arXiv:1603.08156v1 [math.CA] 27 Mar 2016 uddb h cdm fFinland. of Academy the by funded 60G57. 60G46, 42A61, 42A38, 28A78, oceecaso admst n esrs hc snevertheles is which measures, and (suc sets patterns random of of class existence perco concrete t fractal the cartes develop especially study include and we to does fractals it [27], which random apply work inside measures and progressions) forthcoming random order, of a any class In of wider even measure. an pow random cartesian of example given for a applications, in of arise often class, which This fractals fractal others. martingales as among independent such cutouts, spatially literature, Poissonian previous measur and an the random cascades, in model, of dom interest underlying class of the axiomatic th models of an of many features for all) key theory not and of analysis geometric certainly number in (though general small problems many a to that on applications observed n on based we works view [26], a many th In with in At but fields. investigated itself, years. been in last has end the an fractals in random attention of of geometry deal considerable a attracted ..akolde upr rmteCnr fEclec nAnalysis in Excellence of Centre PICT and the 2013-1393 from PICT support Projects acknowledges by V.S. supported partially was P.S. 2010 h td ftegoercmauetertcpoete frand of properties theoretic measure geometric the of study The nti ril,w tpbc rmtemxmmgnrlt n conside and generality maximum the from back step we article, this In ahmtc ujc Classification. Subject Mathematics rbe o rca esrs n h oncinbtenarith between connection to the results decay. and these Fourier Fou measures, of and applications and fractal present self-convolutions for and problem study class, We our in conta and measures measure. applications several percolation for for enough fractal allows general that is setting yet concrete proofs, more a in martingales, independent Abstract. LS FRNO ATRMAUE,WITH MEASURES, CANTOR RANDOM OF CLASS A esre oeo u eetrslso h emtyo spatially of geometry the on results recent our of some survey We AL HEKNADVLESUOMALA VILLE AND SHMERKIN PABLO APPLICATIONS 1. ssiltorsrciet nops oerandom some encompass to restrictive too still is , Introduction rmr:2A0 00;Scnay 12,28A75, 11B25, Secondary: 60D05; 28A80, Primary: 1 n h well-known the ins 0418 APy) . (ANPCyT). 2014-1480 n yaisResearch Dynamics and ei structure metic eil nuhto enough flexible s h restriction the hre,direct shorter, lation. irdcyof decay rier hc etermed we which aetm,the time, same e eclto,ran- percolation, swihinclude which es sage and angles as h r forder of ers mfatl has fractals om ts uhas much so ot eeoe a developed d s ok are works ese a products ian etheory he more a r related ≥ 3 2 PABLOSHMERKINANDVILLESUOMALA obtain many of the applications that motivated our works [26, 27]. Bypassing the general theory provides shorter, more direct proofs of some of our results, especially those concerning self-convolutions. It also allow us to present some of the new ideas in [27] in a concrete setting. In Section 2 we introduce the class of random measures we will work with in the rest of the article, and establish some of their basic properties. We study self- convolutions of the random measures in Section 3, their Fourier decay and restriction estimates in Section 4, and their arithmetic structure in Section 5.
2. The model Throughout the paper, c,C denote positive and finite constants whose values may be different from line to line. Occasionally c,C will be random variables which take values in (0, + ) almost surely. Whenever necessary, we may denote the ∞ dependency of C from various parameters by subscripts, for instance Cε denotes a positive and finite constant whose value depends on ε > 0 but not on other parameters. N N d Fix 2 M and for each n , denote by n the family of all M-adic half open sub-cubes≤ ∈ of the unit cube [0∈, 1)d, that is Q
d d = [j M −n, (j + 1)M −n):0 j M n 1 . Qn i i ≤ i ≤ − (i=1 ) Y 1 For convenience, we also denote n by n. We will consider a sequence ofQ randomQ functions µ : [0, 1)d [0, + ), which in n → ∞ the sequel we will identify with the measures µndx, satisfying the following condi- tions for some deterministic nondecreasing sequence (βn)n∈N:
(M1) µ0 = 1[0,1)d . d (M2) µn = βn1An , where An is a union of cubes in n. (M3) E(µ (x) A )= µ (x) for all x [0, 1)d. Q n+1 | n n ∈ (M4) Conditional on An, the random variables µn+1(Q), Q n+1 are jointly independent. ∈ Q This class of random measures is, essentially, a subclass of the subdivision mar- tingales, which in turn fit into the more general framework of spatially independent martingale measures developed in [26]. However, there is one direction in which our class is more general: in [26] we required µn+1/µn to be uniformly bounded. This condition can be substantially weakened in all applications, so we chose to remove it from the hypotheses. As indicated in the introduction, in order to avoid unnec- essary technicalities, and to draw attention to the main ideas, we will stick the this special class in this survey. A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 3
