ dL dL (4.19) Π(λ ,ω) Π(λ ,ω) C λ λ δ for all λ ,λ I and ω Ω dλL 1 − dλL 2 ≤ δ,I | 1 − 2| 1 2 ∈ ∈ with some suitable constant CI,δ. The following definition is very similar to Definition 2.7, only here we also allow for the slightly more general case of Hölder continuity. Definition 4.8. Suppose Π CL,δ(J) as in 4.7 and let β (0, 1). We say that J is an interval of λ ∈ ∈ transversality of order β for Πif there exists a constant Cβ so that for all λ1,λ2 J and ω1,ω2 Ωthe condition ∈ ∈ Φ (ω ,ω ) + Φ (ω ,ω ) C d(ω ,ω )β | λ1 1 2 | | λ2 1 2 |≤ β 1 2 implies that d (4.20) Φ C d(ω ,ω )β. dλ λ1 ≥ β 1 2 In addition, we say that Πλ is L,δ–regular on J if under the same condition, with some constants Cβ,,Cβ,L,δ, d β (4.21) Φ (ω ,ω ) C d(ω ,ω )− for =1, 2,...,L and dλ λ1 1 2 ≤ β, 1 2 L L d d δ β(L+δ) Φλ1 (ω1,ω2) Φλ2 (ω1,ω2) Cβ,L,δ λ1 λ2 d(ω1,ω2)− . dλL − dλL ≤ | − | Under these conditions one has the following analogue of Theorem 2.8. Since Lemma 4.3 and Proposition 4.4 only require that Πλ is 1–regular on J, we restrict ourselves to a discussion of the Sobolev estimate (2.7) and the dimension bound (2.8) from Theorem 2.8. Notice that the following statements reduce to those estimates if L = . ∞ Theorem 4.9. Let Π:J Ω R be as in Definition 4.7. Assume that J is an interval of transversality of × → order β for some β [0, 1) in the sense of Definition 4.8 and that Πλ is L,δ–regular on J with L + δ>1. Let µ be a finite positive∈ measure on Ω with finite α–energy for some α>1. Then on any compact I J the 1 ⊂ family of measures ν = µ Π− satisfies λ ◦ λ α (4.22) ν 2 dλ C (µ) provided 0 < 1+2γ

As already mentioned above, the ﬁnal statement holds since Proposition 4.4 requires only C1 parameteriza- tions. As in the C∞–case, we will exploit the Sobolev bound in order to obtain the dimension estimates (4.23) and (4.24). This will be accomplished by means of the following proposition which is based on Lemma 3.2, cf. Proposition 4.2.

Proposition 4.10. Let Ω,J and Π:J Ω R be as in Deﬁnition 4.7 for some positive integer L and δ [0, 1).Supposeν as in 4.7 satisﬁes × ν →2 dλ< with some I J and γ> 1/2. ∈ λ I λ2,γ ∞ ⊂ − ((i)) If σ [0 2γ,1 (1 + 2γ)], then ∈ ∧ ∧ (4.25) dim λ I :dim(ν ) σ σ 2γ. ∈ s λ ≤ ≤ − ((ii)) If σ [1, 1+2γ), then ∈ σ 1 1 (4.26) dim λ I :dim(ν ) σ 1 (1 + 2γ σ) 1+ − − . ∈ s λ ≤ ≤ − − L + δ

Proof. Let hj j∞=1 be deﬁned as in (4.3) and set h0 = 1. We will verify the hypotheses of Lemma 3.2 with A = { j} i j 2 and H = 2 − h . Recall that the Littlewood–Paley function ψ is given by ψˆ(ξ)=φˆ(ξ/2) φˆ(ξ)where j i=0 i − φˆ 0, φˆ(ξ) = 1 for ξ 1 and φˆ(ξ) = 0 for ξ > 2; see the proof of Lemma 4.1. In particular, ≥ | |≤ | | j ∞ j 1 2 (4.27) 2 H (λ)=1+ φˆ(2− − ξ) φˆ(ξ) ν (ξ) dξ j − | λ | −∞ =1+ 2j+1φ 2j+1 Π (ω) Π (τ) φ Π (ω) Π (τ) dµ(ω) dµ(τ). λ − λ − λ − λ Ω Ω Let χ (R) satisfy φ χ (for example, set χ(x)=C ∞ max y z 1 φ(z) η(x y) dy where η C∞ is ∈S | |≤ | − |≤ | | − ∈ a standard bump function). Thus −∞ j+1 j+1 Hj (λ) C + C 2 χ 2 Πλ(ω) Πλ(τ) Πλ (ω) + Πλ (τ) dµ(ω) dµ(τ) | |≤ Ω Ω − | | | | ∞ j 1 2 j (4.28) C + C χˆ(2− − ξ) ν (ξ) dξ C 2 H (λ). ≤ | λ | ≤ j −∞ To obtain the ﬁnal inequality in (4.28) notice that

j 1 j 1 j+1 χˆ(2− − ξ) χˆ φˆ(2− − ξ)if ξ 2 . ≤ ∞ | |≤ Hence (4.28) follows from (4.27) and the rapid decay ofχ ˆ. In view of (4.4) and (4.5), inequality (4.25) follows from (3.8) in Lemma 3.2 with n = 1, A = B = 2, R =21+2γ , and r =2σ+ as 0+. Estimate (4.26) follows from Lemma 3.2 (ii) with the same choice of parameters.→ It remains to verify the Hölder estimate (3.9). For simplicity we will only carry this out for L = 1. The general case is only slightly more involved technically. Let Π (ω,τ)=Π (ω) Π (τ). H as defined above clearly satisfies λ λ − λ j j+1 j+1 (4.29) H (λ ) H (λ ) C +2 φ 2 Π (ω,τ) Π (ω,τ) Π (ω,τ) dµ(ω) dµ(τ) | j 1 − j 2 |≤ λ1 λ1 − λ2 Ω Ω j+1 j+1 j+1 (4.30) +2 φ 2 Π (ω,τ) φ 2 Π (ω,τ) Π (ω,τ) dµ(ω)dµ(τ). λ1 − λ2 λ2 Ω Ω In view of (4.19) the same argument as in the first part of the proof shows that right–hand side of (4.29) is j δ j bounded by C 2 H (λ ) λ λ .If λ λ > 2− , then the integral in (4.30) can be again bounded by j 1 | 1 − 2| | 1 − 2| C 2j H (λ )+H (λ ) C 2j(1+δ) λ λ δ H (λ )+H (λ ) . j 1 j 2 ≤ | 1 − 2| j 1 j 2 20 YUVAL PERES AND WILHELM SCHLAG

j It remains to estimate (4.30) if λ λ 2− . As above one constructs χ such that φ χ and | 1 − 2|≤ 1 ∈S | |≤ 1 Cχ1(x) max x y 1 χ1(y) for all x R. The integral in (4.30) is therefore bounded by ≥ | − |≤ ∈ 1 2j j+1 j+1 C 2 φ 2 Π (ω,τ)+2 t(Π Π )(ω,τ) dt Π Π (ω,τ) dµ(ω) dµ(τ) λ1 λ2 − λ1 λ2 − λ1 Ω Ω 0 2j j+1 j ∞ j 1 2 (4.31) 2 λ2 λ1 χ1 2 Πλ1 (ω,τ) dµ(ω) dµ(τ)=C 2 λ 2 λ1 χ1(2− − ξ) νλ(ξ) dξ ≤ | − | Ω Ω | − | | | −∞ C 22j λ λ H (λ ) C 2j(1+δ) λ λ δ H (λ ). ≤ | 2 − 1| j 1 ≤ | 2 − 1| j 1 To pass to (4.31) one uses that 2j Π (ω,τ) Π (ω,τ) C2j λ λ C and the properties of χ . | λ1 − λ2 |≤ | 2 − 1|≤ 1 Proof of Theorem 4.9. By Proposition 4.10 the dimension bounds (4.23) and (4.24) follow from (4.25) and (4.26), respectively, if (4.22) holds. (4.18) shows that this is the case as soon as the inequality (4.14) is valid for some (real) q>1+2γ. In fact, inspection of the proof of Lemma 4.6 reveals that under the conditions of Definition 4.8 one has (4.14) for any q0 and some M>0. Since F C − , as can be seen from the definition of F , Definition 4.7 and Lemma 4.5, ∈

A(1) = F (u)ψ(2jru) du L 1 () 1 L 1 j − F (0) (L 1) (L 1) (1 t) − L 1 (4.32) = ψ(2 ru) u + F − (tu) F − (0) − dt u − du j 1+ ! − (L 1)! u <(2 r)− 0 | | =0 − j M (1) (1) + O (2 r u )− F (u) du = A1 + A2 . u >(2j r) 1+ | | | | | | − (1) j q β Taking M = M(, q) large enough one obtains A2 Cβ,q(2 r)− r− for any q>0. Estimating the contribution of the Taylor polynomial in (4.32) again| involves|≤ the rapid decay of ψ. To bound the error term (L 1) in (4.32) one uses the H¨older condition on F − . 1 L 1 j (L 1) (L 1) (1 t) − L 1 ψ(2 ru) F − (tu) F − (0) − dt u − du u <(2j r) 1+ 0 − (L 1)! | | − − CLβ L 1+δ j 1+Cβ (1 )(L+δ) C ,L r− u − du C,L(2 r )− − . ≤ u <(2j r) 1+ | | ≤ | | − Since >0 is arbitrary, (4.14) holds for any q

5. Applications n 5.1. Classical Bernoulli convolutions. Recall that ν is the distribution of ∞ λ . Here we show that λ 0 ± the density of ν has fractional derivatives in L2 for almost all λ ( 1 , 1) and we estimate the dimension of λ ∈ 2 those λ so that νλ is singular with respect to Lebesgue measure. First we recall the deﬁnition of δ–transversality from [34].

Deﬁnition 5.1. Let δ>0. We say that J R is an interval of δ–transversality for the class of power series ⊂ ∞ (5.1) g(x)=1+ b xn, with b 1, 0, 1 n n ∈{− } n=1 if g(x) <δimplies g(x) < δ for any x J. − ∈ SMOOTHNESS OF PROJECTIONS 21

Lemma 5.3 establishes the connection between Definitions 2.7 and 5.1. A useful criterion for checking δ– transversality was found in [34]. A power series h(x) is called a ( )–function if for some k 1 and a [ 1, 1], ∗ ≥ k ∈ − k 1 − ∞ h(x)=1 xi + a xk + xi. − k i=1 i=k+1 In [39] Solomyak showed that among all convex combinations of series of the form (5.1), the power series with the smallest double zero must be a ( )–function. The following lemma from [34] bypasses this fact and reduces the search for intervals of transversality∗ to finding a suitable ( )–function. ∗ Lemma 5.2. Suppose that a ( )–function h satisfies ∗ h(x ) >δ and h(x ) < δ 0 0 − for some x0 (0, 1) and δ (0, 1). Then [0,x0] is an interval of δ–transversality for the class of power series (5.1) . ∈ ∈

