On arithmetic sums of Ahlfors-regular sets
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ On arithmetic sums of Ahlfors-regular sets © 2022 the Authors Published version Orponen, Tuomas Orponen, T. (2022). On arithmetic sums of Ahlfors-regular sets. Geometric and Functional Analysis, 32(1), 81-134. https://doi.org/10.1007/s00039-021-00589-x 2022
Geom. Funct. Anal. GAFA Geometric And Functional Analysis https://doi.org/10.1007/s00039-021-00589-x c 2022 The Author(s) ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS Tuomas Orponen Abstract. Let A, B ⊂Rbe closed Ahlfors-regular sets with dimensions dimHA=: αand dimHB=: β. I prove that dimH[A+θB]≥α+β·1−α 2−α for all θ∈R\E, where dimHE=0. Contents 1 Introduction ....................................... 2 1.1 Proof outline. ................................... 5 1.2 Notation. ..................................... 7 2 Some Definitions .................................... 7 3 Technical Version of the Main Theorem ....................... 10 3.1 Proof of Theorem 3.6. .............................. 14 4 Proof of Proposition 3.18 ............................... 15 4.1 Choosing constants. ............................... 17 4.2 The counter assumption. ............................. 18 4.3 Pigeonholing a branching scale for ν....................... 19 4.4 High multiplicity sets at scale ¯ δ......................... 24 4.5 Removing very high multiplicity subsets. .................... 27 4.6 The geometry of the set ¯ K............................ 30 4.7 Statement of Shmerkin’s inverse theorem. ................... 35 4.8 Deriving a product structure from ¯ K0...................... 36 4.9 From projections to convolutions. ........................ 40 4.10 Applying Shmerkin’s inverse theorem. ..................... 42 4.11 Lower bounds for branching numbers. ..................... 44 5 Proof of Proposition 4.32 ............................... 48 References .......................................... 53 T.O. is supported by the Academy of Finland via the projects Quantitative rectifiability in Euclidean and non-Euclidean spaces and Incidences on Fractals, Grant Nos. 309365, 314172, 321896. T.O. is also supported by the University of Helsinki via the project Quantitative rectifiability of sets and measures in Euclidean spaces and Heisenberg groups, Project No. 7516125 Keywords and phrases: Ahlfors-regular sets, Sum-product problem, Hausdorff dimension Mathematics Subject Classification: 11B30 Primary; 28A80 Secondary
2T.ORPONENGAFA 1 Introduction This paper contains the following sum-product result for Ahlfors-regular subsets of R: Theorem 1.1. Let A, B ⊂Rbe closed Ahlfors-regular sets with dimHA=α∈ [0,1] and dimHB=β∈[0,1].Then dimH[A+θB]≥α+β·1−α 2−α for all θ∈R\E, where dimHE=0. The paper also contains a δ-discretised version of Theorem 1.1, see Theorem 3.6, and Remark 3.8. I will next review other statements of similar nature in previous literature. Three close relatives are Bourgain’s discretised sum-product theorem [4], Dyatlov and Zahl’s estimate [6] for the additive energy of Ahlfors-regular sets, and Hochman’s projection theorem [9] for self-similar sets. Bourgainin[4], see also his earlier paper [3], proved lower bounds for the dimension of A+θB without assuming that A, B are Ahlfors-regular. He showed that if α=β∈(0,1), and E⊂Rhas dimension dimHE≥κ>0, then dimH[A+θB]≥α+(1.2) for some θ∈E, where >0 only depends on α=β,andκ>0. Without the Ahlforsregularity assumption, the dependence of “”on“κ”in(1.2) is inevitable, see the next example. As Theorem 1.1 shows, the dependence disappears for Ahlfors-regular sets. Example 1.3. Fix n∈Nand κ∈(0,1 2] such that n2κ∈N.LetAn:= {k/n :1≤ k≤n}and En:= {k/n2κ:1≤k≤n2κ}.ThenAn+EnAn⊂{k/n1+2κ:1≤k≤ 2n1+2κ}.Thus,ifδ∈(0,1), and Nδ(A) is the δ-covering number of A,wehave Nδ(An+EnAn)≤2n1+2κ,c∈En.(1.4) For 0 <δ≤n−1−2κ, the same inequality remains true, up to an absolute constant, if we replace An,E nby the δ-neighbourhoods An(δ)andEn(δ). Now, choose δ:= n−2. Then, if Hs ∞stands for s-dimensional Hausdorff content, it is easy to check that H1/2 ∞(An(n−2)) 1andHκ ∞(En(n−2)) 1. Therefore, informally speaking, An(n−2) is the n−2-neighbourhood of a 1 2-dimensio- nal set, and En(n−2) is the n−2-neighbourhood of a κ-dimensional set. Noting that n1+2κ=(n−2)1/2+κ, the inequality (1.4) roughly says that An(n−2)+En(n−2)An (n−2) is at most (1 2+κ)-dimensional. The construction described above can be “iterated” to produce compact, Cantor type, non-Ahlfors-regular sets A, Eκ⊂[0,1] such that dimHA=1 2, and dimHEκ= κ, and dimH(A+EκA)≤1 2+κ. In particular, limκ0dim(A+EκA)=dimA. This shows that the Ahlfors-regularity assumption in Theorem 1.1 is necessary, even when A=B. The “iteration” procedure is explained further in [13, Example 4.1] in a slightly different, but very similar, context.
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 3 We have now seen that the size of “”in(1.2) depends on “κ”, and this dependence vanishes for Ahlfors-regular sets (as long as κ>0). Theorem 1.1 has another key difference compared to Bourgain’s result: Bourgain initially proved (1.2)only in the case α=dim HA=dim HB=β, while Theorem 1.1 makes no assumptions on the relative sizes of αand β. A variant of Bourgain’s result (1.2) (for general sets) is simply false in the total absence of such assumptions: it is not difficult to construct (non-Ahlfors-regular) sets A, B ⊂[0,1] of dimensions dimHA=1 2and dimHB=1 4such that dimH[A+θB]=1 2for all θ∈E, where dimHE=1 4. The construction is based on arithmetic progressions, just like Example 1.3.If An={k/√n:1≤k≤√n}and Bn=En={k/ 4 √n:1≤k≤4 √n}for some n∈Nsuch that 4 √n∈N, then •H 1/2 ∞(An(n−1)) ∼1, •H 1/4 ∞(Bn(n−1)) = H1/4 ∞(En(n−1)) ∼1, and •Nn−1(An(n−1)+En(n−1)Bn(n−1)) ∼√n=Nn−1(An(n−1)). The last point is the scale n−1analogue of the equation dimH(A+EB)=dim HA. Again, an iterative construction as in [13, Example 4.1] is required to pass from the approximating sets An,E n,B nto A, E, B. While Bourgain’s result (1.2) was initially proven only in the case α=β, the case α=βhas been recently studied in [17]. Based on evidence in finite fields [18], the following conjecture seems plausible for general sets: Conjecture 1.5. Let κ>0, and let A, B, E ⊂[0,1] be compact sets with dimHB≥ κ,dimHB+dim HE≥dimHA+κ,anddimHA≤1−κ.Then(1.2)holds for some θ∈E, and for some >0depending only on κ. To understand why the lower bound in Theorem 1.1 might be plausible—while a similar statement for non-Ahlfors-regular sets fails in multiple ways—one should next compare Theorem 1.1 to the bound of Dyatlov and Zahl [6] on the additive energy of Ahlfors-regular sets. In [6, Theorem 6], it was shown that if A⊂[0,1] is an Ahlfors-regular set with dimHA=α∈(0,1), then there exists a constant >0, which depends on both “α” and the Ahlfors-regularity constant of A, such that (Hα)4({(x1,x 2,x 3,x 4)∈A4:|(x1+x2)−(x3+x4)|<δ})δα+,0<δ<1. For more recent improvements and generalisations of the work of Dyatlov and Zahl, see the papers [20] by Rossi and Shmerkin, and [5] by Cladek and Tao. It follows from the estimate above that dimH[A+A]≥α+. More generally, one can show that dimH[A+θA]≥α+for every θ∈[1 2,1], where only depends on α∈(0,1), and the regularity constant of A. Does this imply that the zero-dimensional exceptional set mentioned in Theorem 1.1 is not really necessary? No, because the constant “” in the uniform lower bound essentially depends on the regularity constant of “A”. Example 1.6. Consider a 1 2-dimensional self-similar set ANon [0,1] with “N”generating intervals of common length N−2, placed in arithmetic progression. Then,
4T.ORPONENGAFA ANis 1 2-Ahlfors-regular for every N≥1, but the constant increases when N→∞, and also dimH[AN+AN]→1 2as N→∞. Regardless, Theorem 1.1 shows that dimH[AN+θAN]≥2 3for every θ∈[0,1]\E, where dimHE=0. Even though the largest “”indim H[A+A]≥α+depends on the Ahlforsregularity constant of A, it is known that A+A+···+Acontains an interval, and in particular dimH[A+···+A] = 1, if the number of copies of Ain the sum is sufficiently large, depending on the dimension and Ahlfors-regularity constant of A. This is due to Astels [1], and a recent higher-dimensional generalisation is due to Feng and Wu [8] (Ahlfors-regularity is not sufficient in Rd, but one rather needs to assume that Ahas positive thickness, which is implied by Ahlfors-regularity for A⊂R.) While these results of Astels or Feng-Wu are not used in the proof of Theorem 1.1, an idea of “iteration” is also present, see Section 1.1, and enables one to upgrade -improvements to more substantial ones, at the cost of throwing away a zero-dimensional set of exceptional directions E⊂R. The examples above indicate that Theorem 1.1 illustrates a phenomenon slightly distinct from those discussed by Bourgain, and Dyatlov-Zahl. In some ways, a closer relative is Hochman’s projection theorem [9] for self-similar sets: a special case of his result shows that if A, B ⊂Rare self-similar sets with common contraction ratios, then dimH[A+θB]=min{dimHA+dim HB,1} for all θ∈R\E, where dimHE= 0. In fact, even the packing dimension of Eis zero. The assumption about common contraction ratios is needed to ensure that A×Bis also a self-similar set: this is relevant, because Hochman’s projection theorem states, generally, that if K⊂R2is a self-similar set without rotations, then dimHπθ(K)= min{dimHK, 1}for all θ∈R\E, where dim E=0,andπθ(x, y):=x+θy. I briefly mention that the case with (dense) rotations can also be handled with different methods, see [19]. More generally, the product of self-similar sets is selfaffine, and there are numerous papers containing strong projection theorems for self-affine sets and measures, see [2,7,10,19]. The starting point of the current paper was an attempt to generalise Hochman’s projection theorem from self-similar sets K⊂R2to Ahlfors-regular sets K⊂R2. The attempt failed in various ways, and Theorem 1.1 was the best outcome I could recover. The initial goal still seems plausible, however, so I pose it as a question: Question 1. Let K⊂R2be a closed Ahlfors-regular set. Is it true that dimHπθ(K) =min{dimHK, 1}for all θ∈R\E,wheredimHE=0? An affirmative answer to Question 1would be an improvement over Marstrand’s classical projection theorem [11] for Ahlfors-regular sets. Partial evidence for Question 1can be found in [16], where an analogous result is proved for Assouad dimension in place of Hausdorff dimension. (The result for Assouad dimension does not require “K” to be Ahlfors-regular, which is essentially due to the possibility to shift attention from “K” to its more regular tangents.)