We describe a general construction that yields sequences (µn) satisfying the above d d d hypotheses. Given Q n, we denote by (Q) the family of all M cubes in n+1 which are contained in∈Q Q(thinking of the hierarchyS of M-adic cubes as a tree,Q they d are the offspring of Q). Suppose that for each Q n, n 0, there is a random d ∈ Q ≥ subset SQ (Q), such that SQ : Q n, n 0 are independent and, for each Q′ (Q),⊂ S { ∈ Q ≥ } ∈ S ′ βn P(Q SQ)= . ∈ βn+1 d Then, if we inductively set A0 = [0, 1) , and ′ An+1 = Q , d Q′∈S Q∈Q[n,Q⊂An [Q the random sequence µn = βn1An is easily seen to satisfy (M1)–(M4). The following are two important classes of examples of this construction: −n (i) Fix p (0, 1) and set βn = p . Let SQ be the family obtained by select- ing each∈ Q′ (Q) independently with probability p, with all the choices independent∈ for S different Q. This is the well-known fractal percolation pro- cess, and is a geometric realization of a standard Galton-Watson branching process. d (ii) Pick 0 More precisely, denote A = n∈N An, and let log β log β α = limT inf M n , α = lim sup M n . (2.1) n→∞ n n→∞ n The following standard lemma collects the basic properties of our random model. 4 PABLOSHMERKINANDVILLESUOMALA Figure 1. Illustration for the examples (i) and (ii) in the case d = M = 2. The one on the top shows the first four steps of fractal percolation with p =0.7. The second one is a realization of (ii) with β1 = 1, β2 = β3 = 4 and β5 = 16. Lemma 2.1. The following holds almost surely: (1) The sequence of measures µn converges weakly to a measure µ supported on A [0, 1]d. ⊂ (2) dim (A) d α. B ≤ − (3) If α E 2 2 P −α−ε (Xn)= βn (x, y An) dxdy C x y dx dy . (2.3) d d ∈ ≤ | − | Zx∈[0,1) Zy∈[0,1) ZZ E 2 d This shows that (Xn) is bounded if α E x y −s dµ (y) dµ (x) = β2 x y −sP(x, y A ) dxdy | − | n n n | − | ∈ n ZZ ZZ C x y −s−α−ε dx dy . ≤ | − | ZZ 6 PABLOSHMERKINANDVILLESUOMALA The upper bound is independent of n and it is finite whenever s + α x y −s dµ(y) dµ(x) < (2.5) d d | − | ∞ Zx∈[0,1) Zy∈[0,1) whenever s 3. self convolutions Recall that if µ, ν are two finite Borel measures on Rd, then their convolution is the push-down of µ ν under the addition map (x, y) x + y, that is, × 7→ fd(µ ν)= f(x + y)dµ(x)dν(y) ∗ Z Z for all bounded functions f : Rd R. In this section we investigate self-convolutions→ of the random measures µ. It has been known for a long time that the self-convolution µ µ of a singular measure on R can be absolutely continuous, and the density can even∗ be H¨older continuous. K¨orner [13] obtained an optimal estimate in terms of the dimension of (the sup- port of) µ and the H¨older exponent of the self-convolution on R. Recently, Chen and Seeger [3], by adapting K¨orner’s construction, extended this to Rd and self- convolutions of any order. The constructions of K¨orner, and of Chen and Seeger are ad-hoc and in some sense geometrically irregular, for example they are never Ahlfors- regular. In this section we show that a similar result holds for the measures in our class, including fractal percolation. The result may be equivalently stated in terms A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 7 Rd m Rd m of orthogonal projections: denoting by Π: ( ) , (x1,...,xm) i=1 xi the orthogonal projection onto the d-dimensional diagonal→ of (Rd)m, the result→ says that the push-forward µn Π−1 is absolutely continuous with a H¨older density,P whenever α < (m 1)d/m (where◦ α is defined in (2.1)). For fractal percolation sets A in R2, the study− of orthogonal projections was initiated by Rams and Simon [21, 22] and for the planar fractal percolation measure with almost sure dimension > 1 the H¨older continuity of the projections was first addressed by Peres and Rams [20]. Rams and Simon [21] studied the dimension of the sums of m independent copies of the fractal percolation set for any m 2; their methods do not work for convolu- tions of the natural measure. See also [28,≥ 25] for other related recent work. As will be explained below, the study of self-convolutions, especially of order 3, requires new ideas. ≥ Theorem 3.1. Suppose (µn) satisfy (M1)–(M4), and let α be as in (2.1). If α lim log βn+1/ log βn =1, (3.1) n→∞ then, conditioned on µ > 0, the convolution µ µ is almost surely absolutely continuous, and the densityk k is H¨older continuous∗ with exponent γ for any γ < d/2 α. If−α< 2d/3 the same holds for µ µ µ and H¨older exponents ∗ ∗ d 3 α when d/2 α< 2d/3 γ < − 2 ≤ (3.2) 1 (d α) when 0 < α d/2 ( 2 − ≤ Likewise, if α< (m 1)d/m for some 3 < m N, then the m-fold self convolution µ∗m is a.s. absolutely− continuous and the density∈ is H¨older with a quantitative exponent γ = γ(d, m, α). We make some remarks on the statement. The hypotheses hold, in particular, 1 when log n βn α (0,d/2), and more concretely for our classes of examples (i) and (ii). At least→ for∈ double convolutions, the range of H¨older exponents is optimal (up to the critical exponent), in the sense that if µ is any measure supported on a set of Hausdorff dimension d α, then µ µ cannot have a H¨older density of exponent larger than d/2 α, see− [13, 3]. ∗ For clarity of exposition,− we only present the proof in the case of dimension d = 1 and double and triple convolutions; these cases already contain the main ideas of the general case, while being technically much simpler. The proof of the general case can be found in [26, 27]. The proof has a deterministic and a random component; in order to clarify the ideas, we deal with them separately starting with the deterministic result. We further split this into the cases of µ µ and µ µ µ. ∗ ∗ ∗ 8 PABLOSHMERKINANDVILLESUOMALA d Proposition 3.2. Let (µn) be a sequence of measures in [0, 1) satisfying (M1) and (M2) and suppose that µn weakly converges to a non-trivial measure µ. For each u [0, 2]d, define ∈ u d Yn = µn µn d , −1 × H ZΠ (u) where Π: R2d Rd, (x, y) x + y → 7→ d Let 0 < γ 1 and let Γn [0, 2] be δn-dense for each n N, where δn = M −n(γ+d)β−2 .