2/3 2/3 In [34] a particular ( )–function was found that satisfies h(2− ) > 0.07 and h(2− ) < 0.09, so transver- ∗ 2/3 − sality in the sense of 5.1 holds on [0, 2− ] by this lemma. On the other hand, in [39] Solomyak proved that 2/3 there is a power series of the form (5.1) with a double zero at roughly 0.68, whereas 2− 0.63. We will return to this issue below. It is easy to see that Bernoulli convolutions are a special case of the general results in section 2. Indeed, 1 1 N ∞ let Ω= 1, +1 be equipped with the product measure µ = 0 ( 2 δ 1 + 2 δ1). For any distinct ω,τ Ω we define{− } − ∈ ω τ =min i 0:ω = τ . | ∧ | { ≥ i i} n Fix some interval J =(λ0,λ1) (0, 1) and define Π: J Ω R via Πλ(ω)= n∞=0 ωnλ . The metric on Ω ⊂ ω τ × → 1 (depending on J) is given by d(ω,τ)=λ| ∧ |. One checks that (µ) < if and only if λα > . 1 Eα ∞ 1 2 Lemma 5.3. Suppose J =(λ0,λ1) is an interval of δ–transversality. Then J is an interval of transversality 1+β of order β for Π if λ0 >λ1 .

k ω1 ω2 λ | ∧ | Proof. Since d(ω1,ω2)=λ1 ,Φλ(ω1,ω2)=2λk g(λ)withk = ω1 ω2 and a power series g of the 1 | ∧ | form (5.1). Therefore,

k k () λ − λ () 1 1 βlk Φ C k g + g C(k λ0− (1 λ1)− +(1 λ1)− − ) C,β λ1− , | λ |≤ λk ∞ λk ∞ ≤ − − ≤ 1 1 βk since λ1 < 1 and β>0. Thus condition (2.6) in 2.7 holds. To check (2.5) assume Φλ δbβλ1 where the constant b (0, 1) will be determined below. Then | |≤ β ∈ λ k λ k 2 0 g(λ) 2 g(λ) δb λβk λ | |≤ λ | |≤ β 1 1 1 1 1 1+β k implies g(λ) δb (λ− λ ) . Hence | |≤ 2 β 0 1 1 k k βk k 1 1+β k βk Φ 2 λ λ− g(λ) g(λ) 2λ δ δb (λ− λ ) δλ | λ|≥ 0 1 | |−λ | | ≥ 1 − β 2λ 0 1 ≥ 1 0 0 1+β 1 1 k 1 k − provided bβ 2 1+supk 0 2λ (λ0− λ1 ) . Thus condition (2.5) in Deﬁnition 2.7 holds with Cβ = ≤ ≥ 0 δb . β Theorem 2.8 now implies the following theorem. 22 YUVAL PERES AND WILHELM SCHLAG

1 Theorem 5.4. Suppose J =[λ0,λ0 ] ( 2 , 1) is an interval of δ–transversality for the power series (5.1). log 2 ⊂ Then dims(νλ) log λ for a.e. λ J. Furthermore, ≥ − ∈ 2 log 2 dim λ J : νλ L (R) 2 . { ∈ ∈ }≤ − log λ − 0 Proof. Fix any small β>0 and partition J into subintervals Ji =[λi,λi+1] for i =0,...,m so that λi 1+β ≥ (1 + β)λi+1 . By the previous lemma all Ji are intervals of transversality of order β for Π. Notice that the ω1 ω2 metric on Ωdepends on i, in fact di(ω1,ω2)=λ|i+1∧ |. In particular, µ has ﬁnite αi–energy with respect αi 1 to di if λi+1 > 2 . Theorem 2.8 therefore implies that

log 2 1 dim (ν ) (1 + a β)− for a.e. λ J s λ ≥ log λ 0 ∈ i − i+1 and

2 log 2 1 log 2 1 dim λ Ji : νλ L (R) 2 (1 + a0β)− 2 (1 + a0β)− . { ∈ ∈ }≤ − log λ ≤ − log λ − i+1 − 0 Letting β 0+ ﬁnishes the proof. → 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 fact, νλ is a log 2 1 − Frostman measure on that set which implies that dims(νλ)= log λ for 0 <λ< 2 . In [39] Solomyak showed that the ﬁrst double zero for a power series of the form (5.1) lies− in the interval [0.649, 0.683]. In particular, the previous theorem will apply only up to some point in this interval. It seems natural to conjecture that log 2 1 dims(νλ) log λ for a.e. λ ( 2 , 1), but our methods do not yield this estimate. Nevertheless, one can show ≥ − ∈ 1 that νλ has some smoothness for a.e. λ ( 2 , 1). This follows from Theorem 5.4 by “thinning and convolving”, see [39] and [34]. As one expects, the number∈ of derivatives tends to as λ 1. ∞ → Lemma 5.5. For any >0 there exists a γ = γ() > 0 so that

1 √ 2 2 (5.2) νλ 2,γ dλ< . 1 + ∞ 2 Furthermore, there exists some positive constant γ0 so that

2 k 1 2− − − k+1 2− 2 (5.3) νλ k + νλ 2,γ dλ< for k =1, 2,.... k 2,2 γ0 0 2 2− ∞ − 2/3 Proof. As mentioned above, [0,λ1] is an interval of transversality for the power series (5.1) for some λ1 > 2− . 1 2/3 Fix any λ0 ( 2 , 2− ]. Partitioning the interval [λ0,λ1] as in the proof of Theorem 5.4, one obtains from (2.7) that ∈ λ1 1 (5.4) ν 2 dλ< provided λ1+2γ > . λ2,γ ∞ 0 2 λ0 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. It was shown in [39] and [34] that the class of power| series (5.1) that satisfy either b = 0 for all j 0 or b for all j 0have[0,λ ] as an interval of 3j+1 ≥ 3j+2 ≥ 3 δ–transversality for some λ3 > 1/√2. As for the full series one easily deduces from Theorem 2.8 that

λ3 2 1+2γ 2/3 (5.5) ν dλ< provided λ > 2− . λ2,γ ∞ 2 λ2 SMOOTHNESS OF PROJECTIONS 23

2/3 Since νλ νλ and λ1 > 2− , (5.2) follows from (5.4) and (5.5). Moreover, we have shown that there | |≤| | 1/2 1/4 0 2 exists some (2 , 2 ) and a γ > 0 so that 2 ν dλ< . Using ν (ξ)=ν 2 (ξ)ν 2 (λξ)we 0 − − 0 λ 2,γ0 λ λ λ ∈ 0 ∞ conclude that 1 1 2 2 1 0 0 ∞ 2 2 4γ0 2 2 νλ 2,2γ0 dλ = νλ (ξ)νλ (λξ) ξ dξ dλ | | | | 0 0 −∞ 1 0 2 4 4γ0 (5.6) C νλ(ξ) ξ dξ dλ ≤ 2 | | | | 0 0 2 2γ0 (5.7) C νλ(ξ) ξ dξdλ< . ≤ 2 | | | | ∞ 0 (5.6) follows from Cauchy–Schwarz and a change of variables. To pass from (5.6) to (5.7) one basically observes that the inner integral in (5.6) behaves like a sum over ξ Z. More precisely, one has νλ = νλχ for some ∈ χ C∞(R) with compact support. Consequently, ∈ ∞ ∞ ν (ξ) 4 ξ 4γ0 dξ = ν χˆ 4(ξ) ξ 4γ0 dξ | λ | | | | λ ∗ | | | −∞ −∞ 4 +1 4 ∞ ∞ ∞ = ν (ξ η) χˆ(η) dη ξ 4γ0 dξ = ν (ξ η) χˆ(η) dη ξ 4γ0 dξ | λ − || | | | | λ − || | | | −∞ −∞ ∈Z −∞ 1 4 ∞ = ν (ξ + η) χˆ(η) dη ξ + 4γ0 dξ | λ − || | | | 0 ∈Z −∞ 2 1 2 ∞ ν (ξ + η) χˆ(η) dη ξ + 2γ0 dξ ≤ | λ − || | | | 0 ∈Z −∞ 2 1 ∞ 2 2 2γ0 χˆ 1 ν (η) χˆ(ξ + η) ξ + dη dξ ≤L | λ | | − || | 0 ∈Z −∞ 2 ∞ C ν (η) 2(1 + η )2γ0 dη < . ≤ | λ | | | ∞ −∞ Finally, since νλ(ξ) 1, | |≤ 1 1 2 2 0 0 0 2 2 2γ0 2 2γ0 2 2 νλ 2,γ0 dλ = νλ (ξ)νλ (λξ) ξ dξdλ C νλ(ξ) ξ dξdλ< . | | | | ≤ 2 | | | | ∞ 0 0 0 We have shown that (5.3) holds for k = 1. The general case now follows easily by induction along the same lines.

1 Corollary 5.6. Let S(λ0)=essinfλ [λ ,1] dims(νλ). Then S(λ0) > 1 for any λ0 > and S(λ0) as ∈ 0 2 →∞ λ 1. Furthermore, for any λ > 1 there exists (λ ) > 0 such that 0 → 0 2 0 dim λ (λ , 1) : ν does not have L2–density < 1 (λ ). { ∈ 0 λ } − 0 Proof. This follows immediately from the previous lemma and Proposition 4.2.

As observed by Kahane [22], Erd˝os’s argument yields that (λ0) 1 as λ0 1 . To conclude the discussion of classical Bernoulli convolutions we prove that (2.3) fails. → → − 24 YUVAL PERES AND WILHELM SCHLAG

Lemma 5.7. Suppose 1 is a Pisot number. Then λ0

Proof. Recall that lim sup ξ νλ0 (ξ) > 0, see [6]. Also, | |→∞ | |

iξΠλ(ω) iξΠλ0 (ω) νλ(ξ) νλ0 (ξ) e e dµ(ω) ξ Πλ(ω) Πλ0 (ω) dµ(ω) C ξ λ λ0 . | − |≤ Ω − ≤| | Ω | − | ≤ | || − | Therefore, if Cis a suﬃciently large constant, 2 lim sup ξ νλ(ξ) dλ> 0, ξ | | λ λ (C ξ ) 1 | | | |→∞ | − 0|≤ | | − as claimed. p 5.2. Asymmetric Bernoulli convolutions. In [35] Peres and Solomyak studied the distribution νλ of the sum λn where the signs are chosen independently with probabilities (p, 1 p). For p [1/3, 2/3] ± p p 1 p − q ∈ they showed that νλ is absolutely continuous for a.e. λ p (1 p) − , 1 and has L –density for a.e. 1 ∈ − q q q 1 λ (p +(1 p) ) − , 1 ,whereq (1, 2]. The results from section 2 give bounds on the dimension of∈ the exceptional− parameters for the absolutely∈ continuous case and if q = 2. As in the symmetric case, Ω= 1, 1 N. Here µ is the product measure that assigns weights p and 1 p to +1 and 1, respectively. {− } − − Theorem 5.8. Let p [1/3, 2/3].Forallλ (p2 +(1 p)2, 1] there exists = (λ ) such that ∈ 0 ∈ − 0 (5.8) dim λ (λ , 1) : νp does not have L2-density < 1 (λ ) ∈ 0 λ − 0 p 1 p and such that for all λ (p (1 p) − , 1] 0 ∈ − (5.9) dim λ (λ , 1) : νp is singular < 1 (λ ). ∈ 0 λ − 0 Proof. Since [0, 0.649] is an interval of transversality for the power series (5.1), Theorem 2.8 implies via Lemma 5.3 and a subdivision of the interval as in the proof of Theorem 5.4 that dim λ (λ , 0.649) : νp does not have L2-density < 1 ∈ 0 λ 2 2 p p for any λ0 >p +(1 p) . To extend this up to 1, we follow [35]. The measure νλ νλ is the distribution of the − n 2 ∗ 2 random series 0∞ anλ where an 2, 0, 2 with probabilities p , 2p(1 p), (1 p) . By Lemma 5.2 and an explicit choice of ( )–function Peres∈{ and− Solomyak} showed that [0, 0.5] is− an interval− of transversality for the ∗ 2 11 4 2 4 power series (5.1) with bn 4, 2, 0, 2, 4 .Since(0.649) > 27 = maxp [1/3,2/3](p +(2p(1 p)) +(1 p) ) Theorem 2.8 yields in the∈ same{− fashion− as} before that ∈ − −

p p 2 2 p 4 dim λ (0.649, 1/√2) : νν L (R) =dim λ ((0.649) , 0.5) : ν L (R) < 1. ∈ λ ∗ λ ∈ } { ∈ λ ∈ Splitting the random power series λn into odd and even indices one obtains ± p p p (5.10) νλ(ξ)=νλ2 (ξ)νλ2 (λξ). Therefore, p 2 dim λ (0.649, 1/√2) : ν L (R) < 1. ∈ λ ∈ p p p n The measure ν ν ν is the distribution of the random series ∞ a λ where a 3, 1, 1, 3 with λ ∗ λ ∗ λ 0 n n ∈{− − } probabilities p3, 3p2(1 p), 3p(1 p)2, (1 p)3. It was shown in [35] that [0, 0.415] is an interval of transversality − − − SMOOTHNESS OF PROJECTIONS 25