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 5 1.1 Proof outline. The uniform lower bound dimH[A+A]≥dimHA+for Ahlfors-regular sets, discussed above, can be deduced from certain inverse theorems in additive combinatorics; there are several of these, for example by Hochman [9], Sanders [21], and most recently by Shmerkin [22]. Similar ideas are also present in the works of Bourgain [3,4]. To be precise, Dyatlov-Zahl [6] applied Sanders’s theorem [21], while Rossi-Shmerkin used Shmerkin’s inverse theorem [22]. I deliberately wrote “can” in the opening of the section, since Cladek-Tao [5] avoid the use of inverse theorems. The idea of inverse theorems is, briefly, the following: if dimH[A+B]≤dimHA+ for a sufficiently small constant >0, then Aneeds to have 1-dimensional branching on scales where Bhas positive-dimensional branching.Ifdim HB>0, then B necessarily has positive-dimensional branching on many scales, and hence Ahas 1-dimensional branching on many scales. However, if Ais α-Ahlfors-regular with α<1, then Acannot have 1-dimensional branching on any scales. If follows that dimH[A+B]≥dimHA+. As we have seen in Example 1.6, there is an inevitable dependence between the Ahlfors-regularity constant of A, and the number “”. This dependence is hidden in the argument above: the statement that “Acannot have 1-dimensional branching on any scales” is only true, if the scales are chosen appropriately: the leaps between consecutive scales need to be big enough, depending on the regularity constant of A.Theleap size is one of the parameters in the inverse theorem (of either Hochman or Shmerkin), and the theorems only become applicable if the threshold “” in the inequality dimH[A+B]≤dimHA+is small enough, depending on the leap size. To prove Theorem 1.1, a bootstrapping scheme is required. The following outline is not in 1-to-1 correspondence with what really happens in the proof, but I hope that some ideas are transmitted. The main idea is to assume “inductively” that one has already established a weaker version of Theorem 1.1: one has already managed to show that dimH[A+θB]≥α+χ(1.7) for a certain parameter χ≥0, for all pairs of Ahlfors-regular sets A, B with fixed dimensions α, β, and regularity constants CA,C B, and for all θ∈[0,1] outside a tiny exceptional set E⊂[0,1]. The set Emay naturally depend on the choices of A, B. In the “base case” χ= 0 (or alternatively χ=, depending on CA), one simply has E=∅. To quantify the “tininess” of Ein general, one formulates a δ-discretised version of the theorem, so E=Eδ. Then, one shows that Hτ ∞(E)≤δη, where τ>0 can be chosen arbitrarily small (one eventually wants to let τ→0toestablishthe 0-dimensionality of exceptions), and η>0 is an auxiliary parameter. For the details of how to set this up, see Theorem 3.6. Next, one makes a counter assumption: there exists a pair of Ahlfors-regular sets A, B, with exactly the same parameters α, β, CA,C Bas above, such that dimH[A+θB]≤α+χ+with χ+<β·1−α 2−α(1.8)
6T.ORPONENGAFA for all θ∈E⊂[0,1], where, roughly speaking, Hτ ∞(E)≥δη/2. This means that Eis allowed to be fairly tiny, but is assumed to be substantially larger than the “tininess” of Ein the inductive hypothesis. Hence, combining (1.7)and(1.8), one finds that α+χ≤dimH[A+θB]≤α+χ+(1.9) for a typical direction θ∈E.Now,fixθ0∈Esuch that (1.9) holds, and view A×B as a subset of [A+θ0B]×R=: A×R(after a change of coordinates). Then Ahas the following two properties: (1) α+χ≤dimHA≤α+χ+, (2) Ais a union of translates of A. This implies that Ahas ≥α-dimensional branching on all scales. For a clarification of what this means, see the statement of Theorem 4.76, and the remark after it. Recalling that A×B⊂A×R, up to a change of coordinates, one more precisely rewrites A×B=A×B, where Brepresents the intersection of A×Bwith the “typical fibre” under the map πθ0(x, y)=x+θ0y. From the upper bound in (1.9), one infers that dimHB≥dimH(A×B)−(α+χ+)=β−χ−. Recall from (1.8) that χ+<β(1 −α)/(2 −α), so in fact dimHB≥β/(2 −α). Next, fix another direction θ1∈Esuch that |θ0−θ1|∼1. Then, writing θ:= θ1−θ0, one roughly observes that A+θ1B=A+θB, so in particular dimH(A+θB)=dim H(A+θ1B)(1.9) ≤α+χ+(1) ≤dimHA+. (1.10) This places us in a position to use Shmerkin’s inverse theorem, as described in the second paragraph of this section. See Theorem 4.76 for the precise statement of Shmerkin’s result. The leap size (hence the choice of ) will only depend on CA.The conclusion is that if >0 is small enough, then Ahas 1-dimensional branching on all the scales where Bhas positive-dimensional branching. The fraction of scales such that Bhas positive-dimensional branching is at least dimHB≥β−χ−. Consequently Ahas 1-dimensional branching on at least that many scales. On the other hand, property (2) of the set Asays that Ahas ≥α-dimensional branching on all scales (this is where the dependence between the leap size and the Ahlfors-regularity of Acomes in). Ignoring “”, this allows us to compute the following lower bound on the dimension of A: α+χ(1) ≥dimHA≥1·dimHB+α·(1 −dimHB)≥(β−χ)+α·(1 −β+χ). Rearranging, we find that χ≥β(1 −α)/(2 −α), which contradicts (1.8). Hence the counter assumption in (1.8) must be false, and in fact dimH[A+θB]≥α+χ+ for some θ∈E. Repeating this argument ∼−1times will eventually conclude the proof of Theorem 1.1.
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 7 1.2 Notation. Open balls in either Rdwill be denoted B(x, r); in this paper always d∈{1,2}.Iff,g are real-valued non-negative functions of some parameter x∈X, the notation fgmeans that there exists an absolute constant C≥1 such that f(x)≤Cg(x) for all x∈X.IfA⊂Rdis a bounded set, and δ>0, then Nδ(A) refers to the smallest number of balls B(x, δ)⊂Rdrequired to cover A.IfAis a finite set, its cardinality is denoted |A|.Forδ>0, the open Euclidean δ-neighbourhood of a set A⊂Rdis denoted A(δ). For δ>0, a tube of width δ>0is a set of the form π−1 θ(I), where πθ(x, y)=x+θy,θ∈[0,1], and I⊂Ris an interval of length δ. Note that if T=π−1 θ[x−δ/2,x+δ/2] ⊂R2is a tube of width δ, where θ∈[0,1], then (δ/4) ⊂T⊂(δ/2) with := π−1 θ{x}. 2 Some Definitions We start by recalling the notion of entropy. If μis a probability measure on some space Ω, and Fis a μmeasurable partition of Ω, the F-entropy of μis defined by H(μ, F):= F∈F μ(F)log 1 μ(F). Here 0 ·log 1 0:= 0, and “log” refers to logarithm in base 2. Typically, the measures of interest are probability measures R,andFis the partition into dyadic intervals of length δ>0; this partition is denoted by Dδ:= {[kδ,(k+1)δ):k∈Z}. In addition to F-entropy, we will also need the following conditional entropy of μ: H(μ, F|E):= E∈E μ(E)H(μE,F). Here μE:= μ(E)−1·μ|E,andE,Fare μmeasurable partitions of Ω. In the applications below, every E∈Eis a finite union of certain sets in F. In this special case, the conditional entropy can be rewritten as H(μ, F|E)=H(μ, F)−H(μ, E),(2.1) as an easy calculation shows (or see [14, Proposition 3.3]). The deepest fact we will need to know about entropy is the estimate H(μ, F)≤log |F|, which is an immediate consequence of Jensen’s inequality. We then proceed to other definitions. Definition 2.2 (Projections). For θ∈R, we define the maps πθ:R2→Rby πθ(x, y):=x+θy, (x, y)∈R2. Definition 2.3. Let K⊂R2, let 0 <r≤R≤∞, and let x∈K.Forθ∈[0,1], we define the following multiplicity number: mK,θ(x|[r, R]) := Nr(B(x, R)∩Kr∩π−1 θ{πθ(x)}).
8T.ORPONENGAFA Here Krrefers to the r-neighbourhood of K.Thus,mK,θ(x|[r, R]) keeps track of the (smallest) number of r-balls needed to cover the intersection between B(x, R)∩ Krand the line π−1 θ{πθ(x)}. Often the set “K” is clear from the context, and we abbreviate mK,θ =: mθ. We also allow for the case R=∞: then B(x, R):=R2. Definition 2.4 (High multiplicity sets). Let 0 <r≤R≤∞,M>0, and let θ∈[0,1]. For K⊂R2, we define the high multiplicity set Hθ(K, M, [r, R]) := {x∈K:mK,θ(x|[r, R]) ≥M}. Note that the same latter “H” will stand for both high-multiplicity sets, and entropy; the correct interpretation should always be clear from context. We will next verify some elementary but useful facts about the multiplicity numbers and high multiplicity sets. The first observation is that M≥M>0=⇒Hθ(K, M,[r, R]) ⊂Hθ(K, M, [r, R]) (2.5) for all θ,K,r ≤R, since if mK,θ(x|[r, R]) ≥M, then also mK,θ(x|[r, R]) ≥M. The next lemma answers the questions: what happens to the multiplicity numbers and high multiplicity sets if we change the radii rand R? Lemma 2.6. Let K⊂R2,x∈K,θ∈S1,C≥1, and assume that Cr ≤R. Then, mK,θ(x|[r, R]) ≤C·mK,θ(x|[Cr,R]) and mK,θ(x|[r, R]) ≤mK,θ(x|[r, CR]).(2.7) In particular, Hθ(M,[r, R]) ⊂Hθ(M C,[Cr,R]) and Hθ(M,[r, R]) ⊂Hθ(M,[r, CR]) (2.8) for all M>0. Here we abbreviate Hθ(...):=Hθ(K,...). Proof. Clearly Kr⊂KCr,so mK,θ(x|[r, R]) ≤Nr(B(x, R)∩KCr ∩π−1 θ{πθ(x)}). The right hand side above only differs from mK,θ(x|[Cr,R]) in the appearance of “Nr” instead of “NCr”. But evidently Nr(A∩π−1 θ{t})≤C·NCr(A∩π−1 θ{t}),A⊂R2, and we have proven the first inequality in (2.7). The second inequality is a restatement of Nr(B(x, R)∩Kr∩π−1 θ{πθ(x)})≤Nr(B(x, CR)∩Kr∩π−1 θ{πθ(x)}). The inclusions in (2.8) are direct consequences of the inequalities in (2.7).