≤ Suppose that for⊂ some C < , ∈ n+1 ∞ e e Y u Y u CM nγ (1 + Y u) (3.3) | n+1 − n | ≤ n e p for each n N and each u Γn. Then µ µ is absolutely continuous and its density is H¨older of∈ any exponent γ∈ < γ. ∗ d Proposition 3.3. Let (µn) be ae sequence of measures in [0, 1) satisfying (M1) and (M2) and suppose that µn weakly converges to a non-trivial measure µ. For each u [0, 3]d, define ∈ u 2d Yn = µn µn µn d , −1 × × H ZΠ (u) where Π: R3d Rd, (x, y, z) x + y + z. → d 7→ N −n(γ+2d) −3 Let γ 1 and Γn [0, 3] be δn-dense for each n , where δn = M βn+1. Suppose≤ that for some⊂ C < , ∈ ∞ e e Y u Y u CM nγ (1 + Y u) | n+1 − n | ≤ n e p for each n N and each u Γn. Then µ µ µ is absolutely continuous and its density is H¨older∈ with any exponent∈ γ < γ. ∗ ∗ We will only give a proof for Propositione 3.2; Proposition 3.3 may be proved with minor modifications. Proof of Proposition 3.2 for d =1. Recall that µ µ is the image of µ µ under the addition map Π: (x, y) x + y, R2 R. We∗ identify Π with the orthogonal× 7→ → projection onto the diagonal line x = y R2 multiplied by a factor √2. Thus, { } ⊂ A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 9 for all x R and r> 0, Fubini’s theorem implies ∈ µ µ(B(x, r))=(µ µ)(Π−1B(x, r)) ∗ × −1 lim inf µn µn(Π (B(x, r))) ≤ n→∞ × 2 2 −1 = lim inf βn (Π (B(x, r)) (An An)) n→∞ L ∩ × (3.4) 1 2 1 −1 = lim inf βn (Π (u) (An An))du √2 n→∞ H ∩ × ZB(x,r) 1 u = lim inf Yn du . √2 n→∞ ZB(x,r) u 1 To complete the proof, it is enough to show that Y := lim −1 µ µ d n→∞ Π (u) n× n H is well defined for all u R, and that u Y u is H¨older continuous. Indeed, ∈ 7→ u R once this is verified, it follows that supn∈N,u∈[0,2] Yn is bounded, and (3.4) yields µ µ(B(x, r)) Cr for all x R. Moreover, replacing the open balls in (3.4) with closed∗ balls, it≤ follows that (3.4)∈ is actually an equality. Furthermore, the density dµ µ(x) equals Y x/√2. In particular, Y x is not zero for all x, since µ is non-trivial. The∗ proof of the H¨older continuity of u Y u relies on the following modulus of u 7→ continuity for u Yn : 7→ ′ Y u Y u 3β2M n u u′ . (3.5) | n − n | ≤ n | − | This follows by elementary geometry. Indeed, for each fixed Q 2 , the map ∈ Qn u 1(Π−1(u) Q) is Lipschitz continuous with Lipschitz constant √2. Since each7→ Π H−1(u) intersects∩ at most 2M n such cubes, the estimate (3.5) follows. −n(γ+1) −2 Since Γn is δn-dense with δn = M βn+1, Equation (3.5) implies that for each ′ u u′ 2 n+1 ′ 1−nγ u [0, 2], there is u Γn such that Yen+1 Yn+1 3βn+1M u u 3M , ∈ u ∈ u′ | − | ≤ | − | ≤ and likewise for Yn Yn . e | − |u Let Xn = 1+supu∈[0,2] Yn . Combining the above with (3.3), and using the triangle inequality for u, we get u u Xn+1 Xn sup Yn+1 Yn | − | ≤ u∈[0,2] | − | 2 n+1 u u 6βn+1M δn + max Yn+1 Yn (3.6) ≤ u∈Γn | − | CM −nγ X , ≤ n for all n N. This implies that X convergese to a finite limit X, so in particular ∈ n X = supn Xn is finite, and that (absorbing X into the constant C) X X X X CM −nγ | n − | ≤ | k+1 − k| ≤ Xk≥n e 10 PABLOSHMERKINANDVILLESUOMALA N u holds for all n . Furthermore, the same is true for Yn , that is, the limit Y u = lim Y u exists∈ for all u R, and n n ∈ Y u Y u CM −nγ , (3.7) | n − | ≤ whenever n N and u [0, 2]. e ∈ ∈ u Define γ0 = 0 and γm+1 = γ/(1 + γ γm). We will show that u Y is H¨older with exponent γ for all m N. Since− lim γ = γ, we thus7→ recover all the m ∈ m→∞ m H¨older exponents γ < γ. Wee proceede by induction on m. Since X is bounded, we have a uniform upper bound for Y u (so we may say that u Y u is H¨older with e ′ u u7→ u ′ γm exponent γ0 = 0). Supposee that u Y is γm-H¨older, i.e. Y Y C u u ′ 7→ | − | ≤ | − | for all u,u [0, 2]. Let Q = [a, b] n. Recalling (3.7) and using the triangle inequality, ∈ ∈ Q b a b b b a a a Yn Yn Yn Y + Y Y + Y Yn | − |≤| − | | − | | − | (3.8) CM −nγ + CM −nγm CM n(1−γm )M −n . ≤ ≤ e −1 (Note that γm < γ for all m.) Since each preimage Π (Q), Q n, consists of 2 ∈ Q halves of cubes in n, either above or below the top-left to bottom-right diagonal, Q u we see that the mape u Yn is linear on each Q n (and this holds also for the M-adic subcubes of [1, 7→2)). Hence we arrive at the∈ following Q key estimate: ′ Y u Y u CM n(1−γm) u u′ whenever u u′ M −n . (3.9) | n − n | ≤ | − | | − | ≤ Note that this improves upon the crude estimate (3.5). Now let u,u′ [0, 2] be ∈ arbitrary and let n N such that M −n(1+γ−γm) < u u′ M −(n−1)(1+γ−γm). Then ∈ | − | ≤ (3.7) and (3.9) yield e e ′ ′ ′ ′ Y u Y u Y u Y u + Y u Y u + Y u Y u | − |≤| − n | | n − n | | n − | CM −nγ + CM n(1−γm) u u′ . ≤ | − | e Recalling the choice of n, both terms are bounded by C u u′ γm+1 . This confirms that u Y u is H¨older continuous with exponent γ , finishing| − | the proof. 