3/2 245 6 2 2 2 2 for the corresponding power series. Since 2− > = maxp [1/3,2/3] p +(3p (1 p)) +(3p(1 p) ) + 729 ∈ − − (1 p)6 , Theorem 2.8 again implies that − 3/2 p 6 dim λ (2− , 0.415) : ν L (R) < 1. ∈ λ ∈ Splitting the original series modulo 3 one concludes that 1 p 2 dim λ 1/√2, (0.415) 3 : ν L (R) < 1. ∈ λ ∈ 1 By this procedure we have estimated the dimension of λ inside the interval [λ , (0.415) 3 ] for which ν does 0 λ 2 2 2 5 1 5 not have L –density. Since maxp [1/3,2/3] p +(1 p) = and (0.415) 3 > , inequality (5.8) follows ∈ − 9 9 from this estimate by applying (5.10) repeatedly. By the strong law of large numbers the lower pointwise dimension of µ is

π−(ω)= (p log p +(1 p) log(1 p)) µ–a.e. µ − − − It therefore follows immediately from Corollary 2.9 that dim λ (λ , 0.649) : νp is singular < 1 ∈ 0 λ for any λ0 > exp (p log p +(1 p) log(1 p)) . Since (5.8) is a stronger assertion on the interval [0.649, 1], we finally obtain (5.9).− − − 5.3. 0, 1, 3 –problem. This problem concerns the Hausdorffdimension of the set { } ∞ Λ(λ)= a λi : a 0, 1, 3 . i i ∈{ } i=0 1 1 log 3 Pollicott and Simon [36] showed that for a.e. λ ( 4 , 3 ) one has dim Λ(λ)= log λ and Solomyak [39] 1 2 ∈ − established that for a.e. λ ( 3 , 5 )thesetΛ(λ) has positive Lebesgue measure. In this section we estimate the Hausdorffdimension of∈ the exceptional set of λ with respect to these properties. It is again clear that the general results in section 2 apply to this case. Indeed, let Ω= 0, 1, 3 N be { } equipped with uniform product measure. For any interval J =(λ0,λ1) (0, 1) define the projections Πand the metric d on Ωas in section 5.1. In [39] Solomyak showed that the⊂ smallest double zero of the class of power series ∞ 1 (5.11) g(x)=a + b λn with b 1, a n | n|≤ | |≥3 n=1 occurs in the interval [0.418, 0.437], see also Corollary 5.2 in [35]. In particular, a simple compactness argument shows that [0, 2 ] is an interval of δ–transversality for the class (5.11) for some δ>0 in the sense that g(λ) <δ 5 | | implies g(λ) >δfor any λ in that interval. As in Lemma 5.3, we establish the connection with Definition 2.7. | | Lemma 5.9. Suppose J =(λ0,λ1) is an interval of δ–transversality for the class of power series (5.11). 1+β Then J is an interval of transversality of order β for Π provided λ0 >λ1 .

1 ω1 ω2 Proof. It suﬃces to notice that Φλ(ω1,ω2) = 3(λλ1− )| ∧ |g(λ)whereg is of the form (5.11). Otherwise the proof is identical with that of Lemma 5.3. log 3 Theorem 2.8 now easily leads to the following result. Let ∆(λ)= log λ . − Theorem 5.10. For any λ (0.25, 0.4) 0 ∈ (5.12) dim λ (0.25,λ ):dimΛ(λ) < ∆(λ) ∆(λ ) and ∈ 0 ≤ 0 (5.13) dim λ (λ , 0.4) : 1(Λ(λ)) = 0 2 ∆(λ ). ∈ 0 H ≤ − 0 26 YUVAL PERES AND WILHELM SCHLAG

Proof. Suppose λ ( 1 , 1 ). Fix any β (0, 1). As in the proof of Theorem 5.4 we write [ 1 ,λ ]= m J 0 ∈ 4 3 ∈ 4 0 i=0 i with J =[λ ,λ ] and (1 + β)λ1+β λ . Applying Lemma 5.9 to each J one concludes from Theorem 2.8 i i+1 i i i+1 i that ≤ dim λ J :dim(ν ) < ∆(λ ) 3β ∆(λ ) ∆(λ ). { ∈ i s λ i − }≤ i ≤ 0 (5.12) follows by letting β 0+. If λ ( 1 , 2 ), one partitions→ [λ , 2 ] in the same fashion. Theorem 2.8 then implies 0 ∈ 3 5 0 5 ∆(λi+1) ∆(λ0) dim λ Ji : νλ is singular 2 2 . { ∈ }≤ − 1+a0β ≤ − 1+a0β (5.13) now follows again by letting β 0+. → 5.4. Sums of Cantor sets. Following [35] we consider homogeneous self–similar sets in R

∞ = s (λ)λn : s ,λ J Cλ n n ∈D ∈ n=0 1 where = s1,s2,...,sm is a set of C (J) functions and J =[λ0,λ1] (0, 1). As in [35] we assume the strongD separation{ condition} ⊂

(5.14) min min dist(si(λ)+λ λ,sj(λ)+λ λ) >h0 > 0. 1 i

L,δ Lemma 5.11. Suppose s1,s2,...,sm C (J) for some positive integer L and some δ [0, 1). Then ∈ ∈ 1+β J =[λ0,λ1] is an interval of transversality of order β for Π (in the sense of Deﬁnition 4.8) provided λ1 >λ0.

Proof. Fix some (x, ω) and (y, τ) Ω. Let k = ω τ and r = d((x, ω), (y, τ)). Notice that k>k0 by our assumption above. Then ∈ | ∧ | x y + λkg(λ) Φ ((x, ω), (y, τ)) = − λ x y + λk | − | 1 where g(λ) >h by (5.14). Now suppose | | 0 (5.15) Φ C ( x y + λk)β = C rβ | λ|≤ β | − | 1 β SMOOTHNESS OF PROJECTIONS 27

k where the constant Cβ is to be determined. In fact, Cβ can be chosen so that x y C1λ1 with some constant C that depends only on and λ . More precisely, suppose that | − |≤ 1 D 1 k k 2λ1 2λ1 x y max di dj L (J) = M0. | − |≥1 λ i=j − ∞ 1 λ − 1 − 1 Then k 1 x y λ M (1 λ )− Φ | − |− 1 0 − 1 C | λ|≥ x y + λk ≥ | − | 1 k kβ which contradicts (5.15) if Cβ is sufficiently small. Consequently, r λ1 and Φλ Cβ λ1 . It is then easy to () βk | |≤ see that Φλ Cβ, λ1− for suitable constants Cβ, and all =1, 2,...,L and also that a Hölder condition | |≤(L) of order δ on Φλ holds if δ>0. Hence (4.21) in Definition 4.8 is satisfied. Moreover, k λ k 1 k kh0 M0 + M1 kβ β Φ = g(λ)+g(λ) C(λ λ− ) Cλ C r | λ| r λ ≥ 0 1 λ − (1 λ )2 ≥ 1 ≥ β 1 1 − 1+β if Cβ is sufficiently small, since λ0 >λ1 and k>k0. Thus (4.20) also holds.

Under the strong separation assumption (5.14) Peres and Solomyak showed in [35] that dim(K + λ)= dim K +dim for a.e. λ such that dim K +dim < 1. Furthermore, they also showed that K + C has Cλ Cλ Cλ positive Lebesgue measure for a.e. λ such that dim K +dim λ > 1. The following theorem estimates the dimension of the exceptional values of λ in these statements. ThisC will be a simple consequence of Theorem 4.9.

Theorem 5.12. Suppose K R is compact and the sets λ satisfy the strong separation condition (5.14) on ⊂ C J =[λ ,λ ] (0, 1). Then 0 0 ⊂ (5.16) dim λ J :dim(K + λ) < dim K +dim λ dim K +dim λ . ∈ C C ≤ C 0 L,δ If the symbols s1,...,sm are in C with L + δ>1, then (5.17) dim λ J : 1(K + )=0 2 min(dim K +dim ,L+ δ). ∈ H Cλ ≤ − Cλ0 Proof. Fix some >0 and let ρ be a Frostman measure on K with exponent dim K . Let the measure µ N − on Ωbe define as above. Fix β (0, 1) and partition J = n=1 Ji where each Ji =[λi,λi+1] satisfies 1+β ∈ λi (1 + β)λi+1 . It is easy to see that µ has finite αi–energy with respect to the metric di((x, ω), (y, τ)) = ≥ ω τ | ∧ | x y + λi+1 on Ωif αi < dim K +dim λi+1 .Indeed,settingσi =dim λi+1 which is the same as | σi− | − C C λi+1m = 1, one obtains

dρ(x)dρ(y)dµ(ω)dµ(τ) ∞ k dρ(x)dρ(y) αi (µ)= m− ω τ α k αi E Ω Ω K K ( x y + λ| ∧ |) i ≤ K K (λi+1 + x y ) | − | i+1 k=0 | − | dρ(x)dρ(y) C < ≤ x y αi σi ∞ K K | − | − provided 0 <α σ < dim K . Therefore, by Lemma 5.11 and (2.9), i − i − dim λ J :dim(K + ) < dim K + σ 3β dim λ J :dim(ν ) < dim K + σ 3β ∈ i Cλ − i − ≤ ∈ i s λ − i − dim K +dim λ dim K +dim λ , ≤ − C i+1 ≤ − C 0 and (5.16) follows by letting β 0+ and then 0+. The second statement (5.17) follows in a similar → → fashion from 4.23. 28 YUVAL PERES AND WILHELM SCHLAG

6. Orthogonal projections onto planes in Euclidean space It is easy to see that Theorem 2.8 covers various classical results on the projection of planar sets onto lines, 2 1 see [28], [23], [24], [9], [31]. Let Ω= R and Π(θ, x)=projθ(x)=(x θ)θ, where we consider J S as · ⊂ an interval of angles. Here projθ denotes the projection onto the line tθ : t R . Note that Φθ(x, y)= { ∈ } 2 cos((θ, x y)) for any x = y so that Deﬁnition 2.7 holds with β = 0. Suppose E R is a Borel set. Applying Theorem− 2.8 to a suitable Frostman measure supported on a compact subset⊂ of E one therefore obtains Kaufman’s [24] and Falconer’s [9] theorems in the plane, i.e.,

dim θ S1 :dimproj(E) t t and dim θ S1 : 1(proj (E)) = 0 2 dim E. { ∈ θ ≤ }≤ { ∈ H θ }≤ − Kaufman and Mattila [25] proved that the first bound is optimal. As noted by Falconer [9], their example can be easily modified to show that the second bound is sharp, too. See also Falconer [11], Theorem 8.17. In particular, this implies that the estimates stated in Theorem 2.8 are optimal if β = 0, α 2, and σ 1. Moreover, it is easy to see that in this case (2.7) holds with β = 0 and equality, see also Proposition≤ 2.2. It turns≤ out that the calculation from the proof of that proposition can be generalized to higher dimensions. G(d, k) denotes the Grassmann manifold of all k–planes in Rd passing through the origin. Recall that dim G(d, k)= d k(d k), see [31]. For any finite measure µ on R we define its projections νπ onto any k–plane π through the− origin as usual, i.e., given any continuous f

fdνπ = f(projπ(x)) dµ(x). Proposition 6.1. Let d 2 and k be positive integers. Suppose µ is a compactly supported measure in Rd. Then ≥ (6.1) dim π G(d, k):dim(ν ) <σ k(d k)+σ dim (µ). { ∈ s π }≤ − − s