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 15 such that the following holds for all product measures μ=μα×μβas in (A1)–(A2), and for all 0 <δ≤Δ0: Hτ ∞({θ∈[0,1] : μ(B(1) ∩Hθ(spt μ, δ−σ,[δ, 1])) ≥δη0})≤δη0.(3.20) We claim that in fact Σ = β/(2 −α) for all τ>0, which will prove Theorem 3.6. Assume to the contrary that Σ >β/(2 −α). Then, let σ>Σ be such that σ−ζ<Σ, where ζ:= ζ(α, β, Cα,σ,τ) is the parameter appearing in Proposition 3.18. This can be done, since ζis bounded away from zero when σstays bounded away from β/(2 −α), and now this is true for all σ>Σ (even σ≥Σ), by the counter assumption. Since σ>Σ, there exist parameters Δ0>0andη0>0 such that (3.20)holds for this specific “σ”, and for all product measures μ=μα×μβsatisfying (A1)–(A2). This is precisely what Proposition 3.18 asks for, and therefore the proposition yields new parameters η>0andδ0>0 such that (3.20) holds with “σ” replaced by “σ−ζ”, with “η0” replaced by “η”, for all 0 <δ≤δ0, and again for all product measures μ=μα×μβas in (A1)–(A2). By definition of the number “Σ”, this means thatinfactΣ≤σ−ζ, and a contradiction has been reached. This completes the proofofTheorem3.6. 4 Proof of Proposition 3.18 The proof of Proposition 3.18 uses many rescaling arguments, and we start by checking that the class of “products of regular measures” is invariant under rescaling maps. Remark 4.1. Let Tz0,r0:R2→R2be a rescaling map, with z0=(x0,y 0)∈R2and r0>0. Then Tz0,r0can be written as a product of rescaling maps on R, namely Tz0,r0(x, y)=x−x0 r0,y−y0 r0=(Tx0,r0(x),T y0,r0(y)),(x, y)∈R2. Let μ=μα×μβbe a product of an (α, Cα)-regular measure μαand a (β,Cβ)-regular measure μβ. Then, writing γ:= α+β, the rescaled and renormalised measure μz0,r0:= r−γ 0·Tx0,r0μ=(r−α 0Tx0,r0μα)×(r−β 0Ty0,r0μβ)=:μα,x0,r0×μβ,y0,r0(4.2) can again be expressed as a product of a (α, Cα)-regular and (β,Cβ)-regular measures μα,x0,r0:= r−α 0Tx0,r0μαand μβ,y0,r0:= r−β 0Ty0,r0μβ. In other words, μz0,r0is a product of two new regular measures with precisely the same constants as μαand μβ.
16 T. ORPONEN GAFA Notation 4.3. If μis a (γ,Cγ)-regular measure on R2(nearly always a product of two regular measures on Rin this paper), and B=B(z,r)⊂R2is a disc, we write μB:= r−γ·TBμ, which is another (γ,Cγ)-regular measure on R2. More accurate notation would be μB,γ, but the index “γ” should always be clear from context. We then repeat the statement of Proposition 3.18: Proposition 4.4. Let α, β, τ ∈(0,1],Cα,C β>0, and assume that σ> β 2−α.(4.5) Then, there exist ζ=ζ(α, β, Cα,σ,τ)>0such that ζstays bounded away from zero as long as σstays bounded away from β/(2 −α), and the following holds. Assume that there exists a parameter η0>0, and a scale Δ0>0, such that Hτ ∞({θ∈[0,1] : μ(B(1) ∩Hθ(spt μ, Δ−σ,[Δ,1])) ≥Δη0})≤Δη0,0<Δ≤Δ0, (4.6) whenever μ=μα×μβis a product of an (α, Cα)-regular measure μα,anda(β,Cβ)- regular measure μβ. Then, there exists a parameter η>0and a scale δ0>0,both depending only on α, β, Cα,C β,σ,τ,Δ0,η 0, such that Hτ ∞({θ∈[0,1] : μ(B(1) ∩Hθ(spt μ, δ−σ+ζ,[δ, 1])) ≥δη})≤δη,0<δ≤δ0, whenever μ=μα×μβsatisfies the same hypotheses as above. Remark 4.7. This remark is a continuation of the proof outline presented in Section 1.1. The proof of Proposition 4.4 proceeds roughly in the manner we described there, with A=sptμα,B=sptμβ, and “exceptional set” E={θ∈[0,1] : μ(B(1) ∩Hθ(spt μ, δ−σ+ζ,[δ, 1])) ≥δη}. We make the counter assumption Hτ ∞(E)≥δη, and fix θ0∈E. The purpose here is to point out two technical difficulties which we glossed over in Section 1.1. The first one is related to the following sentence above (1.10): Recalling that A×B⊂A×R, up to a change of coordinates, one more precisely rewrites A×B=A×B,where A=A+θ0B,and Brepresents the intersection of A×Bwith the “typical fibre” under the map πθ0(x, y)=x+θ0y. This was a rather inaccurate description: even though A×B⊂A×R, the fibres (A×B)∩π−1 θ0{t}are not so uniform, in general, that we could rewrite A×B=A×B. We do the following instead, imitating an idea which first appeared in [15]. We fix a small scale δ>0, and let T=π−1 θ0(I) be an arbitrary tube of width δ1/2. Then, it roughly speaking turns out that there exists a product set of the form A×B, where A=πθ0((A×B)∩T), with the property Nδ(A+(θ1−θ0)B)∼Nδ(πθ((A×B)∩T)),|θ−θ0|≤δ1/2.(4.8)
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 17 This corresponds to (4.98) in the actual proof (in reality, one needs to replace A×B by a “fat” subset G⊂A×B, but this is only a minor technical problem). To summarise: rewriting A×B=A×Bis hopeless, but instead it is possible to associate to (A×B)∩Ta product set A×B=πθ0((A×B)∩T)×B, in the sense that (4.8)holds. In Section 1.1, the next step was to fix another direction θ1∈Esuch that |θ0−θ1|∼1. In reality, with (4.8) in mind, we rather need to find θ1∈Esuch that |θ1−θ0|≈δ1/2. This is non-trivial: our only assumption on Eis that Hτ ∞(E)≥δη, and it may be impossible to find a pair of points θ0,θ 1∈Ewith separation |θ0−θ1|≈ δ1/2, for a given scale δ>0. This issue is resolved by pigeonholing another scale ¯ δ∈[δ, 1], which is not too much larger than δ, and for which we can find two points θ0,θ 1∈Esuch that |θ0−θ1|≈¯ δ1/2. This is accomplished in Section 4.3. After such a scale ¯ δhas been located, the steps mentioned above for the pair (δ, ¯ δ1/2) are, in reality, carried out for the pair (¯ δ, ¯ δ1/2). In particular, the tube T=π−1 θ0(I) will have width |I|=¯ δ1/2. It will be chosen (in Section 4.6)insucha manner that N¯ δ(πθ1((A×B)∩T)) ≈N¯ δ(πθ0((A×B)∩T)) = N¯ δ(A),(4.9) see (T2). Via an analogue of (4.8) at scale ¯ δ, this roughly implies that N¯ δ(A+(θ1−θ0)B)(4.8) ≈N¯ δ(πθ1((A×B)∩T)) (4.9) ≈N¯ δ(A). This equation is a more accurate analogue of (1.10) from Section 1.1. It can still be used in the same manner to draw conclusions about the “branching” structure of A and B. We begin the proof of Proposition 4.4. We fix the parameters α, β, τ ∈(0,1], Cα,C β>0, and we let Δ0,η 0>0 be constants such that (4.6) holds for all products μ=μα×μβof an (α, Cα)-regular measure μαand a (β,Cβ)-regular measure μβ. 4.1 Choosing constants. We take a moment to list and specify a few other constants. First of all, we write γ:= α+βand Cγ:= 5CαCβ.Thenμ=μα×μβis a(γ,Cγ)-regular measure on R2. Second, we note that the condition σ>β/(2 −α) is equivalent to D(α, β, σ):=(2−α)σ−β=(1−α)σ+α−γ+σ>0. We specify small constants , ρ, ζ0∈(0,1), and a large integer m0∈N, such that the following inequality holds for all m≥m0: γ−σ+10ζ0<(1 −α−)(σ−10(+ζ0)) + (1 −ρ)(1 −30(+ζ0) αρ −10Cα αρm )α. (4.10) Since γ−σ=(1−α)σ+α−D(α, β, σ)<(1 −α)σ+α, it is qualitatively clear that the constants , ρ, ζ0,andm0can be chosen so that (4.10) holds, but let us be
18 T. ORPONEN GAFA more specific about their dependencies. The constant ρ>0 should be chosen first: if 0 <ρ<1 2·D(α, β, σ), then γ−σ=(1−α)σ+α<(1 −α)σ+(1−ρ)α−D(α,β,σ) 2, since α≤1. Next, the remaining three constants , ζ0∈(0,1), and m0≥1, can be chosen in an arbitrary order such that (4.10) holds for all m≥m0. Evidently , ζ0only depend on α, β and ρ(hence on α, β,andD(α, β, σ)). The constant m0 additionally depends on the regularity constant “Cα”. The notation “ζ0” suggests that this constant should have something to do with the constant ζ=ζ(α, β, Cα,σ,τ)>0, whose existence is the main claim in Proposition 3.18. To specify this connection, we should state Shmerkin’s inverse theorem [22, Theorem 2.1], but the statement is so long that we postpone the full details to Theorem 4.76. However, the theorem begins with the following words: For every >0and m0≥1there exists κ=κ(, m0)>0and m≥m0such that the following holds for large enough N. Now, the number ζ>0 from the claim of Proposition 3.18 can be taken to be any parameter satisfying 0<O·ζ≤min{ζ0,κ(, m0)},(4.11) where , ζ0,m 0are familiar from the discussion above, and O=O(τ)>0 will be a constant depending only τ. In particular, since these constants together only depend on α, β, σ, τ,andCα, the same will be true for the constant ζ>0. It is not a mistake (as far as I know!) that the constants , ρ, ζ0,m 0do not depend on the regularity constant “Cβ” appearing in Proposition 4.4. This constant is present in Cγ=5CαCβ, and will also influence the thresholds for the parameters “η”and“δ0”. 4.2 The counter assumption. Now that we have clarified the roles of future constants, we make a counter assumption: the conclusion of the proposition fails for a certain product measure μ=μα×μβ, where μαis (α, Cα)-regular, μβis (β,Cβ)- regular, for a certain small scale δ∈(0,Δ0], and a certain parameter η∈(0,η 0]: Hτ ∞({θ∈[0,1] : μ(B(1) ∩Hθ(K, δ−σ+ζ,[δ, 1])) ≥δη})≥δη,(4.12) Here K:= spt μ. The reader should think that δΔ0and ηη0, and additionally that ηis small compared to the previously fixed small constants , η0,ζ ≤ζ0,τ.We should be on the safe side, if we assume that 0<η<C −1·(αβρτη0ζ)C(4.13) for a suitable absolute constant C≥1, so in particular “η” will not depend on Δ0. The threshold for δ>0forwhich(4.12) leads to a contradiction will only depend (in principle effectively) on the parameters α, β, Cα,C β,σ,τ,Δ0,andη0, but this dependence will not be tracked explicitly.
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 19 Remark 4.14. On several occasions, we will need to assume that the scale δ>0 has a special form: very commonly δ>0 needs to be a dyadic number, and later on a fortiori we need that δ=(2 −m)Nfor some integer N≥1, where m≥m0 is the integer produced by Shmerkin’s inverse theorem with initial data and m0 (see above (4.11)). Such assumptions are harmless: if our counter assumption (4.12) involves a parameter δ>0, which is not of the correct form, then there exists a parameter δ∈[δ, Cδ] of the correct form, where C≥1 only depends on the usual parameters listed above. But now Hθ(K, δ−σ+ζ,[δ, 1]) ⊂Hθ(K, 1 C·(δ)−σ+ζ,[δ,1]) ⊂Hθ(K, (δ)−σ+2ζ,[δ,1]), where the first inclusion follows from Lemma 2.6, and the second inclusion is true as soon as δ>0 is so small that 1 C≥(δ)ζ. Using this inclusion, we may find arbitrarily small values of δ(a scale of the “correct form”) such that the counter assumption (4.12) holds with “2ζ” in place of “ζ”. We can then use this variant of (4.12), instead, to derive a contradiction. In the sequel, we will assume that δ(and a certain other scales “¯ δ” derived from δ) is of the “correct form” without further remark. Since the set E:= {θ∈[0,1] : μ(B(1) ∩Hθ(K, δ−σ+ζ,[δ, 1])) ≥δη}is assumed, by (4.12), to have τ-dimensional Hausdorff-content bounded from below by δη,we may find (see [12, Theorem 8.8]) a (τ,Cδ−η)-Frostman probability measure νwith spt ν⊂E. Here C>0 is an absolute constant. From this point on, the counter assumption (4.12) will only be used via the following: μ(B(1) ∩Hθ(K, δ−σ+ζ,[δ, 1])) ≥δη,θ∈spt ν. (4.15) Notation 4.16. We will denote by “O” a generic large constant which may depend on α, β, Cα,C β,C γ,σ,τ. In similar spirit, we will denote by “ω” a small positive constant, which is bounded away from zero in a manner depending only on α, β, Cα,C β,C γ,σ,τ.Infact,ω=O−1. If either Oor ωonly depends on a subset of the parameters above, this will occasionally be emphasised by writing, for example, “O(α, β)” instead of “O”. The precise values of the constants “O”and“ω” may vary from line to line. This will often lead to inequalities of the form “2O≤O”, which are not typos. 4.3 Pigeonholing a branching scale for ν.The measure νmay have “no branching” between the scales δ1/2and δ. Defining this defect carefully is not worth the effort, but we roughly mean the possibility that N(spt ν, δ)≈N(spt ν, δ1/2). Fortunately, it follows from the τ-Frostman property of νthat opposite behaviour must occur at some scale ¯ δ>δ, which also satisfies the (roughly) converse inequality ¯ δ≤δω(α,β,σ,τ). Finding the scale ¯ δ, and making these statements more precise, is the goal of this section. The final conclusion will be (4.28).