7→ m+1 In order to establish Theorem 3.1, it then remains to check that the self-convolutions satisfy the assumptions of Proposition 3.2. For this, we will use a generalization of Hoeffding’s inequality due to Janson [11], which allows for some dependencies among the random variables. Recall that a graph with a vertex set I is a depen- dency graph for the random variables X : i I if whenever i I and J I are { i ∈ } ∈ ⊂ such that there is no edge between i and any element of J, the random variable Xi is independent of X : j J . { j ∈ } A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 11 Lemma 3.4. Let Xi : i I be zero mean random variables uniformly bounded by R> 0, and suppose{ there∈ is a} dependency graph with degree ∆. Then 2 P 2̺ Xi > ̺ 2 exp − 2 . (3.10) ! ≤ (∆+1) I R i∈I | | X The case of double convolutions is substantially simpler than the triple ones, so we present its proof first. N u Proof of Theorem 3.1 for µ µ in the case d =1. For u [0, 2] and n , let Yn be defined as in Proposition∗ 3.2. For each fixed u [0, 2],∈ we will consider∈ several ′ ∈ ′ subfamilies of n n as follows: Let Q denote the unique diagonal cube Q = Q × Q −1 Q Q n n which intersects Π (u). ×Further,∈ Q for× each Q n ℓ N, set ≤ ∈ ℓ = Q 2 Q′ : Q A A , M −(ℓ+1) 1(Π−1(u) Q) M −ℓ . Q { ∈ Qn \{ } ⊂ n × n ≤ H ∩ ≤ } Given Q 2 , let e ∈ Qn 1 XQ = (µn+1 µn+1 µn µn) d . −1 × − × H ZQ∩Π (u) For each ℓ we define a graph on the vertices ℓ as follows: there is an edge between (I J ) and (I J ) (where I ,J ) ifQ and only if I ,J I ,J = ∅. It 1 × 1 2 × 2 i i ∈ Qn { 1 1}∩{ 2 2} 6 follows from (M4) that this is a dependencye graph for (XQ An) ℓ . Elementary | Q∈Q geometry shows that the degree of this graph is at most 8: given I n, there are ∈e Q at most two intervals J n such that u Π(I J) and at most two intervals J ′ such that u Π(∈J ′ Q I). ∈ × ∈ Qn ∈ × On the other hand, E(X A ) = 0 for all Q ℓ by (M3). (The reason we Q| n ∈ Q exclude Q′ is that E(X ′ A ) = 0.) Furthermore, Q | n 6 e X √2β2 M −ℓ | Q| ≤ n+1 for all Q ℓ and ∈ Q # ℓ M ℓ+1β−2Y u , Q ≤ n n u e 2 1 −1 since Yn ℓ βn (Q Π (u)). Applying Lemma 3.4, we get ≥ Q∈Q H ∩ e P e cβ2κ2M ℓ−1 P u n ℓ XQ > κℓ Yn 2 exp − 4 (3.11) ≤ βn+1 Q∈Qℓ X p for any κℓ > 0. To be moree precise, this bound holds conditional on An, but as the upper bound does not actually depend on An, it also holds unconditionally. 12 PABLOSHMERKINANDVILLESUOMALA Pick τ > 0 such that 0 < 2τ < 1/2 α and define − n(α+τ−1/2) εn = M , and ε κ = n . ℓ 2(ℓ n + 2)2 − We note the bounds ∞ κ 1, ℓ ≤ nX=ℓ−1 2 −n ′ X ′ √2β M C ε , (3.12) Q ≤ n+1 ≤ n M ℓ−n(ℓ n + 2)−4 c(ℓ n +2) for ℓ > n 2, − ≥ − − β CM (α+τ/4)n, n ≤ β Cβ M nτ/4, n+1 ≤ n where the last one follows from (3.1). Using this, summing up over all ℓ > n 2, − and noting that ℓ = ∅ for ℓ n 2, we arrive at Q ≤ − P Y u Y u >C′ε + Y uε | n+1 − ne| n n n p P X > κ Y u for some ℓ ≤ Q ℓ n ℓ QX∈Q p 2 2 ℓ−1 e cβnκℓ M 2 exp − 4 ≤ βn+1 ℓ≥Xn−1 2 exp cM −(2α+τ/2)nM −τn/2M 2(α+τ−1/2)nM nM ℓ−n+2(ℓ n + 2)−4 ≤ − − ℓ≥n−1 X 2 exp ( cM τn(ℓ n + 2)) ≤ − − ℓ≥Xn−1 C exp( cM τn) . ≤ − (3.13) We emphasize that the constants c,C,C′ are independent of n,u,ℓ and that C′ −1 is the constant from (3.12). Now let Γn [0, 2] be a δn-dense set, with #Γn 2δn −2 −n(α−τ−3/2) ⊂ ≤ for δn = βn+1M . From (3.13), we derive the following estimate P Y u Y u >C′ε + Y u ε for some u Γ < Cδ−1 exp ( cM τn) | n+1 − n | n n n ∈ n n − CM n(3α+2τ+3/2) exp( cMp τn) C exp( M τn/2), ≤ − ≤ − A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 13 for some C < independent of n. The Borel-Cantelli lemma yields a random N N such that∞ 0 ∈ Y u Y u CM n(α+τ−1/2)(1 + Y u) for all n N ,u Γ . | n+1 − n | ≤ n ≥ 0 ∈ n N Making C larger if necessary (depending onpN0), this holds for all n . Thus we have verified the assumptions of Proposition 3.2 for γ = 1/2 α τ∈. Since τ > 0 is arbitrarily small, this finishes the proof. − − We turn to the case of triple convolutions. e Proof of Theorem 3.1 for µ µ µ in the case d =1. We will give a detailed proof for α (1/2, 2/3) and briefly∗ explain∗ the required changes for 0 < α 1/2 at the end.∈ We will skip the details of calculations that are routine or similar≤ to chose carried our in the course of the proof of the double convolution case. u For u [0, 3], let Yn and Π be defined as in the Proposition 3.3. The main reason why we cannot∈ directly apply the same argument as for µ µ is that independence and the martingale condition break down in a much more severe∗ way. For instance, if we let 2 XQ = (µn+1 µn+1 µn+1 µn µn µn) d , (3.14) −1 × × − × × H ZQ∩Π (u) the random variables (XQi ) are not independent conditional on An, whenever Qi 3 −1 ∈ n have the same projection onto one of the coordinate axes. Since Π (u) intersects planesQ of the form x = c , y = c , z = c along a line, there could be many such { } { 3 } { } dependent cubes contained in An. Regarding the failure of the martingale condition, 3 note that if Q n has two common