Proof. Let α =dims(µ). Suppose (6.1) fails. By Frostman’s lemma there exists a nonzero positive measure ρ so that for some >0

k(d k)+σ α+ (6.2) ρ(B(π, r)) r − − for any ball B(π, r) G(d, k) and ρ( π G(d, k):dim(µ ) σ )=0. ≤ ⊂ { ∈ s π ≥ } The measurability hypothesis in Frostman’s lemma can be verified as in Proposition 4.4 and we skip the details. We may assume that supp(µ) B(0, 1). Fix a φ so that φ = 1 on B(0, 1). Hence µ = φµ and ˆ ⊂ ∈S 2 ˆ ˆ 2 thusµ ˆ = φ µˆ. Clearly, νπ(ξ)=ˆµ(projπ(ξ)) and supp(νπ)=projπ(supp(µ)). Also, µˆ φ L1 φ µˆ by Cauchy–Schwarz.∗ Therefore, | | ≤ | |∗| | 2 σ k k 2 σ k k (6.3) ν (ξ) (1 + ξ ) − d (ξ) dρ(π) C ( φˆ µˆ )(ξ)(1 + ξ ) − d (ξ) dρ(π) | π | | | H ≤ | |∗| | | | H G(d,k) π G(d,k) π 2 N σ k k (6.4) CN µˆ(η) (1 + ξ η )− (1 + ξ ) − d (ξ) dρ(π) dη ≤ d | | | − | | | H R G(d,k) π 2 α d (6.5) Cα,d µˆ(η) (1 + η ) − − dη< . ≤ d | | | | ∞ R Here N is a sufficiently large integer. To pass from (6.4) to (6.5) first notice that for any r< η the set 1 | | π G(d, k):dist(π,η) r is a (r η − )–neighborhood of a smooth manifold of codimension d k { ∈ ≤ } | | 1 (k 1)(d k) − 1 in G(d, k) and is therefore contained in the union of no more than ( η r− ) − − balls of radius r η − . Thus, by (6.2) | | | | 1 k σ+α d ρ( π G(d, k):dist(π,η) r ) C( η r− ) − − − . { ∈ ≤ } ≤ | | SMOOTHNESS OF PROJECTIONS 29

Considering the contributions from the dyadic shells ξ π :2j ξ η 2j+1 separately, the inner integrals in (6.4) can therefore be estimated as follows.{ ∈ ≤| − |≤ }

N σ k k σ k (1 + ξ η )− (1 + ξ ) − d (ξ) dρ(π) C(1 + η ) − ρ( π G(d, k):dist(π,η) 1 ) | − | | | H ≤ | | { ∈ ≤ } G(d,k) π j(N k) σ k j +C 2− − (1 + η ) − ρ( π G(d, k):dist(π,η) 2 ) | | { ∈ ≤ } 1 2j < 1 η ≤ 2 | | jN σ k k α d +C 2− (1 + ξ ) − d (ξ) C(1 + η ) − − , ξ 2j+1 | | H ≤ | | 2j 1 η | |≤ ≥ 2 | | if N is suﬃciently large, as claimed. However, the ﬁniteness of (6.3) contradicts (6.2) and the proposition follows. This proposition has the following simple corollary.

Corollary 6.2. Suppose E Rd is a Borel set with dim E>2k. Then for a.e. π G(d, k) the projection of E onto π has nonempty interior.⊂ More precisely, ∈ dim π G(d, k):proj(E) has empty interior k(d k)+2k dim E. { ∈ π }≤ − −

Proof. Let µ be a Frostman probability measure on E,i.e.,µ is supported on a compact subset of E and dims(µ)=dimE.Ifνπ =projπ(µ) has a continuous density its support has nonempty interior and therefore ˆ 2 k so does projπ(E). Applying Cauchy–Schwarz to k f(ξ) dξ shows that for any f L (R ) R | | ∈ k f 2, 1 (k+)< = f C(R ), 2 ∞ ⇒ ∈ see Stein [42], chapter 5, for more precise estimates. The corollary therefore follows by letting σ 2k+ → in (6.1). The following example, described to us by Pertti Mattila, shows that there are Borel sets E of dimension two 3 2 2 in R , so that dim θ S :projθ(E) contains an interval 1. Take a Besicovitch set A in R ,i.e.,aset { ∈ }≤ 2 of measure zero that contains a line in every direction, see [31], Theorem 18.11. Deﬁne E = R r Q2 (r + A) 3 \∪ ∈ as a subset of the (x1,x2)–coordinate plane of R . It clearly has dimension two but its projections onto lines which do not lie in the (x1,x2)–plane do not contain intervals. If k>1 we do not know of an example of a 2k–dimensional set E with the property that γ ( π G(d, k):proj(E) has empty interior ) > 0where d,k { ∈ π } γd,k denotes Haar measure on G(d, k). It seems unlikely that the Sobolev embedding theorem would give the sharp bound for k>1. On the other hand, it is easy to see that Corollary 6.2 allows one to recover Falconer’s theorem [8] about the nonexistence of Besicovitch (d, k)–sets for k>d/2. Recall that a (d, k)–set is deﬁned to be a set of measure zero that contains some translate of every k–plane. Suppose that A Rd is such a set for some choice of d 3 d ⊂ ≥ and k.ThenE = R r Qd (A + r) has dimension d but the projection of E onto any (d k)–dimensional plane has empty interior.\∪ ∈ This contradicts Corollary 6.2 if d>2(d k), as claimed. However,− Bourgain [3] − k 1 proved a much stronger result, namely that (d, k)–sets do not exist for k +2 − d. It seems therefore that the case k>1 in Corollary 6.2 is not optimal. ≥

7. The general projection theorems in higher dimensions In this section we obtain a higher–dimensional version of Theorem 4.9. The proof will closely resemble that of Theorem 4.9 in section 4. In particular, we will need to prove higher–dimensional versions of various technical lemmas. We ﬁrst introduce some terminology. 30 YUVAL PERES AND WILHELM SCHLAG

Deﬁnition 7.1. Let (Ω,d) be a compact metric space, Q Rn an open connected set, and Π: Q Ω Rm ⊂ n × → a continuous map with n m. The length of a multi-index η =(η1,η2,...,ηn) N is deﬁned to be ≥ η ∈ η ∂| | η = η + ... + η and ∂ = η η where λ =(λ ,...,λ ). Let L be a positive integer and 1 n (∂λ1) 1 ... (∂λn) n 1 n | | · · n δ [0, 1). We assume that for any compact Q Q and any multi-index η =(η1,η2,...,ηn) N there exist ∈ ⊂ ∈ constants Cη,Q ,Cδ,Q such that

η η η δ ∂ Π(λ1,ω) Cη,Q provided η L and sup ∂ Π(λ1,ω) ∂ Π(λ2,ω) Cδ,Q λ1 λ2 ≤ | |≤ η =L − ≤ | − | | | L,δ for all λ1,λ2 Q and ω Ω. We denote these conditions by Πλ C (Q). Given any finite measure µ on Ω ∈ 1 ∈ ∈ dµ(ω1)dµ(ω2) let νλ = µ Π− for all λ Q,whereΠλ( )=Π(λ, ). The α–energy of µ is α(µ)= α . ◦ λ ∈ · · E Ω Ω d(ω1,ω2) As in the previous sections we will need a transversality condition. Definition 7.2. Let Π CL,δ be as in Definition 7.1 for some positive integer L and some δ [0, 1). Let λ ∈ ∈ Π(λ,ω1) Π(λ,ω2) Φλ(ω1,ω2)= − . d(ω1,ω2)

For any β [0, 1) we say that Q is a region of transversality of order β for Πif there exists a constant Cβ ∈ β so that for all λ1, λ2 Q and distinct ω1,ω2 Ωthe condition Φλ1 (ω1,ω2) + Φλ2 (ω1,ω2) Cβ d(ω1,ω2) implies that ∈ ∈ | | | |≤ (7.1) det[DΦ (DΦ )t] C2 d(ω ,ω )2β. λ1 λ1 ≥ β 1 2 In addition, we say that Πλ is L,δ–regular on Q if under the same condition, for some constants Cβ,η,Cβ,L,δ, η β η (7.2) ∂ Φ (ω ,ω ) C d(ω ,ω )− | | for any nonzero multi-index η of length η L | λ1 1 2 |≤ β,η 1 2 | |≤ η η δ β(L+δ) sup ∂ Φλ1 (ω1,ω2) ∂ Φλ2 (ω1,ω2) Cβ,L,δ λ1 λ2 d(ω1,ω2)− . η =L − ≤ | − | | | DΦλ above denotes the Jacobi matrix of all first derivatives of Φλ with respect to λ (the number of rows of DΦλ is m). Our main result in this section is the following theorem. L,δ n Theorem 7.3. Let Πλ C be as in Definition 7.1 with L + δ>1.AssumethatQ R is a region of ∈ ⊂ transversality of order β for Π and that Πλ is L,δ–regular on Q in the sense of Definition 7.2. Suppose µ is 1 a finite positive measure on Ω with finite α–energy for some α>0 and let νλ = µ Πλ− be the projection of m ◦ µ onto R under Π for any λ Q. Then for any compact Q Q ∈ ⊂ (7.3) ν 2 dλ C (µ) provided 0 < (m +2γ)(1 + a β) α and 2γ