20 T. ORPONEN GAFA Recall that νis a (τ,Cδ−η)-Frostman probability measure satisfying (4.15). Recall that =(α, β, σ)>0 was one of the constants fixed in Section 4.1.Wedefine the following increasing scale sequence: δ0:= δ(1+)/2,and δj+1 := δ1/(1+) j,j≥1. Thus δ1=δ1/2,andδj=δ1/(1+)j 0=(δ1/2)1/(1+)j−1for j≥0. Recall that Dδjis the partition of [0,1) into dyadic intervals of length δj(if these numbers are not dyadic to begin with, the closest dyadic numbers would work as well). We mention that Dδj+1 consists of intervals longer than those in Dδj. We will prove the following lemma: Lemma 4.17. There exists an index j≤O(α, β, σ, τ), and a Borel set G⊂[0,1] with ν(G)≥ω(α, β, σ, τ)such that the renormalised measure ¯ν=ν(G)−1·ν|Ghas the following properties: •¯νis a (τ,Oδ−η)-Frostman measure with O=O(α, β, σ, τ)>1, •If I∈D j+1 with ¯ν(I)>0, then ¯νI(J)≤δω(α,β,σ,τ),J∈D δj(I).(4.18) Here ¯νI=¯ν(I)−1·¯ν|I,andDδj(I)refers to the intervals in Dδjwhich are contained in I. Note that the length of these intervals is |J|=δj=δ1+ j+1 =|I|1+. The relation between Lemma 4.17, and our search for the scale “¯ δ”, is simply that ¯ δ:= δ2 j+1,hence δj+1 =¯ δ1/2. Since 1 <1+≤2, we then have ¯ δ≤δj¯ δ1/2. The inequality (4.18) informally says that the measure ¯νhas non-trivial branching between the scales δjand δj+1 =¯ δ1/2, and so in particular between the scales ¯ δand ¯ δ1/2. We begin the search for j0and G. Since νisa(τ,Cδ−η)-Frostman probability measure on [0,1], we have the uniform bound ν(I0)≤Cδ−ηδτ 0I0∈D δ0, which gives the following lower bound for the entropy of νat scale δ0: H(ν, Dδ0)= I0∈Dδ0 ν(I0)log 1 ν(I0)≥log δ−τ 0−log C−log δ−η≥log δ−τ/2. (4.19) In the final inequality, we used that δ−τ 0=δ−τ(1+)/2=δ−τ/2·δ−τ/2, and the logarithm of the second factor exceeds the “error term” log C+logδ−ηif δ>0is sufficiently small, and η<τ/2 (as is implied by (4.13)).
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 21 Now, let n∼,τ 1 (therefore n≤O(α, β, σ, τ)) be the smallest integer satisfying 1 2(1 + )n−1≤τ 4.(4.20) This is not too important, but then in fact n∼−1·log(1/τ). Then, observing that Dδnconsists of intervals of length δ1/(1+)n 0=δ1/[2(1+)n−1],wehave H(ν, Dδn)≤log |Dδn|≤log δ−1/[2(1+)n−1]≤log δ−τ/4. Combining this estimate with (4.19), and using the conditional entropy formula (2.1) repeatedly to the nested partitions Dδj, we find that log δ−τ/4≤H(ν, Dδ0)−H(ν, Dδn)= n−1 j=0 H(ν, Dδj|D δj+1 ). Consequently, there exists j∈{0,...,n−1}with the property I∈Dδj+1 ν(I)·H(νI,Dδj)=H(ν, Dδj|D δj+1 )≥log δ−τ/(4n).(4.21) Here νI=ν(I)−1ν|Ifor any I∈D δj+1 with ν(I)= 0. As we discussed after the statement of Lemma 4.17,weset¯ δ=δ2 j+1 (note that δj+1 ≥δ1=δ1/2,so¯ δ≥δ). Let us record that ¯ δ≤δ1/(1+)n−1≤δτ/4,(4.22) by (4.20), or equivalently δ≥¯ δ4/τ . Indeed, since “n” is the smallest number satisfying (4.20), we have 1/(2(1+)n−2)>τ/4, and hence 1/(1+)n−1≥(1+)−1·τ/2≥ τ/4. Thus ¯ δremains somewhat comparable to δ, in a manner depending only on the fixed parameter “τ”. Motivated by (4.21), we set ¯τ:= τ/(4n)∼·τ/log(1/τ)≥τ2, and we note that ¯τ≥ω(α, β, σ, τ). The choice of the index j∈{0,...,n−1}at (4.21) roughly tells us that for many “long” intervals I∈D δj+1 with ν(I)>0, the re-normalised restriction νI is not concentrated on very few “short” sub-intervals of Iin the family Dδj.By restricting νa little bit, we may replace the word “many” by “all”, as we will see next. In fact, since certainly H(νI,Dδj)≤log δ−1,I∈D δj+1 , we may infer from (4.21) that there exists a collection of arcs Gj+1 ⊂D δj+1 of total ν-measure ν(∪Gj+1)≥¯τ/2 (4.23)
22 T. ORPONEN GAFA with the properties ν(I)>0andH(νI,Dδj)≥log δ−¯τ/2for all I∈G j+1. Indeed, if this failed, then I∈Dδj+1 ν(I)H(νI,Dδj)< H(...)≥log δ−¯τ/2 ν(I)·log δ−1+ H(...)<log δ−¯τ/2 ν(I)·log δ−¯τ/2 ≤¯τ 2·log δ−1+logδ−¯τ/2=logδ−¯τ, which contradicts (4.21). We write Gj+1 := ∪Gj+1, and we renormalise νto Gj+1: ¯ν1:= 1 ν(Gj+1)·ν|Gj+1 . Observe, by (4.23), that ¯ν1isa(τ,Oδ−η)-Frostman probability measure with the additional feature that if I∈D δj+1 is an arc with ¯ν1(I)>0, then H(¯ν1,I ,Dδj)=H(νI,Dδj)≥log δ−¯τ/2.(4.24) It is worth noting that νI=¯ν1,I for all I∈G j+1 (= {I∈D δj+1 :ν1(I)>0}). The inequality (4.24) means that the measure ¯ν1cannot be completely concentrated inside any single interval of length δj=|I|1+. What we would prefer is, a fortiori, that νI=¯ν1,I satisfies a Frostman-type estimate at the smaller scale δj, as stated in (4.18). This will be achieved by renormalising ¯ν1further to the measure ¯ν,which finally satisfies Lemma 4.17. Fix I∈D δj+1 with ν1(I)>0, write Dδj(I):={J∈D δj:J⊂I}for I∈D j+1, and then estimate log δ−¯τ/2(4.24) ≤ J∈Dδj(I) νI(J)log 1 νI(J) ≤ νI(J)≥δ¯τ/4 νI(J)logδ−¯τ/4+ νI(J)<δ¯τ/4 νIlog 1 νI(J). Since νIis a probability measure, the first term is bounded from above by 1 2log δ−¯τ/2, and consequently the second term has the lower bound νI(J)<δ¯τ/4 νI(J)log 1 νI(J)≥log δ−¯τ/4.(4.25) On the other hand, if the total νI-measure of the intervals in the sum above, namely GI:= {J∈D δj(I):0<ν I(J)<δ ¯τ/4}⊂D δj, is denoted by mI:= νI(∪GI)wehave log δ−¯τ/4(4.25) ≤ νI(J)<δ¯τ/4 νI(J)log 1 νI(J)=mI J∈GI νI(J) mI log 1 νI(J) ≤mIlog J∈GI 1 mI=mIlog |GI|−mIlog mI,
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 23 by Jensen’s inequality applied to the discrete probability measure J→ νI(J)/mI on GI. Since mI∈(0,1], the second term satisfies |mIlog mI|≤1 2, and on the other hand |GI|≤|D δj(I)|≤δ−1, by a crude estimate. Assuming that δ>0 is so small that log δ−¯τ/4−1 2≥log δ−¯τ/8, we find from the estimate above that log δ−mI≥log |GI|mI≥log δ−¯τ/4+mIlog mI≥log δ−¯τ/8, and consequently νI(∪GI)=mI≥¯τ/8. In other words, if we define Gj:= I∈Gj+1 GI⊂D δjand G:= Gj:= ∪Gj⊂Gj+1, then ¯ν1(G)= I∈Gj+1 ¯ν1(I)·νI(∪GI)≥¯τ/8. Therefore, the measure ¯ν:= 1 ¯ν1(G)·¯ν1|G=1 ν(G)·ν|G remains a (τ,Oδ−η)-Frostman probability measure with the feature that if ¯ν(I)>0 for some I∈D δj+1 , then ¯νI(J)≤δ¯τ/4,J∈D δj(I).(4.26) Note that ¯νwas finally defined by restricting the original measure “ν” to a certain union Gof dyadic intervals (of length δj), whose total νmeasure is bounded from below by ≥ω(α, β, σ, τ). Thus ¯νremains a (τ,Oδ−η)-Frostman probability measure with the property μ(B(1) ∩Hθ(K, δ−σ+ζ,[δ, 1])) ≥δη,θ∈spt ¯ν⊂E. (4.27) We have now proven Lemma 4.17. Since ¯νsatisfies roughly the same hypotheses as ν(and additionally the Frostman property (4.26)), we redefine ν:= ¯νto simplify notation. We also recall that ¯ δ:= δ2 j+1 (so if j=0,simply¯ δ=δ). We also write D1:= Dδjfor the dyadic partition of [0,1) into intervals of length ¯ δ=¯ δ1, and we write D1/2:= Dδj+1 (the dyadic partition to intervals of length ¯ δ1/2=δj+1). Then, (4.26) implies that I∈D 1/2and ν(I)>0=⇒νI(J)≤δω(α,β,σ,τ)≤¯ δω(α,β,σ,τ)for J∈D(1+)/2(I), (4.28) where D(1+)/2(I)⊂D δjare the dyadic intervals of length δj=δ1+ j+1 =¯ δ(1+)/2 contained in I. To close the section, we observe that it is well possible that ¯ δ= δ2 j+1 =δ; this happens if the index j∈{0,...,n−1}fixed at (4.21) happens to be j=0,so¯ δ=δ2 0+1 =(δ1/2)2=δ. This will lead to a simpler special case of the proof