coordinate projections (e.g. Q = Q1 Q1 Q2 ∈ Q 3 E × × for some Q1, Q2 n), and if Q An, then (XQ An) = 0. Again, unlike in the ∈ Q ⊂ 3 | 6 planar case, there can be many such cubes Q An along the semidiagonals x = y , x = z , y = z . ⊂ { } { To overcome} { these} issues, we will do a joint probabilistic induction in n for our u main quantity of interest Yn and for related, two-dimensional quantities (involving the marginals µn µn) that will allow us to find dependency graphs for the XQ with suitably small× degrees, so that ultimately the general scheme in the proof of the double convolution case can be pushed through. We note that for higher order convolutions, a similar argument still works but involves an even more complicated induction also in the order of the convolution. Let us denote by the family of lines in R2 of the form R x + y = u′′ or 2x + y = u′′ or x +2y = u′′ , 14 PABLOSHMERKINANDVILLESUOMALA for some u′′ [0, 3]. For V , define ∈ ∈ R Y V = µ µ d 1 . n n × n H ZV V Since α > 1/2, the random variablese Yn will no longer be uniformly bounded. However, using ideas similar to the case α< 1/2, we will be able to derive a rather V sharp growth estimate for supV ∈R Yn . Toe this end, define V Xn = 1 + sup Yn . e V ∈R Let γ,ε > 0 be such that e e 2 3α 2γ 4ε> 0. (3.15) − − − We considere a very large parameter L< (which remains fixed for now, but will tend to + later on), and claim that e∞ ∞ P(X > LM (n+1)(2α−1+2ε) X LM n(2α−1+2ε)) C exp( cM nε/2) (3.16) n+1 | n ≤ ≤ − for some 0 (3.18), a calculation similar to (3.11)–(3.13) yields the following estimate for each u [0, 3]: ∈ P Y u Y u > CLβ3 β−2M n(2α−2+2ε) + LM −nγ Y u X LM n(2α−1+2ε) | n+1 − n | n+1 n n | n ≤ 2 exp cLβ5β−6 M n(2−2α−2γ−2ε) , ep ≤ − n n+1 e for some constants 0 < c,C < e independent of n, L and u. Note that here β3 β−2M n(2α−2+3ε) CM −2γn due∞ to (3.1), (3.15) and the definition of α and n+1 n ≤ furthermore, e β5β−6 M n(2−2α−2γ−2ε) cM n(2−3α−2γ−3ε) cM nε n n+1 ≥ ≥ for some 0 P Y V Y V > √2β2 M −n + M n(2α−1+2ε)Y V C exp( cM nε) . n+1 − n n+1 n ≤ − q Note thate becausee we are back in the two-dimensionale situation, there is a depen- dency graph of bounded degree, and the martingale condition E(XQ An) = 0 fails at a single cube. | ′′ Let n consist of the lines in corresponding to the parameter values u = kM −2nR, k N, k 3M 2n. Then #R 9M 2n and thus ∈ ≤ Rn ≤ P Y V Y V > √2β2 M −n + M n(2α−1+2ε)Y V for some V n+1 − n n+1 n ∈ Rn+1 q 2n nε εn/2 CMe exp(e cM ) C exp( cM ) . e ≤ − ≤ − Given V corresponding to a parameter u′′, we can pick another line of the same ∈ R ′′ ′′ ′′ −2(n+1) type V0 n+1 corresponding to u0 with u u0 < M . The Lipschitz bound (3.5) holds∈ R also (with a different constant)| for− the| lines x +2y = u′′, 2x + y = u′′, so we can estimate Y V Y V Y V0 Y V0 + Cβ2 M n+1M −2(n+1). | n+1 − n |≤| n+1 − n | n+1 We can deduce that e e e e P Y V Y V >Z for some V C exp( cM nε/2) , (3.20) n+1 − n V ∈ R ≤ − where e e 2 −n 2 n+1 −2(n+1) n(2α−1+2ε) V ZV = √2βn+1M + Cβn+1M M + M Y n . q If X LM n(2α−1+2ε) (and L is large enough), then n ≤ Z CM n(2α−1+ε) + √LM n(2α−1+2ε) e V ≤ L M (n+1)(2α−1+2ε) M n(2α−1+2ε) , ≤ − using that β2 CM n(2α+ε) . Combining this with (3.20) yields (3.16). n+1 ≤ A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 17 z ℓ (0, 0,u′ u′′) − V ′′ ′′ (0,u , 0) y (0,u′, 0) x Figure 2. Illustration for the application of the coarea formula. The picture shows the plane Π−1(u′), the line ℓ = z = u′ u′′, x + y = u′′ and its projection V ′′ onto the (x, y)-plane,{ along with− the cubes Q1},...,QK and their projections. Lemma 3.6. Conditioned on An, there is a dependency graph for (XQ : Q u,n) −2 n ∈ Q with degree at most CXnβn M , where C is independent of n, An and u. ′ −2 n Moreover, # u,n CXnβn M . Q ≤ e e Proof. We define a graph u,n with vertex set u,n as follows: let Qi = Ii Ji Ki, i =1, 2, with I ,J ,K G . Then there is anQ edge between Q and Q if× and× only i i i ∈ Qn 1 2 if I1,J1,K1 I2,J2,K2 = ∅. It is immediate from (M4) that this is indeed a dependency{ }∩{ graph, so our task} 6 is to bound its degree. ′ Let Q = I1 I2 I3 u,n, with Ii n. If there is an edge from Q to Q × × ∈ Q ∈ Q ′ in u,n, there exist 1 i, j 3 such that the i-th coordinate projection of Q is G ≤ ≤ 1 K Ij. Hence, it is enough to show the following: if Q ,...,Q u,n have a joint −2 n ∈ Q coordinate projection, then K Cβn M Xn. Without loss of generality, we may ≤ i −n assume that each Qi is of the form Q I, where I = [a, a + M ) n is fixed i 2 × ∈ Q and Q n depends on i. e ∈ Q e e 18 PABLOSHMERKINANDVILLESUOMALA Applying the linear coarea formula (see [8, Lemma 1 in Section 3.4] and Figure 2 V for an illustration), and recalling the definition of Yn , we estimate u+3M −n K K e KM −3n = 3 Qi = C 2 Qi x + y + z = u′ du′ L H ∩{ } i=1 ! i=1 ! [ u′=u−Z3M −n [ −n ′ u+3M u −a K = C 1 Qi z + u′′ = u′ , x + y = u′′ du′′ du′ H ∩{ } i=1 ! u−3ZM −n u′−a−ZM −n [ u+3M −n u′−a ′′ Cβ−2 Y V du′′ du′ Cβ−2M −2nX , ≤ n n ≤ n n u′=u−Z3M −n u′′=u′−Za−M −n e e where V ′′ R2 is the line x + y = u′′. Hence, (3.17) follows. ⊂ V R2 Note that here we have used the bound Yn Xn only for lines V of the form x + y = u′′. Since the same estimate holds≤ also for the lines