It is easy to check that projections onto planes in Rd satisfy Definition 7.2 with Q G(d, k) a coordinate ⊂ chart, Πλ the Euclidean projection from Ω= supp(µ) onto the plane given by λ, and L = , β = 0. Hence Proposition 6.1 follows from Theorem 7.3 with n = k(d k) and m = k.Ifβ = 0 and L = ∞, Proposition 2.2 shows that (7.3) is sharp. − ∞ Remark 7.4. (7.3) implies the following statement from the introduction: Let µ be a Borel probability measure on Rd with correlation dimension greater than m +2γ. Then for a prevalent set of C1 maps f : Rd Rm the image of µ under f has a density with at least γ fractional → derivatives in L2(Rm). 1 d m 1 2,γ m Let A = f C (R , R ):µ f − has a density in L (R ) and let be Lebesgue measure on the d ∈m ◦ d L m matrices M × (R)withentriesin[ 1/2, 1/2] (we identify linear maps R R with their matrices). We − → need to check that (f + A) = 0 for any f C1(Rd, Rm). Fix some f C1(Rd, Rm). To apply Theorem 7.3, L ∈ dm ∈ d m we may assume that supp(µ)=Ωis compact. Let Q =[ 1, 1] M × (R) and Πλ(ω)= f(ω)+λω. Since (µ) < by assumption, one has − ⊂ − Em+2γ ∞ ν 2 d (λ) < λ2,γ L ∞ as desired, provided transversality holds with β = 0 and regularity with L = . Regularity is obvious. To check transversality, notice that ∞ ω τ D Φ (ω,τ) λ = λ − λ λ · 0 0 · ω τ d m | − | for all λ0 M × (R). In particular, the linear map λ0 DλΦλ(ω,τ) λ0 is surjective and therefore ∈ → · rank DλΦλ(ω,τ) = m. In fact, estimate (7.1) holds with β = 0. The following proposition is a higher–dimensional analogue of Proposition 4.10. L,δ Proposition 7.5. Let Πλ C (Q) be as in Definition 7.1 for some positive integer L and δ [0, 1).Suppose ∈ m 2 ∈ the family of measure νλ λ Q on R as given by 7.1 satisfies νλ 2,γ dλ < with some Q Q and { } ∈ Q ∞ ⊂ γ> m/2. − ((i)) If σ [0 2γ,m (m +2γ)], then ∈ ∧ ∧ (7.7) dim λ Q :dim(ν ) σ n + σ (m +2γ). ∈ s λ ≤ ≤ − ((ii)) If σ [m, m +2γ), then ∈ 1 σ m − (7.8) dim λ Q :dim(ν ) σ n (m +2γ σ) 1+ − . ∈ s λ ≤ ≤ − − L + δ Proof. Let ψ be the Littlewood–Paley function from Lemma 4.1. For any j =1, 2,... we define

jm 2 j j hj(λ)=2− ψ2− (ξ) νλ(ξ) dξ = ψ 2 Π(λ,ω1) Π(λ,ω2) dµ(ω1)dµ(ω2), | | Ω Ω − j m(j i) whereas h0 = 1. Let Hj = i=0 2 − hi. As in the proof of Proposition 4.10 one checks that these functions satisfy the hypotheses of Lemma 3.2 with A = 2 and B =2m. In fact, up to replacing 2 with 2m in various places the argument remains the same and we omit the details. In view of (4.4) and (4.5) the proposition now follows from Lemma 3.1 with r =2σ+ and R =22γ+m as 0+. → Roughly speaking, transversality means that Φλ can be inverted where it is small. However, one needs to interpret this statement more carefully in the higher dimensional case. Firstly, we are dealing with maps from Rn Rm with n m. Secondly, even if m = n one cannot expect to break up a region of transversality → ≥ into disjoint cubes on which either Φλ is large or invertible as in the one–dimensional case, see Lemma 4.3. And thirdly, any decomposition of Q has to provide estimates involving the parameters in Deﬁnition 7.2. The following quantitative version of the inverse function theorem will be used for that purpose. It is undoubtedly well–known, but we provide a proof for the sake of completeness. 32 YUVAL PERES AND WILHELM SCHLAG

m m 1 1 Lemma 7.6. Suppose f : U R R is C and that Df(x) I 2 on B(x0,r) U for some r>0. Then f : B(x ,r/3) f(B(x⊂,r/3))→is a diﬀeomorphism and f(B(−x ,≤ρ)) B(f(x ),ρ/2)⊂ for any 0 <ρ r. 0 → 0 0 ⊃ 0 ≤

Proof. W.l.o.g. x = f(x ) = 0. Define T (x)=y f(x)+x and fix some ρ (0,r]. We claim that 0 0 y − ∈ T : B(0,ρ) B(0,ρ) is a contraction provided y ρ/2. Indeed, if x, x B(0,ρ), y → | |≤ ∈ 1 1 T (x) ρ + I Df(sx) ds x ρ and | y |≤2 − | |≤ 0 1 1 T (x) T (x) = x x (f(x) f(x)) I Df(x + s(x x)) ds x x x x . | y − y | | − − − |≤ − − | − |≤2| − | 0 Hence, for any y B(0,ρ/2) there is a unique x B(0,ρ) so that f(x)=y. In particular, f is one-to-one on B(0,ρ)iff(B(0,ρ∈)) B(0,r/2). Since ∈ ⊂ 1 3 f(x) Df(sx) ds x x , | |≤ | |≤2| | 0 it suffices to take ρ = r/3, as claimed.

The following lemma is a precise statement to the effect that Φλ is invertible where it is small. This will be true only locally. However, one can cover the set of small values of Φ by a collection of balls B whose λ { j} size and number can be controlled and on each of which Φλ can be inverted in the following sense: there exist m coordinate directions (depending on Bj) so that the restriction of Φλ to the intersection of Bj with any hyperplane in those directions is invertible. Moreover, one has uniform bounds on the derivatives of the inverse. 1,δ n Lemma 7.7. Let Πλ C (Q) for some 0 <δ<1.SupposethatQ R is a region of transversality of order β for Π and that∈Π is 1,δ–regular on Q, see Definition 7.2. Let⊂ U Q be open and bounded with λ ⊂ dist(U,∂Q) > 0. Then there exist constants C0,C1,C2 depending only on β, n, m, U, Q so that for any distinct ω ,ω Ω there exist λ ,...,λ Q with the following properties: 1 2 ∈ 1 N ∈ Writing r = d(ω1,ω2) for simplicity, one has N (7.9) λ U : Φ C rb0β B(λ ,C rb0β) { ∈ | λ|≤ 0 }⊂ j 1 j=1 N (7.10) B(λ , 2C rb0β) λ Q : Φ C rβ , j 1 ⊂{∈ | λ|≤ β } j=1 1 b0n where b0 =(2+m)δ− and N C2r− . Cβ is the constant from Definition 7.2. Moreover, on each ball b0β ≤ Bj = B(λj, 2C1r ) one can select n m coordinate directions 1 i1 <...

Fy =Φλ λ Bj : λi1 = y1,...,λin m = yn m { ∈ − − } 1 β is a diﬀeomorphism satisfying det(DF )− C r− and | y |≤ 2 1 mβ (7.11) (DF )− C r− . y ≤ 2

Proof. Fix distinct ω ,ω Ω. By Deﬁnition 7.2 one can choose C and C so that 1 2 ∈ 0 3 E = λ U : Φ C r(2+m)β/δ satisﬁes { ∈ | λ|≤ 0 } (2+m)β/δ β (7.12) E = λ Q :dist(E,λ) C r λ Q : Φ C r . { ∈ ≤ 3 }⊂{ ∈ | λ|≤ β } SMOOTHNESS OF PROJECTIONS 33

Now fix any λ E. By Definition 7.2, (7.1) and the Cauchy–Binet formula there exist m coordinate 0 ∈ directions, say the first m, so that the determinant of the first m m–minor of DΦλ0 is bounded below β × by Cr . Moreover, by (7.12) and the Hölder bounds on DΦλ in (7.2), this continues to hold on all of (2+m)β/δ B(λ0,C4r ) for an appropriate choice of C4. Let sm(λ) sm 1(λ) ... s1(λ) > 0 be the singular values of the first m m–minor of DΦ . We have shown that ≥ − ≥ ≥ × λ β β sm(λ) Cr− and (smsm 1 ... s1)(λ) Cr ≤ − · · ≥ on B(λ ,C rb0β). Thus s Crmβ on B(λ ,C rb0β). As above, let 0 4 1 ≥ 0 4 b0β Fy (λ1,...,λm)=Φλ λ B(λ0,C4r ):λm+1 = y1,...,λn = yn m { ∈ − } 1 mβ for any choice of y =(y1,...,yn m). By the lower bound on the singular values, (DFy )− Cr− on the − ≤ domain of Fy . In particular, with λ =(λ1,...,λm), 1 (m+1+δ)β δ (DFy (λ0 ))− DFy (λ) Im m Cr− λ λ0 . ◦ − × ≤ | − | In view of Lemma 7.6 there exists a constant C1 so that Fy is a diffeomorphism on

b0β λ B(λ0, 2C1r ):λm+1 = y1,...,λn = yn m { ∈ − } for any choice of y . The lemma follows by applying Wiener’s covering lemma to a covering of E with balls of 1 b0β size 3 C1r . We now prove the higher–dimensional version of Proposition 4.10. 1,δ n Proposition 7.8. Let Πλ C (Q) for some 0 <δ<1.SupposethatQ R is a region of transversality ∈ ⊂ of order β for Π and that Πλ is 1,δ–regular on Q, see Definition 7.2. Assume that µ has finite α–energy for some α (0,m]. Then ∈ σ+n m − λ Q :dim(ν ) <σ = 0(7.13) H { ∈ s λ } for any σ (0,α a β] with a constant a depending only on m, n,andδ. ∈ − 0 0 Proof. It suffices to prove (7.13) for any subcube Q Q with dist(Q,∂Q) > 0. As in the proof of Proposi- ⊂ tion 4.4 one assumes that (7.13) fails. By Frostman’s lemma there exists a nonzero measure ρ on Q so that σ+n m ρ(V ) diam(V ) − for all Borel sets V Q and ≤| | ⊂ ρ( λ Q :dim(ν ) >σ ) = 0(7.14) { ∈ s λ − 0} for an appropriate choice of 0 > 0. Fix any small >0 and let r = d(ω1,ω2). In view of Lemma 7.7, dν (x)dν (y) dµ(ω )dµ(ω ) λ λ dρ(λ)= 1 2 dρ(λ) σ σ m m x y Π(λ,ω1) Π(λ,ω2) Q R R | − | − Q Ω Ω | − | − (σ ) dµ(ω1)dµ(ω2) = χ b β Φ (ω ,ω ) − − dρ(λ) [ Φλ C0r 0 ] λ 1 2 σ | |≥ | | d(ω1,ω2) Ω Ω Q − (σ ) dµ(ω1)dµ(ω2) (7.15) + χ b β Φ (ω ,ω ) − − dρ(λ) [ Φλ C0r 0 ] λ 1 2 σ | |≤ | | d(ω1,ω2) Ω Ω Q − N dµ(ω1)dµ(ω2) dρ(λ) dµ(ω1)dµ(ω2) (7.16) C (σ )(1+b β) + C σ σ ≤ d(ω ,ω ) 0 b β Φ d(ω ,ω ) Ω Ω 1 2 − Ω Ω j=1 B(λj ,C1r 0 ) λ − 1 2 − | | dµ(ω )dµ(ω ) (7.17) C 1 2 . ≤ d(ω ,ω )α Ω Ω 1 2 To pass from (7.15) to (7.16) first notice that

dρ(λ) ∞ i(σ ) b0β i σ 2 − ρ( λ B(λj,C1r ): Φλ 2− ). b β Φ ≤ { ∈ | |≤ } B(λj ,C1r 0 ) λ − i= | | −∞ 34 YUVAL PERES AND WILHELM SCHLAG

To bound the measure on the right–hand side one uses (7.11) and (7.2). In fact, those estimates imply that n m b0β b0β i r − the set λ B(λj,C1r ): Φλ 2− is the union of 1+ i β many balls of diameter no larger { ∈ | |≤ } 2− r b β i mβ than C min(r 0 , 2− r− ). Thus

dρ(λ) ∞ σ+n m n m ˜ i(σ ) b0β i mβ − (b0 1)β i − Cβ σ C 2 − min(r , 2− r− ) 1+r − 2 r− b β Φ ≤ B(λj ,C1r 0 ) | λ| − i= −∞ where C˜ = C˜(m, n, σ,δ). Inserting this bound into (7.16) and using the bound on N given by Lemma 7.7 implies (7.17) for an appropriate choice of a0. However, (7.17) contradicts (7.14) if <0. In view of Propositions 7.5 and 7.8, Theorem 7.3 will follow once the Sobolev estimate (7.3) is established. As in section 4, this will require a technical lemma about averages involving Littlewood–Paley functions, see Lemmas 4.6 and 7.10. The proof of that lemma will use the following calculus fact.