24 T. ORPONEN GAFA below. On the the other hand, if ¯ δ>δ, then ¯ δ=δ2 j+1 for some j≥1. Consequently ¯ δ≥δ2 2=δ1/(1+), and hence ¯ δis “much longer” than δ:infact δ ¯ δ≤δ1−1/(1+).(4.29) Note that the right hand side is a positive power of δ. 4.4 High multiplicity sets at scale ¯ δ.Now we have found a scale ¯ δ∈[δ, δω] such that the measure νhas non-trivial “branching” between the scales ¯ δand ¯ δ1/2,as quantified in (4.28). The following problem now materialises: our counter assumption (4.15) concerned the scale δ, and there is a risk that all the information is lost when replacing “δ”by“ ¯ δ”. We resolve the issue by proving the following lemma: Lemma 4.30. There exists a subset S⊂[0,1] with ν(S)≥ω·δηwith the property μB(5)(B(1) ∩Hθ(TB(5)(K),¯ δ−σ+O(τ)ζ,[¯ δ,1])) ≥ω·δη,θ∈S. (4.31) Thus, modulo replacing “ζ”by“O·ζ”, which is harmless for our purposes, the counter assumption (4.15) at scale “δ” can be used (in cooperation with our hypothesis (4.6)) to infer similarly bad behaviour at scale “¯ δ” for the dilated regular set TB(5)(K), and the renormalised measure μB(5) supported on TB(5)(K). As we discussed in Remark 4.1, the measure μB(5) is of the same form as μ, with precisely the same constants. In particular our inductive hypothesis (4.6) may later be applied to μB(5) and TB(5)(K). Note that if δ=¯ δ, then every θ∈spt νalready satisfies (4.31)(withO(α, β, σ, τ) =1andμ, K in place of μB(5),T B(5)(K)) by virtue of our initial counter assumption (4.15). In this case the argument in the present section will not be needed. So, for the time being, we will assume that ¯ δ>δ, which implies by (4.29) that ¯ δis substantially larger than δ. In this case, we will apply Proposition 4.32, whose statement we include here, but whose lengthy proof is postponed to Section 5: Proposition 4.32. Let θ∈[0,1], and let 1≤M≤N<∞be constants, let 0<r≤R≤1, and let μbe a (γ,Cγ)-regular measure with K:= spt μ⊂R2. Abbreviate μs:= μ|B(s)for s>0. Then, there exist absolute constants c, C > 0 such that μ1(Hθ(K, CN, [r, 1])) ≤μ1(Hθ(K, cM, [4R, 5])) +CC2 γ·μ4(Hθ(K, c N M,[4r, 7R])).(4.33) We will apply the proposition to the (γ,Cγ)-regular measure μ=μα×μβwith the parameters M≤Nsuch that CN =δ−σ+ζand c·N M=δ/¯ δ−σ,
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 31 recall (4.55). We start by claiming the following: N¯ δ(πθj(¯ K)) Cγ·¯ δσ−γ−¯ ζ,j∈{0,1}.(4.57) To see (4.57), fix θ∈{θ0,θ 1}, and let Tθbe a minimal cover of ¯ Kby tubes of the form T=π−1 θ{I}, where I∈D ¯ δ(R). (We will write “T”for¯ δ-tubes and “T”for ¯ δ1/2-tubes.) Then, each T∈T θcontains a point xT∈¯ K⊂Hθ(K, ¯ δ−σ+¯ ζ,[¯ δ,1]), hence B(x, 1) ∩π−1 θ{πθ(xT)}⊂B(2) ∩T,and N¯ δ(K¯ δ∩B(2) ∩T)≥mK,θ(xT|[¯ δ,1]) ≥¯ δ−σ+¯ ζ. This implies that |Tθ|·¯ δ−σ+¯ ζN¯ δ(K¯ δ∩B(2)) Cγ·¯ δ−γ, and (4.57) follows by rearranging. We next consider the projections of ¯ Kat scale ¯ δ1/2, but we first do an initial reduction. Let K¯ δ1/2be a minimal cover of ¯ Kby discs of radius ¯ δ1/2, so in particular ¯ K∩B=∅for all B∈K¯ δ1/2. Since ¯ K⊂K∩B(1), and μ(¯ K)≥ω·δ4η,wehave ω·δ4η·(¯ δ1/2)−γ≤|K¯ δ1/2|≤N¯ δ1/2(K∩B(1)) ≤Cγ·(¯ δ1/2)−γ.(4.58) AdiscB∈K¯ δ1/2is called heavy if μ(B∩¯ K)≥(μ(¯ K)/2Cγ)·(¯ δ1/2)γ≥ω·δ4η·(¯ δ1/2)γ.(4.59) Then, the total μmeasure of the light (that is, non-heavy) discs if bounded from above by |K¯ δ1/2|·(μ(¯ K)/2Cγ)·(¯ δ1/2)γ≤μ(¯ K)/2. Therefore, if we replace ¯ Kby the intersection ¯ K∩ B∈K¯ δ1/2heavy B, (4.60) then μ(¯ K)≥(ω/2) ·δ4η, and the key property (4.56)of ¯ Kremains valid (it is worth emphasising here that the high-multiplicity set in (4.56) is defined relative to K, not ¯ K). These are all the properties we will need in the sequel, so, without loss of generality, we may assume that all the discs in K¯ δ1/2are heavy. Now, in order to consider the πθ0and πθ1projections of ¯ Kat scale ¯ δ1/2, let T¯ δ1/2 be a minimal cover of ¯ Kby tubes T=π−1 θ0(I)withI∈D¯ δ1/2(R). It will be good to keep in mind that |θ0−θ1|≤¯ δ1/2by (4.54), so the πθ0and πθ1projections of ¯ Kare virtually indistinguishable at scale ¯ δ1/2: for example N¯ δ1/2(πθ0(¯ K)) ∼N¯ δ1/2(πθ1(¯ K)). Also, even though the tubes in T¯ δ1/2were defined via the projection πθ0, they still satisfy the following property for both j∈{0,1}: the sets πθj(B(1)∩T) have bounded overlap as T∈T¯ δ1/2varies.
32 T. ORPONEN GAFA We now claim the following upper bound, assuming that δ>0andη>0are sufficiently small: N¯ δ1/2(πθ0(¯ K)) ∼|T¯ δ1/2|Cγ·(¯ δ1/2)σ−γ−4¯ ζ.(4.61) The proof goes as follows. Every tube T∈T ¯ δ1/2meets at least one (heavy!) ball BT∈K¯ δ1/2. We will shortly see that N¯ δ(πθj(¯ K∩B)) ≥(¯ δ1/2)σ−γ+2¯ ζ,j∈{0,1},B∈K¯ δ1/2,(4.62) so in particular this holds with B=BT. Since the sets πθ0(¯ K∩BT) have bounded overlap as Tvaries in T¯ δ1/2, it follows from (4.57)and(4.62) that Cγ·¯ δσ−γ−¯ ζN¯ δ(πθ0(¯ K)) |T¯ δ1/2|·(¯ δ1/2)σ−γ+2¯ ζ, and then (4.61) follows by rearranging terms. Let us then prove (4.62). Fix B∈K ¯ δ1/2, and keep in mind that μ(B∩¯ K)≥ ω·δ4η·¯ δγ/2, since all the balls in K¯ δ1/2are heavy. Let us first check that (4.62) follows if we manage to show the next claim: if T¯ δ=π−1 θj(I) is an arbitrary tube of width |I|=¯ δ, then N¯ δ(B∩¯ K∩T¯ δ)≤(¯ δ1/2)−σ−¯ ζ,B∈K¯ δ1/2.(4.63) Indeed, since μ(B∩¯ K)≥ω·δ4η·(¯ δ1/2)γfor all (heavy) discs B∈K¯ δ1/2,wehave N¯ δ(B∩¯ K)≥ω·δ4η·(¯ δ1/2)−γ by the (γ,Cγ)-regularity of μ. Combining this lower bound with (4.63) implies that it takes ≥ω·δ4η·(¯ δ1/2)σ−γ+¯ ζtubes of the form T¯ δ=π−1 θj(I), with |I|=¯ δ,tocover B∩¯ K. This implies (4.62)forδ, η > 0 small enough that ω·δ4η≥¯ δ¯ ζ/2. It remains to establish (4.63). Fix a tube T¯ δ:= π−1 θj(I)with|I|=¯ δ, and assume to the contrary that B∈K¯ δ1/2is a disc with N¯ δ(B∩¯ K∩T¯ δ)≥(¯ δ1/2)−σ−¯ ζ.(4.64) Then, if x0∈B∩¯ K∩T¯ δis arbitrary, note that B⊂B(x0,5¯ δ1/2). From this, combined with (4.64), and assuming δ>0 sufficiently small in terms of ¯ ζ, it follows easily that mK,θj(x0|[5¯ δ,5¯ δ1/2]) := N5¯ δ(B(x0,5¯ δ1/2)∩K5¯ δ∩π−1 θj{πθj(x0)}) ≥(¯ δ1/2)−σ,(4.65) see Figure 1for further details. In other words x0∈¯ K∩Hθj(K, (¯ δ1/2)−σ,[5¯ δ,5¯ δ1/2]). However, by the definition in (4.55), the set ¯ Kis a subset of Kθj, and this Kθj,by its definition in (4.53), contains no points of Hθj(K, (¯ δ1/2)−σ,[5¯ δ,5¯ δ1/2]). Therefore (4.63) holds, and this completes the proof of (4.61).