x⊂+2y = u′′, 2x + y = u′′, a similar application of the coareae formulae implies that each Π−1(u) −2 n intersects at most Cβn M Xn cubes in u,n intersecting one of the semidiagonals x = y , x = z or y = z , giving theQ second claim. { } { } { } e 4. Fourier decay, and restriction 4.1. Fourier decay and Salem sets and measures. Let µ be a compactly sup- ported probability measure on Rd. The Fourier transform of µ is defined as µ(ξ)= e−2πix·ξ dµ(x) , ξ Rd . ∈ Z The speed of decay of µ(ξb) as ξ (if any) gives important information about µ. Very roughly speaking, slow or| no|→∞ decay indicates that µ has “arithmetic structure” while fast decay indicatesb “pseudo-randomness”. In more quantitative terms, a first question is whether there is any power decay µ(ξ) C (1 + ξ )−σ/2 (4.1) | | ≤ σ | | for some σ> 0, and if so, what is the largest such σ. The reason for looking at σ/2 (rather than σ) comes from theb following universal upper bound: if the t-energy dµ(x)dµ(y) I (µ)= = , t x y t ∞ ZZ | − | A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 19 then (4.1) cannot hold for any t<σ I (µ)= C ξ t−d µ(ξ) 2 dx. (4.2) t t,d | | | | Z See [17, Lemma 12.12] for the proof of this identity.b In particular, if the topological support of µ has dimension t, then (4.1) cannot hold for any σ > t. These observa- tions lead to the following definition: the Fourier dimension of a measure µ on Rd is dim (µ) = sup σ : µ(ξ) C (1 + ξ )−σ/2 , F | | ≤ σ | | and the Fourier dimension of a set A bRd is ⊂ dim (A) = sup dim (µ): µ(A)=1 , F { F } where the supremum is over Borel probability measures. See [7] for a discussion of the properties of Fourier dimension and some variants. Our previous discussion shows that one always has an inequality dim (A) dim (A). Sets for which F ≤ H dimF (A) = dimH (A) are called Salem sets, as it was Salem [23] who first constructed examples of such sets. Salem sets abound as random sets (see e.g. [15] and references there), but few deterministic fractal Salem sets are known (curved manifolds such as the sphere are Salem sets - this is proved with standard stationary phase methods). Salem sets, therefore, should be thought of as pseudo-random in some sense. The next well-known lemma gives a concrete manifestation of this. Recall that the arithmetic sum of two sets A, B Rd is A + B = x + y : x A, y B . ⊂ { ∈ ∈ } Lemma 4.1. Let A Rd be a Salem set. Then for any Borel set B Rd, ⊂ ⊂ dimH (A + B) = min(dimH (A) + dimH (B),d). Moreover, if dimH (A) + dimH (B) >d, then A + B has positive Lebesgue measure. ′ Proof. Pick t < dimH (A), t < dimH (B). By the definition of Salem set, there is a Borel probability measure µ such that µ(A) = 1 and dimF (µ) > t. By Frostman’s Lemma (see e.g. [17, Theorem 8.8]), we can also find a Borel probability measure ν d with ν(B) = 1, such that It′ (µ) < + . Note that µ ν(R (A + B)) = 0. Using the expression for the energy in terms∞ of the Fourier transform,∗ \ Equation (4.2), we 20 PABLOSHMERKINANDVILLESUOMALA obtain t+t′−d 2 I ′ (µ ν)= C ′ ξ µ[ν(ξ) dξ t+t ∗ t,t ,d | | | ∗ | Z t+t′−d 2 2 = C ′ ξ µ(ξ) ν(ξ) dξ t,t ,d | | | | | | Z ′ t+t′−d −t 2 Ct,t′,dC ξ b ξ bν(ξ) dξ ≤ t | | | | | | ′ Z C ′ C I ′ (ν) < + . ≤ t,t ,d t t ∞ b ′ ′ If t + t < d, the finiteness of the energy implies that dimH (A + B) t + t (see e.g. [17, Theorem 8.7]), while if t + t′ d, then the above calculation≥ together with Parseval’s Theorem show that µ ν ≥has a density in L2, so A + B has positive Lebesgue measure. Letting t dim (A∗), t′ dim (B) finishes the proof. ↑ H ↑ H 4.2. Fourier decay of the random measures. There is no universally agreed definition of Salem measures (partly because there are many notions of dimension one could use). However, it is clear from the previous discussion that if a measure σ/2 satisfies µ(ξ) Cσ ξ , then σ dimH (supp(µ)). We will see that for the class of random| measures| ≤ | studied| in this≤ article, this holds for any σ [15], and there are strong parallels with the proof of Theorem 3.1 in the simpler double-convolution case. Proof of Theorem 4.2 in the case d =1. Fix σ < 1 α. It is enough to show that −σ/2 − µ(k) Cσ k for k Z, as this implies decay over real frequencies, see [30, Lemma≤ 9A4].| | ∈ b For fixed k, we have µ (k) µ (k)= X , n+1 − n Q QX∈Qn where b b X = (µ (x) µ (x)) exp( 2πikx) dx . Q n+1 − n − ZQ Then X Cβ M −n. Let N be the number of cubes making up A . Using | Q| ≤ n+1 n n Lemma 3.4 (with ∆ = 0) for the real and imaginary parts of XQ and property (M2), we obtain P µ (k) µ (k) > M −σn/2 µ 1/2 A C exp cM (2−σ)n µ β−2 N −1 | n+1 − n | k nk | n ≤ − k nk n+1 n (1−σ)n −2 C exp( cM βnβn+1) . b b ≤ − Since lim inf log β / log β−2 α and σ < 1 α, we get n M n M n+1 ≥ − − P µ (k) µ (k) > M −σn/2 µ 1/2 for some k < M n+1 ε , (4.3) | n+1 − n | k nk | | ≤ n n (1−σ)n −2 where ε n C M exp( cM βnβn+1) is summable. ≤ − −n−1 Let Q b n+1. Sinceb the endpoints of the interval Q are of the form jM , it follows that∈ Q for k < M n+1, 0 = ℓ Z, | | 6 ∈ k 1 (k + M n+1ℓ)= 1 (k) . Q k + M n+1ℓ Q Since µn+1 is a linearb combination of the functionsb1Q, the same relation holds n+1 between µn+1(k + M ℓ) and µn+1(k) (and this holds also for µn in place of µn+1). ′ ′ n+1 ′ n+1 