m m L Lemma 7.9. Suppose h =(h1,...,hm):U R R is a C diffeomorphism on the open set U and let ⊂ → m H denote the inverse of h. For any nonzero multi-index η =(η1,...,ηm) N with η L, ∈ | |≤ η 1 p | |− η p (7.18) ∂ηH = Dh H −| |− v (σ ,...,σ , ) ∂σi h H ◦ η,p 1 p i ◦ t=0 p i=1 σ1,...,σp, m p where the third sum runs over multi-indices σi N and 1, 2,...,m .Ifη 2 the vectors m ∈ ∈p{ } | |≥ v η,p(σ1,...,σp, ) R vanish unless η σi 2 for all i and σi 2( η 1). ∈ | |≥| |≥ i=1 | |≤ | |− η Proof. If η =(1, 0,...,0) = e1, one clearly has ∂ H = Dh H e1. Hence the Lemma holds with η = 1. For the inductive step one differentiates (7.18): ◦ | | η 1 | |− η p 1 ∂η+e1 H = ( η + p) Dh H −| |− − ∂e1 Dh H − | | ◦ ◦ t=0 p σ1,...,σp, p η p v (σ ,...,σ , ) ∂σi h H + Dh H −| |− v (σ ,...,σ , ) η,p 1 p i ◦ ◦ η,p 1 p i=1 p p ∂σi h H ∂e1 (∂σj h ) H . i ◦ j ◦ j=1 i=1,i=j Since for any multi-index σ 1 ∂e1 ∂σh H = Dh H − a ∂τ h H ◦ ◦ τ ◦ τ = σ +1 | | | | m with suitable constant vectors aτ R , (7.18) holds for all multi-indices η with η = η + 1. Also notice that p σ has increased by at∈ most two, as claimed. | | | | i=1 | i| L,δ n Lemma 7.10. Let Πλ C (Q) for some positive integer L and some δ [0, 1).SupposethatQ R is a ∈ ∈ ⊂ n region of transversality of order β for Π and that Πλ is L,δ–regular on Q, see Definition 7.2. Let ρ C∞(R ) ∈ be supported inside Q.Ifψ is the Littlewood–Paley function in Rm from Lemma 4.1, then for any distinct ω ,ω Ω, any integer j,andany0 q

Proof. Fix distinct ω ,ω Ωand j, q as above. We may assume that 2jr>1wherer = d(ω ,ω ). Let 1 2 ∈ 1 2 φ C∞ be nonnegative with φ = 1 on [ 1, 1] and supp(φ) [ 2, 2]. Then ∈ − ⊂ − ρ(λ)ψ 2j[Π(λ,ω ) Π(λ,ω )] dλ 1 − 2 j 1 β j 1 β (7.20) = ρ(λ)ψ(2 rΦ (ω ,ω ))φ(C− r− Φ ) dλ + ρ(λ)ψ(2 rΦ (ω ,ω )) 1 φ(C− r− Φ ) dλ. λ 1 2 β λ λ 1 2 − β λ Here Cβ is the constant from Deﬁnition 7.2. By the rapid decay of ψ

j 1 β j 1+β q j 1+β q ρ(λ)ψ(2 rΦ (ω ,ω )) 1 φ(C− r− Φ ) dλ C ρ(λ) (1 + 2 r )− dλ C (1 + 2 r )− . λ 1 2 − β λ ≤ q,β | | ≤ q,β Thus it suﬃces to estimate the ﬁrst integral in (7.20). To this end we introduce a partition of unity subordinate to the cover given by Lemma 7.7 with U = Q : ρ>0 . More precisely, by the standard construction of a n { } b0β partition of unity there exist χj C∞(R ) so that supp(χj) Bj = B(λj, 2C1r ) for j =1,...,N and so N ∈ ⊂ that χ = 1 on λ supp(ρ): Φ C rb0β , see (7.9). Moreover, i=1 j { ∈ | λ|≤ 0 } η η b0β (7.21) sup ∂ χj Cη r−| | j ∞ ≤

n 1 β for all multi-indices η N . By (7.10), χj(λ)=χj(λ)φ(C− r− Φλ) for all j. We can therefore write the ∈ β ﬁrst integral in (7.20) as follows.

N j 1 β j ρ(λ)ψ(2 rΦλ)φ(Cβ− r− Φλ) dλ = ρ(λ)χj(λ)ψ(2 rΦλ) dλ i=1 N N j 1 β (j) + ρ(λ) 1 χ (λ) ψ(2 rΦ )φ(C− r− Φ ) dλ = A + B. − j λ β λ i=1 i=1 b0β To estimate the second term B notice that Φλ C0r on the support of the integrand. The rapid decay j 1+|b0β |≥q of ψ therefore implies B Cq(1 + 2 r )− . For simplicity, we assume henceforth that i = 1. By | |≤ m Lemma 7.7 the map λ Φ(λ,λ) = Fλ (λ) is a diﬀeomorphism on λ R :(λ, λ) Bj ,where → { ∈ ∈ } λ =(λ1,...,λm) and λ =(λm+1,...,λn) (possibly after a permutation of the coordinates). Let Hλ denote

the inverse of Fλ . Assuming as we may that ρ(λ)=ρ1(λ)ρ2(λ)welet G (u)=ρ (H (u))χ (H (u)) det DH (u) . λ 1 λ 1 λ | λ | L 1,δ Clearly, Gλ C − .Sinceq0 such∈ that (M + m + δ)(1 ) >qand− rewrite A(1) as follows. − − ≤ − − (1) j A = ρ2(λ) Gλ (u)ψ(2 ru) du dλ n m m R − R η j ∂ Gλ (0) η = ρ2(λ) ψ(2 ru) u n m u <(2j r) 1+  η! R − | | − η M | |≤ 1 η η η M u + ∂ G (tu) ∂ G (0) (1 t) dt du dλ λ − λ − η!  η =M 0 | |  j 2q m + ρ2(λ) O (2 r u )− − Gλ (u)du dλ n m u >(2j r) 1+ | | | | R − | | − (1) (1) = A1 + A2 . 36 YUVAL PERES AND WILHELM SCHLAG

β (1) j q m β By Lemma 7.7, Gλ Cr− . In particular, A2 Cq(2 r)− − r− .Sinceψ has vanishing moments of all orders ∞ ≤ | |≤ η (1) j ∂ Gλ (0) η A1 = ρ2(λ) ψ(2 ru) u du dλ − n m u >(2j r) 1+ η! R − | | − η M | |≤ η j (M+m+δ)(1 ) δ (7.22) + ρ2(λ)O sup ∂ Gλ C (2 r)− − dλ. n m η =M R − | | η It remains to estimate ∂ Gλ (u), which can be done uniformly in λ. Fix any λ and let u = Fλ (λ). By Lemma 7.9, (7.2) in Deﬁnition 7.2, and Lemma 7.7,

η 1 | |− η 1 η +p 2β( η 1) ∂ H (u) C v (σ ,...,σ , ) DF − | | r− | |− | λ |≤ η | η,p 1 p | λ t=0 p σ1,...,σp, 2 η (m+1)β C r− | | . ≤ η η C η β Therefore, by Leibnitz’s rule and (7.21), ∂ Gλ Cηr− | | where C = C(b0,m). Plugging this into ∞ ≤ (1) a0Mβ j (M+m+δ)(1 ) (7.22) and exploiting the rapid decay of ψ one obtains A1 Cq r− (2 r)− − .SinceN b nβ | |≤ ≤ Cr− 0 and (M + m + δ)(1 ) >q, this ﬁnally yields (7.19). − Proof of (7.3). Fix some γ with 0 < (m+2γ)(1+a β) α and 2γ

2 ∞ 2jγ ∞ ν ρ(λ) dλ 2 (ψ j ν )(x) dν (x) ρ(λ) dλ λ2,γ 2− ∗ λ λ j= −∞ −∞ ∞ 2j(m+2γ) ψ 2j[Π(λ,ω ) Π(λ,ω )] ρ(λ) dλ dµ(ω )dµ(ω ) ≤ 1 − 2 1 2 Ω Ω j= −∞ ∞ j(m+2γ) j 1+a0β q Cγ 2 (1 + 2 d(ω1,ω2) )− dµ(ω1)dµ(ω2) ≤ Ω Ω −∞ dµ(ω1)dµ(ω2) Cγ C α(µ) < , ≤ d(ω ,ω )(1+a0β)(m+2γ) ≤ E ∞ Ω Ω 1 2 as claimed.

8. Applications of the higher–dimensional projection theorems 8.1. Self–similar sets in the complex plane. In [41] Solomyak considered sets of the form

∞ (8.1) CS = a λn : a S λ n n ∈ n=0 n where S = s1,...,s is a set of digits and λ D = z C : z < 1 (λ of course denotes complex powers). Let{ Γ= S S} and ∈ { ∈ | | } − ∞ = b λn : b Γ BΓ n n ∈ n=0 Γ = λ D : there exists a nonzero f Γ with f(λ)=0 M { ∈ ∈B } Γ = λ D : there exists a nonzero f Γ with f(λ)=f (λ)=0 . M { ∈ ∈B } SMOOTHNESS OF PROJECTIONS 37

S If λ D Γ it is easy to see that Cλ satisfies a strong separation condition as in (5.14) and that there- ∈ \MS log fore dim Cλ = log λ . The case λ Γ was studied in [41]assuming the transversality condition λ Γ. − | | ∈M ∈ M It was shown there that for any fixed choice of digit set S C a.e. λ Γ Γ satisfies ⊂ ∈M \ M S log 1/2 2 S 1/2 dim C = if λ <− and (C ) > 0if λ >− . λ log λ | | H λ | | − | | It is straightforward to see that Theorem 7.3 applies to this case. First we verify that conditions (7.2), (7.1) in Definition 7.2 are satisfied with L = . This is very similar to Lemma 5.3. ∞ To specialize from section 7 let 0

Q λ D : λ Γ,R1 < λ R1. Then Q is a region of transversality of order 2β for Π in the sense of Deﬁnition 7.2. Proof. There exists δ = δ(Q) (0, 1) such that for any g and λ Q ∈ ∈BΓ ∈ g(λ) <δ implies g(λ) >δ. | | | | 1 ω τ This follows since Γ is a normal family. Clearly, Φλ(ω,τ)=(λR2− )| ∧ |g(λ) with some g = gω,τ Γ.Fix ω,τ Ωand let k B= ω τ . Therefore, ∈B ∈ | ∧ | η η k η k k k σ ∂ Φλ Cη k| | λ −| |R2− g + λ R2− sup ∂ g | |≤ | | ∞ | | σ = η ∞ | | | | η η 1 η 1 β η k C k| |R−| |(1 R )− +(1 R )−| |− C R− | | , ≤ η 1 − 2 − 2 ≤ η,β 2 βk since R2 < 1 and β>0. Thus condition (7.2) in 7.2 holds with L = . To check (7.1) assume Φλ δbβR2 where the constant b (0, 1) will be determined below. Then ∞ | |≤ β ∈ 1 k 1 k βk R R− g(λ) λ R− g(λ) δb R 1 2 | |≤ | | 2 | |≤ β 2 1 1+β k implies g(λ) δb R− R . Hence (here Φ denotes the complex derivative) | |≤ β 1 2 λ 1 k 1 βk 1 1 1+β k βk βk Φ R R − g(λ) R− k g(λ) R δ δb R− k R− R δR /2 δb R | λ|≥ 1 2 | |− 1 | | ≥ 2 − β 1 1 2 ≥ 2 ≥ β 2 1 1 k 1 1+β k − 2 if bβ 2 1+supk 0 R R1− R2 . Since the Cauchy–Riemann equations imply that det[DΦλ]= Φλ , ≤ ≥ 1 | | 2 condition (7.1) in Deﬁnition 7.2 holds with 2β instead of β and Cβ =(δbβ) . Theorem 8.2. Let 0