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 33 B T¯ δ x0 Figure 1: One of the heavy discs B∈K¯ δ1/2intersected with a tube T¯ δ=π−1 θj(I), and a point x0∈B∩¯ K∩T¯ δ. The grey discs form a minimal ¯ δ-cover for B∩¯ K∩T¯ δ,asin(4.64), and their 5-times enlargements, shown in dotted lines, all intersect the red line π−1 θj{πθj(x0)}. This gives (4.65). We have now shown, in (4.61), that ¯ Kcan be covered by Cγ·(¯ δ1/2)σ−γ−4¯ ζ tubes of the form T=π−1 θ0(I), I∈D¯ δ1/2(R), whose collection we denoted T¯ δ1/2.This implies that the average tube in T¯ δ1/2meets C−1 γ·|K¯ δ1/2|·(¯ δ1/2)γ−σ+4¯ ζ(4.58) ≥ω·δ4η·(¯ δ1/2)−σ+4¯ ζ discs in K¯ δ1/2. For technical convenience, we wish to arrange that the statement above holds for every tube in T¯ δ1/2, and this can be accomplished by another pruning argument, as follows. We say that a tube T∈T¯ δ1/2is heavy if |{B∈K¯ δ1/2:B∩T=∅}| ≥ ω·δ4η·(¯ δ1/2)−σ+4¯ ζ, for a suitable constant ω>0. Since |K¯ δ1/2|≥ω·δ4η·(¯ δ1/2)−γby (4.58), and |T¯ δ1/2|Cγ·(¯ δ1/2)σ−γ−4¯ ζas we just argued, the union of the light (non-heavy) tubes in T¯ δ1/2intersects at most half of the discs in K¯ δ1/2, assuming that the “ω” constant in the definition of heaviness above is chosen appropriately. We now redefine K¯ δ1/2 to be those discs in (former) K¯ δ1/2which intersect a heavy tube, and we redefine ¯ K to be the part of ¯ Kcovered by the union of the (remaining) discs in K¯ δ1/2.Wealso restrict T¯ δ1/2to the heavy tubes. With this new notation, and taking δ, η > 0 small enough so that ω·δ4η≥(¯ δ1/2)¯ ζ, |{B∈K¯ δ1/2:B∩T=∅}| ≥ ω·δ4η·(¯ δ1/2)−σ+4¯ ζ≥(¯ δ1/2)−σ+5¯ ζ,T∈T¯ δ1/2. (4.66) Moreover, K¯ δ1/2continues to satisfy the estimates from (4.58), namely |K¯ δ1/2|≈ (¯ δ1/2)−γ. The tubes in T¯ δ1/2need not quite cover ¯ Kany longer. This is because some
34 T. ORPONEN GAFA disc in K¯ δ1/2might have been initially half-half covered by a light tube and a heavy tube; then the ball was selected to the new K¯ δ1/2, but is not covered by the heavy tubes it touches. To fix this, we inflate the tubes in T¯ δ1/2by a factor of 3 without changing notation: then T¯ δ1/2consists of tubes of width 3¯ δ1/2,and ¯ K= B∈K¯ δ1/2 ¯ K∩B⊂ T∈T¯ δ1/2 T.(4.67) We also write ¯ KT:= B∈K¯ δ1/2 B⊂T ¯ K∩B⊂T,T∈T¯ δ1/2.(4.68) Since the tubes in T¯ δ1/2are heavy (and since “T” now already refers to the 3-times inflated tubes), the cardinality of this union is bounded from below by (4.66), for every T∈T¯ δ1/2. We would next like to show that for “most” of the tubes T∈T¯ δ1/2, the projections πθj(¯ KT), for both j∈{0,1}, are fairly small at scale ¯ δ,say N¯ δ(πθj(¯ KT)) ≤(¯ δ1/2)σ−γ−6¯ ζ,j∈{0,1}.(4.69) The only clue available is (4.57), which controls the πθj-projections of the whole set ¯ K.Forj∈{0,1}fixed, this can be easily used to show that there are only few tubes T∈T¯ δ1/2such that (4.69) fails. Namely, if T¯ δ1/2,j ⊂T¯ δ1/2is the sub-family of “bad” tubes for which (4.69) fails, then it follows from (4.57), and the bounded overlap of the projections πθj(¯ KT), T∈T¯ δ1/2, that |T¯ δ1/2,j|·(¯ δ1/2)σ−γ−6¯ ζN¯ δ(πθj(¯ K)) Cγ·¯ δσ−γ−¯ ζ,j∈{0,1}, and hence |T¯ δ1/2,j|≤(¯ δ1/2)σ−γ+3¯ ζ,(4.70) assuming that ¯ δis so small that the constants (implicit and Cγ) are bounded from aboveby( ¯ δ1/2)−¯ ζ. So, now we know that (4.69) can only fail for few tubes in T¯ δ1/2. What we really wanted was, instead, that (4.69) holds for “most” tubes in T¯ δ1/2. To deduce the latter statement from the former, we need to show that the families T¯ δ1/2,j of “bad” tubes above only constitute a small fraction of all the tubes in T¯ δ1/2. According to (4.70), this follows if we manage to show that |T¯ δ1/2|≥(¯ δ1/2)σ−γ+2¯ ζ.(4.71) The proof of (4.71) is extremely similar to the proof of the lower bound in (4.62). Instead of showing (4.71) directly, we prove that N¯ δ1/2(¯ K∩T)≤(¯ δ1/2)−σ−¯ ζ,T∈T¯ δ1/2,(4.72)
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 35 which is roughly a reverse of (4.66). Once (4.72) has been established, (4.71) follows (assuming that δ, η > 0 are small enough, as usual), since the union of the tubes in T¯ δ1/2covers all the discs in K¯ δ1/2by (4.67), and |K¯ δ1/2|≥ω·δ4η·(¯ δ1/2)−γby (4.58). To prove (4.72), fix x0∈¯ K∩Tarbitrary, and note that mK,θ0(x0|[5¯ δ1/2,5]) = N5¯ δ1/2(B(x0,5) ∩K5¯ δ1/2∩π−1 θ0{πθ0(x0)})N¯ δ1/2(¯ K∩T), (4.73) using that ¯ K⊂K⊂B(1), and the width of Tis 3¯ δ1/2. The geometry of the inequality in (4.73) is similar to the one depicted in Figure 1, the main difference being that the scales “¯ δ”and“ ¯ δ1/2” are replaced by “¯ δ1/2” and “1”. Since ¯ K⊂ Kθ0⊂Gθ0by (4.56), we have x0/∈Hθ0(K, (¯ δ1/2)−σ,[5¯ δ1/2,5]), and hence the left hand side in (4.73) is no larger than (¯ δ1/2)−σ.Forδ>0 sufficiently small, this yields (4.72). Combining (4.70)–(4.71), we see that if ¯ δ>0 is small enough, then only a small fraction of the tubes in T¯ δ1/2lies in T¯ δ1/2,0∪T¯ δ1/2,1. In particular, we may find a tube T0∈T¯ δ1/2\[T¯ δ1/2,0∪T¯ δ1/2,1]. We gather the relevant properties of T0for future reference: Lemma 4.74. There exists a tube T0⊂R2of the form T0=π−1 θ0(I0), where I0⊂R is an interval of length 3¯ δ1/2, such that T0has the following properties: (K1) |{B∈K ¯ δ1/2:B⊂T0}| ≥ (¯ δ1/2)−σ+5¯ ζby (4.66), and all the balls B∈K ¯ δ1/2 here are heavy, that is, μ(B∩¯ K)≥ω·δ4η·¯ δγ/2, recall (4.59). (K2) The set ¯ K0:= ¯ KT0is a subset of Gθ0∩Gθ1, and has small πθj-projections at scale ¯ δin the sense that N¯ δ(πθj(¯ K0)) ≤(¯ δ1/2)σ−γ−6¯ ζfor both j∈{0,1},see (4.69). The properties (K1)–(K2) are, finally, the precise versions of (T1)–(T2). 4.7 Statement of Shmerkin’s inverse theorem. We pause the main line of the proof for a moment to introduce Shmerkin’s inverse theorem, and some associated notation. Definition 4.75 (δ-measures and L2-norms). Let δ∈2−Nbe a dyadic rational. Then, any probability measure supported on the discrete set δ·Z∩[−1,1] is called aδ-measure.TheL2-norm of a δ-measure μis defined by μL2:= z∈δ·Z μ({z})21/2 .
36 T. ORPONEN GAFA Theorem 4.76 (Shmerkin). Given >0and m0∈N, there are κ=κ(, m0)>0 and m≥m0such that the following holds for all large enough N∈N.Letδ= (2−m)N, and let η1,η 2be δ-measures such that η1∗η2L2≥δκη1L2.(4.77) Then, there exist (δ-separated) sets U⊂spt η1and V⊂spt η2such that η1|UL2≥δη1L2and η2(V)≥δ, and the following properties hold: (A) there is a sequence (R1 s)N−1 s=0 ⊂{1,...,2m}N−1, such that N(U∩I,2−(s+1)m)=R1 s for all dyadic intervals Iof length 2−ms intersecting U, (B) there is a sequence (R2 s)N−1 s=0 ⊂{1,...,2m}N−1, such that N(V∩I,2−(s+1)m)=R2 s for all dyadic intervals Iof length 2−ms intersecting V. For each s∈{0,...,N −1}, either R2 s=1or R1 s≥2(1−)m, and the set S={s: R1 s≥2(1−)m}satisfies m|S| ≥ log η2−2 L2−log2δ. (4.78) Remark 4.79. The numbers Rj sare the branching numbers of the measures ηj; this terminology is heuristically useful, but imprecise, since the numbers Rj smay depend on the specific choices of Uand V. If this imprecision is tolerated, and if R1 s≥2αm for all s∈{0,...,N −1}, then we might say that “η1has α-dimensional branching at all scales”. This terminology was used in the proof outline, Section 1.1. 4.8 Deriving a product structure from ¯ K0.We the return to the main line of the argument. Based on the properties (K1)–(K2) of the set ¯ K0=¯ KT0,wewill construct a pair of ¯ δ-measures η1,η 2which eventually contradict the statement of Shmerkin’s inverse theorem. This contradiction will show that our counter assumption (4.12) must be false, and the proof of Proposition 3.18 will be completed. To simplify notation a little, we assume without loss of generality that T0is the tube T0=π−1 θ0(I0),where I0:= [0,3¯ δ1/2]. We will also assume that 3¯ δ1/2is a dyadic rational. Let T¯ δbe a minimal cover of ¯ K0 with tubes of the form π−1 θ0(I), where I∈D¯ δ(R); a more accurate notation would be Tθ0,¯ δ, but the choice between θ0and θ1at this point is completely arbitrary. Then
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 37 the tubes in T¯ δare all contained in T0, or in other words I⊂I0, since ¯ K0⊂T0, recall (4.68). Moreover, |T¯ δ|≤(¯ δ1/2)σ−γ−6¯ ζ(4.80) according to property (K2). We also let K¯ δ1/2(T0):={B∈K¯ δ1/2:B⊂T0},andwe write M:= |K¯ δ1/2(T0)|(K1) ≥(¯ δ1/2)−σ+5¯ ζ.(4.81) For purposes in the distant future, we next want to remove from T¯ δa few tubes which have a relatively sparse intersection with ¯ K0. To make this precise, and motivate the numerology, we introduce some notation. For B∈K¯ δ1/2(T0), let K¯ δ(B) be a minimal cover of ¯ K0∩Bby discs of radius ¯ δ, (4.82) so in particular B¯ δ∩¯ K∩B=∅for all B¯ δ∈K¯ δ(B). Recall that each disc B∈K¯ δ1/2 satisfies μ(B∩¯ K)≥ω·δ4η·¯ δγ/2by (K1), so the (γ,Cγ)-regularity of μimplies ω·δ4η·(¯ δ1/2)−γ≤|K¯ δ(B)|≤Cγ·(¯ δ1/2)−γ,B∈K¯ δ1/2(T0).(4.83) By discarding at most 1 2of the μmeasure of ¯ K0∩B, we may assume that all the discs in K¯ δ(B)areheavy, meaning this time that μ(B¯ δ)≥ω·δ4η·¯ δγ,B ¯ δ∈K¯ δ(B).(4.84) We then define ¯ KBto be the union of the heavy discs in K¯ δ(B) intersected with ¯ K0. We let K¯ δto be the union of all the families K¯ δ(B), with B∈K¯ δ1/2(T0). Thus ω·δ4η·M·(¯ δ1/2)−γ≤|K¯ δ|≤Cγ·M·(¯ δ1/2)−γ.(4.85) We also recall from (4.63) that if T=T¯ δ=π−1 θ0(I) is an arbitrary tube with |I|=¯ δ, in particular if T∈T¯ δ,andifB∈K¯ δ1/2(T0)⊂K¯ δ1/2, then |{B¯ δ∈K¯ δ(B):T∩B¯ δ=∅}| N¯ δ(B∩¯ K∩T)≤(¯ δ1/2)−σ−¯ ζ.(4.86) Therefore, it is reasonable to define that T∈T¯ δis B-dense if |{B¯ δ∈K¯ δ(B):T∩B¯ δ=∅}| ≥ (¯ δ1/2)−σ+7¯ ζ,(4.87) and otherwise Tis B-sparse. In particular, a B-dense tube satisfies μ(¯ KB∩2T)≥ω·δ4η·(¯ δ1/2)−σ+7¯ ζ·¯ δγ,(4.88) since 2Tcontains (¯ δ1/2)−σ+7¯ ζdiscs in K¯ δ(B), all of which are heavy in the sense (4.84). Then, we say that T∈T¯ δis ¯ K0-sparse if |{B∈K¯ δ1/2(T0):Tis B-dense}| ≤ M·¯ δ8¯ ζ.