n+1 Fix k bwith k M , and write k = k + M ℓ, whereb k < M , kℓ 0, and ℓ = 0. For such| | ≥k′, we have | | ≥ 6 b b b b k µ (k′) µ (k′) | | µ (k) µ (k) | n+1 − n | ≤ k + M n+1ℓ | n+1 − n | | | M n+1 b b < µ (kb) µ (k) b. k′ | n+1 − n | | | Combining this with (4.3), we arrive at the following key fact: P(E ) < ε , where b b n n En is the event M n+1 µ (k) µ (k) > µ 1/2 min 1, M −σn/2 for some k Z . | n+1 − n | k nk k ∈ | | b b 22 PABLOSHMERKINANDVILLESUOMALA Since ε was summable, and µ is a.s. bounded, it follows from the Borel-Cantelli n k nk Lemma that a.s. there are C and n0 such that M n+1 µ (k) µ (k) C min 1, M −σn/2 for all k Z, n n . | n+1 − n | ≤ k ∈ ≥ 0 | | Thus, choosing n N such that M n1 k < M n1+1, and telescoping, we have b 1b∈ ≤| | µ (k) µ (k) 2CM n(1−σ/2) k −1 + CM −nσ/2 | m − n0 | ≤ | | n0≤n≤n1 X max(n1X,n0) fdµ p d C f q . (4.5) k kL (L ) ≤ p,q,µk kL (µ) The name comes from the dual formulation of (4.5), namely d ′ f q′ C f p′ d . k kL (µ) ≤ p,q,µk kL (L ) In other words, the goal is to understand for which functions f it is meaningful to restrict f to the support of µ.b Classically, the restriction problem has been studied for surfaces such as the sphere or the paraboloid. In the case of the sphere and q = 2, ab famous theorem of Stein and Tomas gives the sharp range of p for which (4.5) holds: p (2d + 2)/(d 1). The problem of finding all the pairs (p, q) is still open and is closely≥ connected− to other well-known problems such as the Kakeya problem. The study of the restriction problem for fractal measures was initiated by Mockenhaupt [19] (see also [18]), who extended the Stein-Tomas argument to very general measures satisfying suitable mass and Fourier decay: if a measure µ on Rd satisfies µ(B(x, r)) C rs , (4.6) ≤ 1 µ(ξ) C (1 + ξ )−σ/2 , (4.7) | | ≤ 2 | | then (4.5) with q = 2 holds whenever b 2(2d 2s + σ) p>p = − . s,σ,d σ Bak and Seeger [1] proved that (4.5) also holds at the endpoint p = ps,σ,d. (Note that taking s = σ = d 1, this recovers the Stein-Tomas estimate in the case of the sphere). Hambrook and− Laba [9] (see also [5] for a generalization) constructed, for a dense set of t [0, 1], measures µ on the real line satisfying (4.6) and (4.7) for ∈ A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 23 s, σ arbitrarily close t, and supported on sets of Hausdorff dimension t, for which the restriction estimate (4.5) does not hold for any p 5. Arithmetic structure A basic problem in additive combinatorics is to understand what properties of a set A 1,...,N imply that A has an arithmetic progression of a given length k. If A⊂ { δN, and} N is sufficiently large in terms of k, then Szemer´edi’s famous | | ≥ 24 PABLOSHMERKINANDVILLESUOMALA theorem asserts that A does contain a progression of length k. On the other hand, as we will see below, for every ε> 0 and large enough N, there are sets of size N 1−ε that do not even contain progressions of length 3. It is natural to ask similar questions for subsets of [0, 1]. A simple application of the Lebesgue density theorem shows that if A has positive Lebesgue measure, then A contains arithmetic progressions of any length and, more generally, con- tains an homothetic copy of any finite subset of R. On the other hand, Keleti [12] constructed a compact set of Hausdorff dimension 1 that does not contain any rect- angles x, x + r,y,y + r, and in particular contains no progressions of length 3. In a different direction, Davis, Marstrand and Taylor [6] constructed a compact set of zero Hausdorff dimension which contains a similar copy of all finite sets. It then appears that Hausdorff dimension by itself is insufficient to detect the presence, or lack thereof, of finite patterns. In the discrete setting, it is well-known that Fourier uniformity is enough to guarantee the existence of three-term arithmetic progressions even for rather sparse (although not extremely sparse) sets. It is natural to ask whether a similar result holds for subsets of [0, 1]. I.Laba and M. Pramanik [15] proved a result in this direction: Theorem 5.1 (Laba and Pramanik). Given C1,C2 > 0,σ > 0, there exists ε0 = ε0(C1,C2, σ) > 0 such that the following holds: if s> 1 ε0, and µ is a measure on [0, 1] such that − s (i) µ(x, x + r) C1r for all x R/Z and all r (0, 1), (ii) µ(ξ) C ≤ξ −σ/2 for all ξ =0∈ , ∈ | | ≤ 2| | 6 then supp(µ) contains a 3-term arithmetic progression. In fact,b the original result from [15] requires σ > 2/3 in the Fourier decay as- sumption, while the relaxation to any σ > 0 is due to [10, Theorem 10.1], where generalizations to certain polynomial patterns and higher dimensions are also ob- tained. We emphasize that the mass decay must be fast enough not just in terms of the Fourier decay, but also in terms of the constants C1,C2. This makes the assump- tions hard to verify in practice. This then leads to the question of whether the mass and Fourier decay conditions (i), (ii) with sufficiently large s,σ < 1 (independent of the constants) are enough to guarantee the presence of arithmetic progressions in the support of µ. Results such as Lemma 4.1 show that, for certain problems, it is only the polynomial exponent of decay that matters, and not the