1+β and Qj rj < λ Rj . By the previous lemma each Qj is a region of transversality of ⊂{ | | } ω τ order 2β for Π. It is easy to check that with the choice of metric d (ω,τ)=R| ∧ | one has (µ) < if j j Eα ∞ Rαl>1. Since supp(ν )=CS, the theorem now follows by applying Theorem 7.3 and letting β 0+. j λ λ → d 8.2. Intersections of sets with spheres. Let S0 R , d 2, be a closed, strictly convex C∞–hypersurface ⊂ ≥ d 1 surrounding the origin (the diﬀerentiability assumption will be relaxed later). Thus, S0 = ρ(u)u : u S − d 1 d x { ∈ } for some function ρ : S − (0, )inC∞.Deﬁnef C∞(R 0 )tobef(x)= ρ(x/| |x ) . In particular, f is → ∞ d ∈ \{ } | | homogeneous of degree one and x R : f(x)=r = rS0. Moreover, by the assumption of strict convexity there exists κ>0 so that for any{ x∈= 0 and any tangent} vector v of S 0 v 2 (8.2) D2f(x)v, v κ| | , | |≥ x | | whereas D2f(x)x = 0 by homogeneity. For any Borel set E Rd and σ<1let ⊂ d Bσ = Bσ(E)= x R :dimf( x + E) <σ . { ∈ − } For σ =1weset d 1 B1 = B1(E)= x R : (f( x + E)) = 0 . { ∈ H − } Our main result in this section concerns the dimension of Bσ.

Theorem 8.3. Let S0 be a strictly convex, C∞–hypersurface surrounding the origin and deﬁne f and Bσ as above. If E Rd is a Borel set and σ [0, 1 dim E], then ⊂ ∈ ∧ (8.3) dim B d + σ max(1, dim E). σ ≤ − Furthermore, for any hyperplane H Rd, ⊂ (8.4) dim(B H) d 1+σ max(1, dim E). σ ∩ ≤ − −

Proof. (8.3) follows from (8.4) by classical theorems of Marstrand and Mattila, see Theorem 10.10 in [31]. For the sake of illustration, however, we begin by showing directly that (8.3) follows from Theorem 7.3. In fact, we d 1 first consider the case S0 = S − because it is very easy to check the conditions in Definition 7.2 for Euclidean spheres. In this special case, it is more convenient to work with the square of the Euclidean norm, i.e., 2 d 2 1 2 2 f(x)= x .DefineΠ:R E R tobeΠ(λ,x)= x λ and let Φλ(x, y)= x y − [ x λ y λ ]= x y | | × → η | − | | − | | − | −| − | x−y ,x+ y 2λ . Therefore, ∂ Φλ(x, y) 2 for any nonzero multi–index η, and condition (7.2) holds | − | − x y ≤ with β = 0. Moreover, DΦλ(x, y) =2 x−y = 2, and (7.1) is also satisfied with β = 0. Hence Definition 7.2 d | − | holds with Q = R ,Ω E an arbitrary compact set, L = , and β = 0. For any >0 one can choose a ⊂ ∞ d 1 Frostman measure µ on Ωwith exponent > dim E .Sincef( λ + E) supp(νλ), (8.3) with S0 = S − follows from Theorem 7.3 as 0+. − − ⊃ → For general S0 as described above, fix a small >0. Dilating, if necessary, one has dim(E Q0) > dim E d \ − for any cube Q0 of side length 2. Now fix a cube Q R of side length 1. To bound dim(Bσ Q)wemay therefore assume that E 2rQ (rQ) for some r 2.⊂ Thus, if v = 1, x E, and λ Q, (8.2)∩ implies that ⊂ \ ≥ | | ∈ ∈ (8.5) f(x λ),v =0= D2f(x λ)v ρ |∇ − | ⇒| − |≥ for some ρ>0 depending on r and κ. Now define Πλ : E R to be Πλ(x)=f(x λ) and let Φλ(x, y)= 1 → − x y − [f(x λ) f(y λ)]. It is clear that condition (7.2) holds with β = 0. To check (7.1) notice that | − | − − − 1 Φ (x, y)= x y − [f(x λ) f(y λ)] = f(x λ),v + O( x y ) λ | − | − − − ∇ − | − | x y where v = x−y . Moreover, | − | DΦ (x, y)= D2f(x λ)v + O( x y )(8.6) λ − − | − | SMOOTHNESS OF PROJECTIONS 39

where D denotes differentiation with respect to λ. In view of (8.5) there exists some small constant C0 such that Φ (x, y) ρ/2(8.7) | λ | 0 ⇒| λ | provided λ Q and x, y E satisfy x y dim E and letting µ be an appropriate Frostman measure we conclude that (8.3) follows∩ from (7.4) and (7.6) with− L = , β = 0, n = d, m = 1, and α =dimE as 0+. ∞ The proof of (8.4) proceeds− by induction→ in the dimension d. First we need to verify that transversality d holds in all dimensions. Fix a hyperplane H R . W.l.o.g. 0 H. Let e1,...,ed 1 be a basis in H and define d 1 ⊂ 1 ∈ −d 1 d Πλ(x)=f(x − λiei),Φλ(x, y)= x y − Πλ(x) Πλ(y) for any λ R − , and distinct x, y R . − i=1 | − | − ∈ ∈ Clearly, with p = x λ1e1 ... λd 1ed 1, − − − − − Φ (x, y)= f(p),v + O( x y )(8.8) λ ∇ | − | 2 2 (8.9) DΦλ(x, y)= D f(p)v, e1 ,..., D f(p)v, ed 1 + O( x y ), − − | − | x y d 1 where v = − and D denotes differentiation with respect to λ. We claim that for any v S − and any x y ∈ z H | − | ∈ d 1 − 2 (8.10) f(z),v =0= D2f(z)v, e > 0. ∇ ⇒ i i=1 In fact, we will show that (8.10) can only fail if z H. More precisely, suppose the sum in (8.10) vanishes d d 1 ∈ 2 for some z R , z = 0 and some v S − . W.l.o.g. f(z) = 1. Then the functionals u D f(z)u, ei , i =1,...,d∈ 1, on the tangent space∈ to S at z are linearly dependent. Thus → − d 1 − ξ D2f(z)e f(z)(8.11) i i ∇ i=1 d 1 for some (ξ1,...,ξd 1) =(0,...,0). Let e = − ξiei. Clearly, e = 0. It is easy to verify that (8.11) implies − i=1 that D2f(z)e, e = 0. First notice that by strict convexity of S, there is a parameterization z + te = r(t)γ(t) for small t where γ is a smooth curve in S with γ(0) = z and r is a smooth positive function with r(0) = 1. Let ∂ f = f,e be the directional derivative. Since ∂ f is homogeneous of degree zero, e ∇ e 2 2 d d 2 d D f(z)e, e = f(z + te) = ∂ef(γ(t)) = D f(z)e, γ(0) =0, dt2 t=0 dt t=0 dt by (8.11). In view of (8.2) and D2f(z)z = 0, we conclude that e z and thus z H, as claimed. To check ∈ transversality, we will need the following quantitative version of (8.10), cf. (8.5). For any ρ0 > 0thereexists d ρ1 > 0 depending on κ from (8.2) so that for any z R with f(z) = 1 and dist(z,H) >ρ0 one has ∈ d 1 − 2 (8.12) f(z),v <ρ = D2f(z)v, e >ρ . |∇ | 1 ⇒ i 1 i=1 Indeed, by (8.10) and compactness, d 1 − 2 2 (8.13) min min D f(z)v, ei >ρ2 > 0 v T S : v =1 z S0:dist(z,H)>ρ0 ∈ z 0 | | i=1 ∈ (TzS0 denotes the tangent space to S0 at z). If f(z),v <ρ1,thenthereexists˜ v TzS0, v˜ = 1, so that ρ1 |∇ | ∈ | | v v˜ C f(z) Cρ1 and (8.12) follows from (8.13) provided ρ1 is small compared to ρ2. | − |≤ |∇ | ≤ 40 YUVAL PERES AND WILHELM SCHLAG

We start by proving (8.4) for d = 2. Fix some e S1 and let R2 be the line through the origin in direction e.Ifdim(E )=dimE,then ∈ ⊂ ∩ dim r>0:(x + rS) E = =dimE { ∩ ∅} for all x and (8.4) holds with H = . Otherwise, for any >0thereexistsρ0 > 0 and R>0 so that ∈ ρ0 ρ0 dim(E B(0,R) ) > dim E where denotes a ρ0–neighborhood of . Now fix such >0 and ρ0 > 0 and let∩J be\ an interval of− length one. In view of (8.8), (8.9), and (8.12) there exist ρ>0 and C ⊂ 0 (depending on J, κ, ρ0, R) so that Φ (x, y) <ρ= DΦ (x, y) 2 >ρ | λ | ⇒| λ | ρ0 ρ0 for any x, y B(0,R) satisfying x y dim E . We| conclude− | that with this choice∩ of Ωand Π\ , J satisfies Definition∈ 2.7 − d 1 with L = and β = 0. Since f i=0− λiei + E supp(νλ), (8.4) therefore follows from Theorem 2.8 with α =dimE∞ as 0+. − ⊃ − → Now suppose that (8.4) holds up to d 1 for some d 3 and fix a hyperplane H Rd passing through the origin. If dim(E H)=dimE, then− (8.4) follows from≥ (8.3) in dimension d 1⊂ (recall that the latter ∩ − estimate follows from (8.4) in dimension d 1). Otherwise, for any >0thereexistsρ0 > 0 so that ρ0 ρ0 − dim(E H ) > dim E where H denotes a ρ0–neighborhood of H. By the same argument we gave for d = 2, one\ concludes from− (8.8), (8.9), and (8.12) that Definition 7.2 holds for any bounded cube Q H and a suitable choice of Ω. Hence (8.4) follows from Theorem 7.3 with L = , β = 0, n = d 1, m =⊂ 1, and α =dimE as 0+. ∞ − − → Suppose E Rd for some line is a set of dimension 1 but with 1(E) = 0. Then it is easy to see that d ⊂ ⊂ d 1 H for all x R ,(x + rS − ) E = for a.e. r. This implies that there are no nontrivial bounds for σ = 1 and ∈ ∩ ∅ dim(E) 1 in Theorem 8.3. In [10] Falconer showed that for any Borel set A Rd with dim A>(d + 1)/2 the distance≤ set D(A)= x y : x, y A has positive measure. In [32] Mattila⊂ and Sjölin showed that under the same assumption{| D−(A)| has nonempty∈ } interior. On the other hand, Bourgain [4] proved that D(A) has positive measure if dim A>(d+1)/2 d for some d > 0ifd =2, 3. (8.3) implies the following sharpened version of Falconer’s theorem concerning− “pinned” distance sets D (A)= x y : y A : x {| − | ∈ } Corollary 8.4. Suppose A Rd with dim A>(d + 1)/2. Then ⊂ d dim x R : Dx(A) has Lebesgue measure zero d +1 dim A<(d + 1)/2. { ∈ }≤ − Proof. Apply (8.3) to f(x)= x and E = A. | | Similarly, we obtain a sufficient condition for pinned distance sets to contain an interval. Using (7.4) with σ>2 in the proof of Theorem 8.3, yields the following statement: Let A Rd with d 3 satisfy dim A>(d + 2)/2. Then ⊂ ≥ d dim x R : Dx(A) has empty interior d +2 dim A<(d + 2)/2. { ∈ }≤ − We now discuss the case where S Ck,δ for some positive integer k and δ [0, 1). 0 ∈ ∈ k,δ Theorem 8.5. Let S0 be a strictly convex, C –hypersurface surrounding the origin with k + δ>2 and d define f and Bσ as above. If E R is a Borel set and σ [0, 1 dim E], then ⊂ ∈ ∧ (8.14) dim B d + σ max 1, min(dim E,k + δ 1) . σ ≤ − − Furthermore, for any hyperplane H Rd, ⊂ (8.15) dim(B H) d 1+σ max 1, min(dim E,k + δ 1) . σ ∩ ≤ − − − In particular, if k + δ dim E +1, then (8.3) and (8.4) remain valid. ≥ SMOOTHNESS OF PROJECTIONS 41