38 T. ORPONEN GAFA Otherwise Tis ¯ K0-dense. This numerology is also sensible, because each tube T∈T¯ δ can meet at most Mdiscs in K¯ δ1/2(T0) (namely all of them). We next claim that only a small fraction of all the discs B∈K ¯ δintersect some ¯ K0-sparse tube, which will allow us to restrict attention to ¯ K0-dense tubes in the sequel. Using the uniform upper bound (4.86), every fixed ¯ K0-sparse tube T∈T¯ δsatisfies |{B¯ δ∈K¯ δ:T∩B¯ δ=∅}| ≤ B∈K¯ δ1/2(T0) Tis B-dense |{B¯ δ∈K¯ δ(B):T∩B¯ δ=∅}|+ B∈K¯ δ1/2(T0) Tis B-sparse ... [M·¯ δ8¯ ζ]·(¯ δ1/2)−σ−¯ ζ+M·(¯ δ1/2)−σ+7¯ ζ ≤2·M·(¯ δ1/2)−σ+7¯ ζ. Since the number of ¯ K0-sparse tubes is bounded from above by |T¯ δ|≤(¯ δ1/2)σ−γ−6¯ ζ by (4.80), we conclude that the number of discs in K¯ δwhich meet some sparse tube is bounded from above by M·(¯ δ1/2)−γ+¯ ζ. Recalling from (4.85) that |K¯ δ|≥ω·δ4η·M·(¯ δ1/2)−γ, we may finally infer that if δ, η > 0 are small enough, there exist ≥1 2·|K¯ δ|discs in K¯ δwhich intersect some ¯ K0-dense tube in T¯ δ. After this observation, we discard all ¯ K0-sparse tubes from T¯ δwithout changing notation; in other words, we assume in the sequel that all the tubes in T¯ δare ¯ K0-dense, and in particular if T∈T ¯ δ, then (4.88) holds for at least M·¯ δ8¯ ζchoices of discs B∈K¯ δ1/2(T0). Before proceeding, we claim the following almost converse to (4.80): |T¯ δ|≥(¯ δ1/2)σ−γ+2¯ ζ,(4.89) assuming that δ, η > 0 are sufficiently small. Indeed, according to (4.81)and(4.86), every tube T∈T ¯ δintersects M·(¯ δ1/2)−σ−¯ ζdiscs in K¯ δ. But since the tubes in T¯ δin total intersect ≥1 2·|K¯ δ|≥ω·δ4η·M·(¯ δ1/2)−γdiscs in K¯ δ, the lower bound (4.89) follows. Recall that the tubes in T¯ δhave the form T=π−1 θ0(IT), where IT⊂I0=[0,3¯ δ1/2] is a dyadic interval of length ¯ δ. Therefore, the left end-points of the intervals IT, with I∈T¯ δ, form a certain finite subset A1⊂¯ δ·Z∩I0. In other words, A1consists of those points x∈¯ δ·Z∩I0such that Tx:= π−1 θ0([x, x +¯ δ)) ∈T¯ δ. In particular, keep in mind that such tubes Txare all ¯ K0-dense (this will be needed in Section 4.11, which is still relatively far away). We record the following corollary of (4.80)and (4.89): (¯ δ1/2)σ−γ+2¯ ζ(4.89) ≤|A1|=|T¯ δ|(4.80) ≤(¯ δ1/2)σ−γ−6¯ ζ.(4.90) Let Π1be the uniformly distributed probability measure on A1.Then spt Π1=A1⊂[0,3¯ δ1/2]∩¯ δ·Z.(4.91)
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 39 This measure is a ¯ δ-measure in the sense of Definition 4.75.Wenotethat Π1L2= a∈A1 Π1({a})21/2 =|A1|−1/2=|T¯ δ|−1/2(4.89) ≤(¯ δ1/2)(γ−σ−2¯ ζ)/2.(4.92) We next define another discrete measure, associated to the y-coordinates of the discs B∈K¯ δ1/2(T0). In fact, for every B∈K¯ δ1/2(T0), let yB∈¯ δ1/2·Z∩π∞(B), where π∞(x, y)=yis the projection to the y-coordinate. This point exists, since π∞(B) is an interval of length 2¯ δ1/2.Wealsonotethatπ∞(B)⊂[−2,2], since B intersects ¯ K, hence B(1). Next, let A2:= {yB:B∈K ¯ δ1/2}, and let Π2be the uniformly distributed probability measure on A2.Then spt Π2=A2⊂[−2,2] ∩¯ δ1/2·Z.(4.93) Thus Π2is a ¯ δ1/2-measure, and Π2L2= a∈A2 Π2({a})21/2 =|A2|−1/2∼M−1/2(4.81) ≤(¯ δ1/2)(σ−5¯ ζ)/2.(4.94) Now Π := Π1×Π2is a discrete probability measure supported on A1×A2, but it is not evident that Π has anything to do with the set ¯ K0=¯ KT0. To clarify the connection, we need to define a certain subset of A1×A2of substantial Π-measure. Recall that that every point x∈A1was the left end-point of a certain ¯ δ-interval Ix=[x, x +¯ δ)⊂[0,3¯ δ1/2] such that Tx=π−1 θ0(Ix)∈T ¯ δ. Similarly, recall that every point y∈A2was contained in the π∞-projection of a certain disc By∈K ¯ δ1/2(T0). With this notation, we define G:= {(x, y)∈A1×A2:Tx∩B¯ δ=∅for some B¯ δ∈K¯ δ(By)}.(4.95) Morally, Gconsists of those tube-disc pairs (T,B)∈T ¯ δ×K¯ δ1/2(T0), where Tintersects (the ¯ δ-neighbourhood of) ¯ Kinside B. Note that if x∈A1, then Txis a ¯ K0-dense tube, so Tx∩B¯ δ=∅for some B¯ δ∈K ¯ δ1/2(B)(see(4.87)) for ≥M·¯ δ8¯ ζ distinct discs B∈K¯ δ1/2(T0). In other words, |π−1 1{x}∩G|≥M·¯ δ8¯ ζ,x∈A1, and consequently (recall that |A2|∼M) Π(G)= |G| |A1||A2|¯ δ8¯ ζ.(4.96)
40 T. ORPONEN GAFA 4.9 From projections to convolutions. While constructing the sets A1,A 2,G above, all the arguments were based on the structure of ¯ K0=¯ KT0relative to tubes which were pre-images of intervals under the πθ0-projection. Next, we exploit the information available for the πθ1-projection in (K2), namely that N¯ δ(πθ1(¯ K0)) ≤(¯ δ1/2)σ−γ−6¯ ζ.(4.97) The plan is, roughly speaking, to use (4.97) to show that the convolution between Π1and (θ1−θ0)Π2has nearly the same L2-normasΠ 1, where (θ1−θ0)Π2refers to the push-forward of Π2under the map y→ (θ1−θ0)y. This will be quantified in (4.101). To get started, we claim that there exists an absolute constant C>0such that πθ1−θ0(G)⊂[πθ1(¯ K0)]C¯ δ,(4.98) where G⊂A1×A2is the set defined in (4.95), and the right hand side refers to the C¯ δ-neighbourhood. To prove (4.98), fix (x, y)∈G, so that Tx∩B¯ δ=∅for some B¯ δ∈K ¯ δ1/2(By), where By∈K ¯ δ1/2(T0). Here Tx=π−1 θ0(Ix) for some interval Ix∈D¯ δ(R), whose left end-point is x. Then we know the following: (1) x∈Ixand y∈π∞(By) by definitions of Txand By. (2) There exists a point (x0,y 0)∈Tx∩B¯ δ,andB¯ δ∈K¯ δ(By), so B¯ δ∩¯ K0∩By=∅ (recall from (4.82) that K¯ δ(By) was a minimal ¯ δ-cover of ¯ K0∩By). In particular, dist((x0,y 0),¯ K0)≤¯ δand dist((x0,y 0),B y)≤¯ δ. From (2), we first deduce that πθ0(x0,y 0)∈Ix, hence |(x0+θ0y0)−x|≤¯ δ. Also, from (1)–(2) it follows that |y0−y|¯ δ1/2. Therefore, recalling also from (4.54) that |θ0−θ1|≤¯ δ1/2, we find |πθ1(x0,y 0)−πθ1−θ0(x, y)|=|(x0+θ1y0)−(x+(θ1−θ0)y)| ≤|(x0+θ0y0)−x|+|θ1−θ0|·|y0−y|¯ δ. Since πθ1(x0,y 0)∈[πθ1(¯ K0)]¯ δby (2), we conclude the proof of (4.98). Combining (4.97)–(4.98), we obtain N¯ δ(πθ1−θ0(G)) (¯ δ1/2)σ−γ−6¯ ζ.(4.99) We abbreviate θ:= θ1−θ0from now on. Combined with the lower bound (4.96)for the Π-measure of G, we will shortly infer from (4.99) a lower bound for the L2-norm of the projection πθΠ. To make this perfectly precise, we will need an additional piece of notation. We would prefer πθto map R2inside the discrete set ¯ δ·Z. So, let us, in place of πθ, consider the map ¯πθ:R2→¯ δ·Z, ¯πθ:= [x]+[θy],
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 47 We can then prove Lemma 4.110, or in other words (4.111)–(4.112). Recall from (4.118) that ¯ U⊂¯ Isatisfied μA(¯ U)≥(¯ δ1/2)α+27¯ ζ+4=: (¯ δ1/2)α+ω, where ω:= 27¯ ζ+4. Since ¯ Iis an interval of length ¯ δ1/2, the rescaled and re-normalised measure μ¯ I:= (¯ δ1/2)−α·T¯ IμA is an (α, Cα)-regular measure, and T¯ I(¯ U)⊂[0,1] is a Borel set with μ¯ I(T¯ I(¯ U)) ≥ (¯ δ1/2)ω. Therefore, Proposition 4.120 can be applied to μ¯ Iand the set T¯ I(¯ U). The scale “δ” at which the proposition is applied is now ¯ δ1/2, which in our notation (namely ¯ δ=2 −mN )canbewrittenas ¯ δ1/2=2 −m·(N/2). Now, the conclusion (4.121) of Proposition 4.120 would literally say something about the intersections of T¯ I(¯ U) with dyadic intervals of lengths between ¯ δ1/2and 1. In the following, we already translate this information back to ¯ U, and scales between ¯ δand ¯ δ1/2(these correspond to indices s∈{N/2,...,N−1}): there exists a family of good indices G⊂{N/2,...,N −1}such that max I∈D2−ms N2−(s+1)m(¯ U∩I)≥2(1−ρ)αm+3,s∈G,(4.126) and |G| (4.122) ≥1−27¯ ζ+4 αρ +5Cα αρm ·N 2. This is the lower bound we claimed in (4.111). Finally, a combination of (4.126)and the inclusion (4.119) finally allows us to estimate from below the branching numbers R1 s. Assume that s∈G⊂{N/2,...,N −1}, so that (4.126) holds, and let I∈D sm be some dyadic interval with N2−m(s+1) (U2¯ δ∩(I+y0θ0)) (4.119) ≥N2−m(s+1) (( ¯ U+y0θ0)∩(I+y0θ0)) =N2−m(s+1) (¯ U∩I)≥2(1−ρ)αm+3. Since 2−m(s+1) ≥¯ δ, this implies that max I∈D2−ms N2−m(s+1) (U∩I)≥2(1−ρ)αm. The left hand side is a lower bound for R1 s, by the definition of these branching numbers. This proves (4.112) for all s∈G, and hence completes the proof of Proposition 3.18.