constant. This suggests the question of whether the dependence of ε0 on the constants (and particularly on C1) is really needed in Theorem 5.1. Recently, the first author [24] used a random construction closely related to the class studied in this note to show that the answer is yes, in a strong sense: A CLASS OF RANDOM CANTOR MEASURES, WITH APPLICATIONS 25 Theorem 5.2. For every s (0, 1) there exists a Borel probability measure µ on [0, 1] such that: ∈ (i) µ(x, x+r) Crs for all x [0, 1] and all r (0, 1), and some C > 0 that depends on s. ≤ ∈ ∈ −σ/2 (ii) For all σ (0,s), there exists a constant Cσ > 0 such that µ(ξ) Cσ ξ for all ξ =0, ∈ | | ≤ | | 6 (iii) The topological support of µ does not contain any arithmeticb progressions of length 3. In fact, an even stronger statement is obtained: one can additionally choose µ to either be Ahlfors-regular, or to satisfy the first condition for all s< 1 (with the constant C depending on s). Proof of Theorem 5.2. We will construct measures supported on sets of dimension arbitrarily close to 1, and leave to the reader the small modifications needed to construct measures of arbitrary dimension, or see [24]. Let ZM be the classes of residues modulo M. A classical example due to Behrend [2] shows the existence of a set E 0, 1,...,M 1 with no 3-term arithmetic progressions and size at least exp(⊂c√ { log M)M.− An} easy argument (see [24] for − details) shows that one can do the same with E ZM , where moreover M is even and all the elements of E are even as well. In particular,⊂ given ε> 0, we can fix a large enough even number M and a set E ZM with no progressions, all elements even, and #E M 1−ε. ⊂ Now given Q≥ and the base M, we can label the set of offspring intervals ∈ Qn (Q) by 0, 1,...,M 1 , and in turn identify this with ZM . Let aQ : Q S , n N{ be i.i.d. random− } variables chosen uniformly in 0, 1,...,M{ 1 , and∈ Qn ∈ } { − } set SQ = E + aQ mod M. We can then carry out the construction described in n Section 2, to obtain a sequence (µn) satisfying (M1)–(M4) with βn = #E . Note that E +aQ does not contain progressions as a subset of ZM , and hence also a subset of 0, 1,...,M 1 . { − } Let µ be the limit measure. Since #SQ = #E is deterministic and constant, it is easy to see that (i) holds with s = logM #E (1 ε, 1). The Fourier decay (ii) is direct from Theorem 4.2. ∈ − Let us see, then, that A = supp(µ) does not contain any 3-term progressions. Suppose to the contrary that y1 < y2 < y3 A is such a progression. Let Q be a minimal M-adic interval{ with y ,y ,y } ⊂ Q. By the self-similarity of the { 1 2 3} ⊂ construction, we may assume that Q = [0, 1). Write yi = xi + δi, where xi is the left-endpoint of the M-adic interval containing yi, and δi [0, 1/M). From 1 ∈ y2 = 2 (y1 + y3), we get 1 1 (δ + δ ) δ = x (x + x ) 2 1 3 − 2 2 − 2 1 3 26 PABLOSHMERKINANDVILLESUOMALA The left hand-side is at most 1/M. On the other hand, by construction the right- hand size is not zero (since x1, x2, x3 are not all equal and do not form an arithmetic progression). Also, the numbers Mxi are either all even or all odd, so the right-hand side is at least 1/M. Hence the only option is that both the left and right-hand sides are equal to 1/M, but this forces all of the yi to be endpoints of M-adic intervals, which almost surely does not happen (since a fixed point has probability zero of belonging to A, and there are countably many such endpoints). This contradiction finishes the proof. We finish the article by commenting on the opposite problem of finding sets with many patterns. As mentioned before, it follows from [6] that there are compact subsets of the real line of zero Hausdorff dimension which contain an homothetic copy of all finite sets. However, such a set necessarily has packing and box-counting dimension 1: Lemma 5.3. If A R contains a homothetic image of all m-element sets, then m ⊂ dimH (A ) m 2, and dimP (A) 1 2/m, where dimP denotes packing dimen- sion. ≥ − ≥ − m Proof. Let Xm = (x1,...,xm) R : x1 <... −1 to do a similar study for the fibers Π (t3,...,tm), where Π is as in the proof of Lemma 5.3. Additional complications are caused by the non-linearity of Π. Also note that Theorem 5.2 shows that one cannot hope to have a similar result of the same generality as Theorem 3.1. References [1] Jong-Guk Bak and Andreas Seeger. Extensions of the Stein-Tomas theorem. Math. Res. Lett., 18(4):767–781, 2011. [2] F. A. Behrend. 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In Fractals, wavelets, and their applications, volume 92 of Springer Proc. Math. Stat., pages 175–190. Springer, Cham, 2014. [29] Claude Tricot, Jr. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc., 91(1):57–74, 1982. [30] Thomas H. Wolff. Lectures on harmonic analysis, volume 29 of University Lecture Series. American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and preface by IzabellaLaba, Edited byLaba and Carol Shubin. Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina E-mail address: [email protected] URL: http://www.utdt.edu/profesores/pshmerkin Department of Mathematical Sciences, University of Oulu, Finland E-mail address: [email protected] URL: http://cc.oulu.fi/~vsuomala/