L,δ k 1,δ Proof. Since f C by assumption, (8.6), (8.8), and (8.9) show that Φ C − in the sense of Defi- ∈ λ ∈ nition 7.1. The proof of Theorem 8.3 therefore yields that Φλ as defined above satisfies Definition 7.2 with L = k 1 and β = 0 and as before, Theorem 7.3 implies (8.14) and (8.15). − Finally, we give some examples to show that Theorem 8.3 can fail under weaker assumptions. Suppose d d 1 S0 = ∂[ 1, 1] is the boundary of the unit cube. Let K [0, 1] be a Cantor set and let E = K [ 1, 1] − . − d 1 ⊂ × − It is easy to see that B (E) [3, ) [ 1, 1] − for any dim K<σ 1. The bound in (8.3) would therefore σ ⊃ ∞ × − ≤ require that d 1+σ dim K, which is clearly false. A similar example shows that (8.3) fails if S0 contains an arbitrarily small≤ piece− of a hyperplane. In view of (8.4) one might ask whether (8.16) dim(B π) 1+dimπ dim E 1 ∩ ≤ − d d 1 for any plane π R . It turns out that this fails if dim π d 2evenifS0 = S − . For simplicity, let d =3 and let ⊂ ≤ − (8.17) E = (0,rcos θ, r sin θ):θ [0, 2π],r K { ∈ ∈ } where K [0, 1] is a Cantor set. Clearly, for any x R,dim r>0:(x, 0, 0) + rS2 E =dimK, whereas ⊂ ∈ { ∈ } (8.16) would require that dim(B1 (x, 0, 0) : x R ) 2 dim E =1 dim K. ∩{ ∈ } ≤ − − 9. Questions and Comments 9.1. Bernoulli Convolutions. 4 ((i)) For which intervals J (1/2, 1) is J νλ 2 dλ< ? ((ii)) Is the distribution of the⊂ zeros of Π (ω) Π (τ) on∞ an interval of transversality absolutely contin- λ − λ uous? 9.2. Geometric problems. ((i)) According to Corollary 6.2, a.e. projection of a Borel set E Rd with dim E>2k has nonempty interior. Can 2k be replaced by a smaller threshold if k>1?⊂ ((ii)) Suppose that µ is a finite planar measure satisfying the Frostman condition µ(B(x, r)) rα for all 2 ≤ x R and r>0. Let νθ be the projection of µ onto a line in direction θ. For p [1, 2/(2 α)), ∈ ∈ −p the Sobolev embedding theorem implies that the projected measures νθ have a density in L for a.e. θ S1. Can the threshold 2/(2 α) be increased? ((iii)) Is Theorem∈ 8.3 optimal? Heuristically− speaking, estimates for distance sets are related to the Erd˝os problem of bounding the minimum number gd(n) of different distances determined by n points in d 1 4/5 n R . It is known that C − n − 5/4= 1(D(E)) > 0. Generally speaking, however, it is presently unclear how to make such a deduction.⇒H Nevertheless, T. Wolff[44, 45] has successfully applied sophisticated methods from combinatorial geometry to obtain Hausdorffdimension estimates; in particular, he showed that a Borel set in the plane that contains a circle of every radius must have Hausdorffdimension 2. For pinned distance sets as in Corollary 8.4, the relevant combinatorial problem seems to be estimating the minimum number of distinct distances between n points and one “typical” point. More precisely, it is known, see 2 Corollary 12.11 in [1], that for any n points p1,p2,...,pn R , one has card pi pj : j = ∈ | − | 1, 2,...,n Cn3/4 for at least n/2 choices of i (this appears to be the best known bound. In particular, the≥ authors of [5] point out that their method does not show that there exists a point 4/5 from which there are n − different distances). The analogy between cardinality and dimension alluded to above then suggests that dim(E) > 4/3= 1(D (E)) > 0 for most points x E. Note ⇒H x ∈ that Corollary 8.4 requires dim(E) > 3/2 for the same conclusion in R2. In the recent preprint [46] 42 YUVAL PERES AND WILHELM SCHLAG

Wolﬀshows that dim(E) > 4/3 implies that 1(D(E)) > 0. His methods are in the spirit of [4] and involve Bochner–Riesz type arguments. H ((iv)) Does Theorem 8.3 hold if S Ck,δ with k + δ

References 1. P. Agarwal, J. Pach, Combinatorial Geometry. John Wiley & Sons Inc., 1995. 2. J. C. Alexander, J. A. Yorke, Fat Baker’s transformations. Ergodic Th. Dyn. Syst. 4 (1984), 1–23. 3. J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis. Geom. and Funct. Anal. 1 (1991), 147–187. 4. J. Bourgain, Hausdorffdimension and distance sets. Israel J. Math. 87 (1994), 193–201. 5. F. Chung, E. Szemerédi, W. Trotter Jr.. The number of distinct distances determined by a set of points in the Euclidean plane. Discrete and Computational Geometry 7 (1992), 1–11. 6. P. Erd˝os, On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 (1939), 974–975. 7. P. Erd˝os, On the smoothness properties of Bernoulli convolutions. Amer. J. Math. 62 (1940), 180–186. 8. K. Falconer, Continuity properties of k–plane integrals and Besicovitch sets. Proc. Cambridge Phil. Soc. 87, 1980. 221–226. 9. K. Falconer, Hausdorffdimension and the exceptional set of projections. Mathematika 29, 1982. 109–115. 10. K. Falconer, On the Hausdorffdimension of distance sets. Mathematika 32 (1985), 206–212. 11. K. Falconer, The geometry of fractal sets. Cambridge Tracts in Mathematics, 85. C.U.P., Cambridge–New York, 1986. 12. K. J. Falconer, The Hausdorffdimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103 (1988), 339–350. 13. M. Frazier, B. Jawerth, G. Weiss, Littlewood–Paley theory and the study of function spaces. CBMS Regional Conference Ser. 79, American Math. Soc. (1991). 14. D. Funaro, Polynomial Approximation of Differential Equations. Lecture Notes in Physics, Springer-Verlag, Heidelberg (1992). 15. A. M. Garsia, Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409–432. d 16. X. Hu and S. J. Taylor, Fractal properties of products and projections of measures in R . Math. Proc. Camb. Phil. Soc. 115 (1994), 527–544. 17. B. Hunt, V. Y. Kaloshin, How projections affect the dimension spectrum of fractal measures. Nonlinearity 10 (1997), 1031– 1046. 18. B. Hunt, T. D. Sauer and J. A. Yorke, Prevalence: a translation-invariant almost-every for infinite dimensional spaces. Bull. Amer. Math. Soc. 28 (1992), 217–238. Prevalence: an addendum. Bull. Amer. Math. Soc. 28 (1993) 306–307. 19. J.E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747. 20. M. Jarvenpää, On the upper Minkowski dimension, the packing dimension, and orthogonal projections. Ann. Acad. Sci. Fenn. Ser. A I, Math. Dissertationes, 1994. 21. B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38 (1935), 48–88. 22. J. P. Kahane, Sur la distribution de certaines series aleatoires. Colloque Th. Nombres [1969, Bordeaux], Bull. Soc. math. France, Mémoire 25 (1971), 119–122. 23. R. Kaufman, On Hausdorffdimension of projections. Mathematika 15 (1968), 153-5. 24. R. Kaufman, An exceptional set for Hausdorffdimension. Mathematika 16 (1969), 57-8. 25. R. Kaufman, P. Mattila, Hausdorffdimension and exceptional sets of linear transformations. Ann. Acad. Sci. Fenn. Ser. A IMath.1(1975),387–392. 26. M. Keane, M. Smorodinsky, B. Solomyak, On the morphology of γ–expansions with deleted digits. Trans. Amer. Math. Soc. 347, no. 3 (1995), 955–966. 27. F. Ledrappier, On the dimension of some graphs. Contemporary Math. 135 (1992), 285–293. 28. J. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London Math. Soc. 4, 1954. 257–302. 29. P. Mattila, Hausdorffdimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. A Math. 1(1975),227–244. 30. P. Mattila, Orthogonal projections, Riesz capacities and Minkowski content. Indiana Univ. Math. J. 39 (1990), 185–198. 31. P. Mattila, Geometry of sets and measures in Euclidean spaces. C.U.P., Cambridge (1995). 32. P. Mattila, P. Sjölin, Regularity of Distance Measures and Sets. To appear. 33. Y. Peres, W. Schlag, B. Solomyak, Sixty years of Bernoulli convolutions. Fractals and Stochastics II, Proceedings of the Greifswald 1998 Conference,Bandt,GrafandZähle(Editors),toappear. SMOOTHNESS OF PROJECTIONS 43

34. Y. Peres, B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof. Math. Research Letters 3:2 (1996), 231–239. 35. Y. Peres, B. Solomyak, Self–similar measures and intersections of Cantor sets. Trans. Amer. Math. Soc. 350 (1998), 4065– 4087. 36. M. Pollicott, K. Simon, The Hausdorffdimension of λ-expansions with deleted digits. Trans. Amer. Math. Soc. 347, no. 3 (1995), 967–983. 37. F. Przytycki, M. Urbansky, On Hausdorffdimension of some fractal sets. Studia Math. 93 (1989). 155–186. 38. T. D. Sauer and J. A. Yorke, Are the dimensions of a set and its image equal under typical smooth functions? Ergod. Th. Dynam. Sys. 17 (1997), 941–956. 39. B. Solomyak, On the random series λi (an Erd˝osproblem). Annals of Math. 142 (1995), 611–625. ± 40. B. Solomyak, On the measure of arithmetic sums of Cantor sets. Indagationes Math., N.S. 8 (1997), 133–141. P 41. B. Solomyak, Measure and dimension of some fractal families. Proc. Cambridge Phil. Soc. 124 (1998), 531–546. 42. E. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, 1970. 43. E. Stein, Harmonic Analysis. Princeton University Press (1993). 44. T. Wolff, A Kakeya type problem for circles. Amer. J. Math. 119 (1997), no. 5, 985–1026. 45. T. Wolff, Recent work connected with the Kakeya problem. 1996 Princeton Conference Proceedings: “Prospects in Mathe- matics”. To appear. 46. T. Wolff, Decay of circular means of Fourier transforms of measures. Preprint (1999). To appear in Int. Math. Res. Notices.

Department of Mathematics, Hebrew University, Jerusalem, Israel and Department of Statistics, University of California, Berkeley, CA 94720-3860, U.S.A. E-mail address: [email protected]

Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, U.S.A. Current address:DepartmentofMathematics,PrincetonUniversity,FineHall,PrincetonN.J.08544,U.S.A. E-mail address: [email protected]