48 T. ORPONEN GAFA 5 Proof of Proposition 4.32 We repeat the statement: Proposition 5.1. Let θ∈[0,1], and let 1≤M≤N<∞be constants, let 0<r≤R≤1, and let μbe a (γ,Cγ)-regular measure with γ∈[0,2],Cγ>0, and K:= spt μ⊂R2. Abbreviate μs:= μ|B(s)for s>0. Then, there exist absolute constants c, C > 0such that μ1(Hθ(CN,[r, 1])) ≤μ1(Hθ(cM, [4R, 5])) + CC2 γ·μ4(Hθ(cN M,[4r, 7R])).(5.2) Here we abbreviated Hθ(K, M, [r, R]) =: Hθ(M,[r, R]). During the proof, we will also abbreviate mK,θ =: mθ. These notions were introduced in Definitions 2.3 and 2.4 . ProofofProposition5.1.For t∈R, write Fr(t):=Nr(B(2) ∩Kr∩π−1 θ{t})andFR(t):=NR(B(3) ∩KR∩π−1 θ{t}),(5.3) where Ksrefers to the s-neighbourhood of K. We also define the following variant of FR: FR(t):=N4R(B(4) ∩K4R∩π−1 θ{t}),t∈R. The first point to observe about the definition fo Fris that if x∈B(1), then B(x, 1) ⊂B(2), and hence mθ(x|[r, 1]) = Nr(B(x, 1) ∩Kr∩π−1 θ{πθ(x)})≤Fr(πθ(x)). In particular, x∈B(1) ∩Hθ(CN,[r, 1]) =⇒Fr(πθ(x)) ≥CN. (5.4) Similarly, x∈B(1)\Hθ(cM, [4R, 5]) =⇒ FR(πθ(x)) ≤cM, (5.5) because if x∈B(1), then B(4) ⊂B(x, 5), and hence FR(πθ(x)) = N4R(B(4) ∩K4R∩π−1 θ{πθ(x)})≤mθ(x|[4R, 5]). There is also a useful relationship between Fr(t) and the push-forward measure μθ:= πθμ1, which reads as follows: if I⊂Ris an interval of length r, then μθ(I)Cγrγ·sup t∈3I Fr(t).(5.6) Indeed, by the (γ,Cγ)-regularity of μ, an upper bound for μθ(I) is given by 4Cγrγ·n, where “n” is the largest number of disjoint r-discs centred at B(1)∩K∩π−1 θ(I). All of these r-discs are contained in B(2) ∩Kr∩π−1 θ(3I). Now, the average line π−1 θ{t},
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 49 with t∈3I, meets ≥n/3 of these discs, since the probability of hitting each disc individually is 1 3. Hence, for some t∈3I,itholds Fr(t)=Nr(B(2) ∩Kr∩π−1 θ{t})n. This proves (5.6). Let KRbe a boundedly overlapping cover of Kr∩B(2) by discs of radius R, centred at K.ThusBR⊂KR∩B(3) for all BR∈K R.ForBR∈K Rfixed, we define Fr(BR)(t):=Nr(BR∩Kr∩π−1 θ{t})and Fr(BR)(t):=N4r(4BR∩K4r∩π−1 θ{t}). We claim the following inequality for every t∈R: Fr(t)≤C1 BR∈KR Fr(BR)(t),(5.7) where C1>0 is an absolute constant. This inequality means that an upper bound for the r-discs intersecting π−1 θ{t}can be obtained by finding an upper bound on both R-discs intersecting π−1 θ{t}, and an upper bound for r-discs intersecting π−1 θ{t} inside any given R-disc. To prove (5.7), fix t∈R, and let {x1,...,x m}be a maximal r-separated subset of B(2) ∩Kr∩π−1 θ{t}, so that Fr(t)m. For every 1 ≤j≤m, the point xj∈Krlies in BR∩Kr∩π−1 θ{t}for some BR∈K R. Consequently, Fr(t)m≤ BR∈KR |{1≤j≤m:xj∈BR∩Kr∩π−1 θ{t}}| BR∈KR Fr(BR)(t), where the final inequality used the r-separation of the points xj.Thisproves(5.7). Next we claim that, for every t∈R, |{BR∈K R:Fr(BR)(t)=0}| ≤ C2FR(t),(5.8) where C2>0 is another absolute constant. Indeed, for every BR∈K Rwith Fr(BR)(t)= 0, there exists xBR∈BR∩Kr∩π−1 θ{t}⊂B(3) ∩KR∩π−1 θ{t}. Since the discs BRhave bounded overlap, the points xBRobtained this way are essentially R-separated (more precisely: contain an R-separated subset of comparable cardinality), and consequently FR(t)=NR(B(3) ∩KR∩π−1 θ{t})|{BR∈K R:Fr(BR)(t)=∅}|, as stated in (5.8). We combine (5.7)and(5.8) to reach the following useful inequality, for any H≥1: Fr(t)≤C1 BR∈KR Fr(BR)(t)≥H Fr(BR)(t)+C1C2HFR(t),t∈R.
50 T. ORPONEN GAFA In particular, choosing H:= N/M and C≥2C1C2(this “C” is the absolute constant referred to in the statement of the proposition), we find that if FR(t)≤Mfor some t∈R, then Fr(t)≤C1 BR∈KR Fr(BR)(t)≥N/M Fr(BR)(t)+CN/2. In particular, we derive the following key observation, again valid for any t∈R: Fr(t)≥CN and FR(t)≤M=⇒Fr(t)≤2C1 BR∈KR Fr(BR)(t)≥N/M Fr(BR)(t). (5.9) To apply (5.9), we start by making the “trivial” estimate μ1(Hθ(CN,[r, 1])) ≤μ1(Hθ(cM, [4R, 5])) + μ1(Hθ(CN,[r, 1])\Hθ(cM, [4R, 5])). Consequently, (5.2) will follow once we manage to prove that μ1(Hθ(CN,[r, 1])\Hθ(cM, [4R, 5])) C2 γ·μ4(Hθ(cN M,[4r, 7R])).(5.10) Let x∈B(1) ∩Hθ(CN,[r, 1])\Hθ(cM, [4R, 5]) be arbitrary. Then, we infer from (5.4)–(5.5) that Fr(πθ(x)) ≥CN and FR(πθ(x)) ≤cM. In particular, the left hand side of (5.10) satisfies μ1(Hθ(CN,[r, 1])\Hθ(cM, [4R, 5])) ≤1{Fr(t)≥CN and FR(t)≤cM}(t)dμθ(t).(5.11) Let Dr(R) be the collection of dyadic intervals of Rof length r. We decompose the integral on the right as I∈Dr(R)I 1{t:Fr(t)≥CN and FR(t)≤cM}(t)dμθ(t). Fix I∈D r(R), and assume that there exists at least one point t0∈Iwith Fr(t0)≥ CN and FR(t0)≤cM (otherwise the corresponding term is zero). For such an interval I∈D r(R), we simply apply the estimate (5.6)toevaluate I 1{t:Fr(t)≥CN and FR(t)≤cM}(t)dμθ(t)≤μθ(I)Cγrγ·sup t∈3I Fr(t).(5.12) Let t1∈3Ibe a point which nearly attains the supremum on the right, say, up to a constant 2, and moreover Fr(t1)≥Fr(t0)≥CN. Recall from (5.3) that FR(t):=NR(B(3) ∩KR∩π−1 θ{t}). We claim that if the absolute constant c>0 is chosen small enough, then FR(t1)≤M. Indeed, let
GAFA ON ARITHMETIC SUMS OF AHLFORS-REGULAR SETS 51 {x1,...,x m}⊂B(3)∩π−1 θ{t1}∩KRbe a maximal R-separated set, with m∼FR(t1). Then, since |t1−t0|≤2r, the line π−1 θ{t0}intersects B(5) ∩K4Rin mpoints, which are 4R-separated. Consequently cM ≥ FR(t0)=N4R(B(5) ∩K4R∩π−1 θ{t0})m∼FR(t1), and the claim follows. Therefore, Fr(t1)≥CN and FR(t1)≤M,andweareina position to apply (5.9) to the point t1: Fr(t1)≤2C1 BR∈KR Fr(BR)(t1)≥N/M Fr(BR)(t1).(5.13) We next claim that if BR∈K Ris one of the discs appearing in the sum in (5.13), that is, Fr(BR)(t1)≥N/M, then Fr(BR)(t1)Cγ rγ5I 1{s: Fr(BR)(s)≥cN/M}(s)dπθ(μ|3BR)(s).(5.14) To prove this, we first claim that πθ(μ|3BR)([t1−2r, t1+2r]) rγ Cγ·Nr(BR∩Kr∩π−1 θ{t1})= rγ Cγ·Fr(BR)(t1).(5.15) To see this, note that every point x∈BR∩Kr∩π−1 θ{t1}lies at distance ≤rfrom apointin2BR∩K∩π−1 θ([t1−r, t1+r]), and B(x, r)⊂3BR∩π−1 θ([t1−2r, t1+2r]), since r≤R. Hence, πθ(μ|3BR)([t1−2r, t1+2r]) exceeds, by a constant factor, the total μ-measure of discs B(x, r) obtained in this way. This measure is bounded from below by the right hand side of (5.15). To deduce (5.14)from(5.15), it remains to observe that [t1−2r, t1+2r]⊂{s: Fr(BR)(s)≥cN/M},(5.16) assuming that the absolute constant c>0 was chosen small enough. To see this, recall that Fr(BR)(s)=N4r(4BR∩K4r∩π−1 θ{s}). The point is that since Fr(BR)(t1)≥N/M (we are only considering these terms in (5.13)), there exist N/M points in BR∩Kr∩π−1 θ{t1},whicharer-separated. Now, if s∈[t1−2r, t1+2r], then π−1 θ{s}intersects K4rwhenever π−1 θ{t1}intersects Kr, and these intersections occur inside 4BR. Hence Fr(BR)(s)Fr(BR)(t1)≥N/M for all s∈[t1−2r, t1+2r]. This completes the proof of (5.16), hence (5.14). We record at this point that x∈3BRand FR(BR)(πθ(x)) ≥cN/M =⇒x∈Hθ(cN M,[4r, 7R]).(5.17)
52 T. ORPONEN GAFA This is so, because if x∈3BR, then B(x, 7R)⊃4BR, and hence mθ(x|[4r, 7R]) ≥N4r(4BR∩K4r∩π−1 θ{πθ(x)})= Fr(BR)(πθ(x)). Combining (5.12)–(5.14), we first learn that I 1{t:Fr(t)≥CN and FR(t)≤cM}(t)dμθ(t) C2 γ BR∈KR5I 1{s: Fr(BR)(s)≥cN/M}(s)dπθ(μ|3BR)(s). Summing over I∈D r(R), using the bounded overlap of the intervals 5I, applying (5.17), and finally using the bounded overlap of the discs 3BR⊂B(4), BR∈K R, we find that 1{t:Fr(t)≥CN and FR(t)≤cM}(t)dμθ(t) C2 γ BR∈KRR 1{s: Fr(BR)(s)≥cN/M}(s)dπθ(μ|3BR)(s) ≤C2 γ BR∈KR μ(3BR∩Hθ(cN M,[4r, 7R])) C2 γ·μ(B(4) ∩Hθ(cN M,[4r, 7R])). Recalling (5.11), this concludes the proof of (5.10), and the proof of the proposition. Acknowledgments I would like to thank the referee for a very careful reading of the manuscript, and for making a large number of helpful suggestions. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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54 T. ORPONEN GAFA T. Orponen Department of Mathematics and Statistics, University of Jyv¨askyl¨a, P.O. Box 35 (MaD), 40014 Jyv¨askyl¨a, Finland. [email protected] Received: July 12, 2021 Revised: November 15, 2021 Accepted: November 27, 2021