Structure of sets with nearly maximal Favard length
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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/ Structure of sets with nearly maximal Favard length © 2024 MSP (Mathematical Sciences Publishers) Published version Chang, Alan; Dąbrowski, Damian; Orponen, Tuomas; Villa, Michele Chang, A., Dąbrowski, D., Orponen, T., & Villa, M. (2024). Structure of sets with nearly maximal Favard length. Analysis and PDE, 17(4), 1473-1500. https://doi.org/10.2140/apde.2024.17.1473 2024
ANALYSIS & PDE msp Volume 17 No. 4 2024 ALAN CHANG , DAMIAN D˛ABROWSKI , TUOMAS ORPONEN AND MICHELE VILLA STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH
ANALYSIS AND PDE Vol. 17 (2024), No. 4, pp. 1473–1500 DOI: 10.2140/apde.2024.17.1473 msp STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH ALAN CHANG , DAMIAN D˛ABROWSKI , TUOMAS ORPONEN AND MICHELE VILLA Let E⊂B( 1 )⊂R2 be an H1 measurable set with H1(E) < ∞ , and let L⊂R2 be a line segment with H1(L)=H1(E) . It is not hard to see that Fav(E)≤Fav(L) . We prove that in the case of near equality, that is, Fav(E)≥Fav(L)−δ, the set E can be covered by an ϵ -Lipschitz graph, up to a set of length ϵ . The dependence between ϵ and δis polynomial: in fact, the conclusions hold with ϵ=Cδ1/70 for an absolute constant C>0. 1. Introduction 1473 2. Measure-theoretic preliminaries 1475 3. Proof of Theorem 1.1 in two main steps 1477 4. Proof of Proposition 3.3 1481 5. Proof of Proposition 3.11 1488 6. The grid example 1495 Appendix: Lines spanned by rectifiable curves 1498 Acknowledgements 1499 References 1499 1. Introduction Let E⊂R2be H1measurable with H1(E) < ∞. We recall the definition of Favard length: Fav(E)= Z π 0 H1(πθ(E)) dθ. Here πθ:R2→R is the orthogonal projection πθ(x)=x·(cos θ, sin θ) . The definition of Fav(E) can be posed without the assumption H1(E) < ∞ , but this hypothesis will be crucial for most of the statements below, and it will be assumed unless otherwise stated. A fundamental result in geometric measure theory is the Besicovitch projection theorem [1939] which relates Favard length and rectifiability: Fav(E) > 0 if and only if H1(E∩0) > 0 for some Lipschitz graph 0⊂R2 — in other words, E is not purely 1-unrectifiable. D ˛abrowski and Orponen are supported by the Academy of Finland via the project Incidences on Fractals, grant 321896. Orponen is also supported by the Academy of Finland via the project Quantitative rectifiability in Euclidean and non-Euclidean spaces, grants 309365, 314172. Villa was supported by a starting grant of the University of Oulu. MSC2020: primary 28A75; secondary 28A78. Keywords: Favard length, Besicovitch projection theorem, Lipschitz graph. © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.
1474 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA The proof of the Besicovitch projection theorem is famous for being difficult to quantify, partly because of its reliance on the Lebesgue differentiation theorem: it is hard to decipher from the argument just how large the intersection E∩0 is, and what the Lipschitz constant of 0 is. In fact, it is nontrivial to even find the right question: for example, if E⊂B( 1 ) , H1(E)= 1, and Fav(E)≥δ for some small but fixed constant δ > 0, then it is not true that H1(E∩0) ≥ϵ for some ϵ−1 -Lipschitz graph 0⊂R2 , where ϵ=ϵ(δ) > 0. We construct a relevant counterexample in Section 6. In Theorem 1.1, we show that similar counterexamples are no longer possible if the assumption “ Fav(E)≥δ ” is upgraded to “ Fav(E)≥ 2 H1(E)−δ ” for a sufficiently small constant δ > 0. The number 2 comes from the fact that Fav([ 0 , 1 ]×{ 0 })= 2 and that [ 0 , 1 ]×{ 1 } has the maximal Favard length among sets of length unity (see (2.4)). Theorem 1.1. For every ϵ > 0there exists δ > 0such that the following holds : Let E⊂B( 1 ) be an H1measurable set with H1(E) < ∞,and assume that Fav(E)≥Fav(L)−δ, (1.2) where L⊂R2 is a line segment with H1(L)=H1(E) . Then , there exists an ϵ -Lipschitz graph 0⊂R2 such that H1(E∩0) ≥H1(E)−ϵ. One can take δ=ϵ70/C for an absolute constant C >1. By an ϵ -Lipschitz graph we mean a set of the form R(Graph f) , where R:R2→R2 is a rotation, and Graph f= {(x,f(x)) :x∈R}is the graph of an ϵ-Lipschitz function f:R→R. This means that |f(x)−f(y)| ≤ ϵ|x−y| for all x,y∈R . It is easy to check that the intersection of an ϵ -Lipschitz graph with B( 1 ) is contained in the 2 ϵ -neighborhood of some line ℓ⊂R2 , so in particular the same is true of E∩0 (as in Theorem 1.1). Theorem 1.1 shows that if Fav(E) is nearly maximal, the Besicovitch projection theorem can be quantified in a very strong way, whereas the example constructed in Section 6 shows that any similar conclusion fails completely if we make the weaker assumption Fav(E)≥δ . However, it remains plausible that the assumption Fav(E)≥δ is sufficient to guarantee a quantitative version of Besicovitch’s theorem under the additional assumption that E is 1-Ahlfors regular, or satisfies other multiscale 1-dimensionality hypotheses. For recent partial results, and more discussion on this question; see [Davey and Taylor 2022; Martikainen and Orponen 2018;Orponen 2021;Tao 2009]. The problem is closely related to Vitushkin’s conjecture [1967] on the connection between analytic capacity and Favard length; see [Chang and Tolsa 2020;D ˛abrowski and Villa 2022]. We briefly mention another closely related topic: if E⊂R2 is self-similar and purely 1-unrectifiable, then Fav(E)= 0 by the Besicovitch projection theorem. It is an interesting and very popular question to attempt quantifying the (sharp) rate of decay at which Fav(En)→ 0, where En is the n -th iteration of the self-similar set. For recent developments; see [Bateman and Volberg 2010;Bond et al. 2014;Bond and Volberg 2010;2012;Bongers and Taylor 2023;Cladek et al. 2022;Łaba and Zhai 2010;Łaba 2015;Łaba and Marshall 2022;Nazarov et al. 2010;Peres and Solomyak 2002].
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1475 It is tempting to consider the following scale-invariant version of Theorem 1.1:for any ϵ1, ϵ2> 0there exists δ > 0such that if E ⊂B(1)satisfies H1(E) < ∞and Fav(E)≥(1−δ) Fav(L), then there exists an ϵ1 -Lipschitz graph 0⊂R2 such that H1(E\0) ≤ϵ2H1(E) .Note that for sets E with H1(E)∼ 1 this statement is equivalent to Theorem 1.1; however, in general, the statement is false. Consider a set En consisting of four horizontal segments of length 1 /n placed in the corners of [ 0 , 1 ]2 . Clearly, one may cover at most half of En using a single 1-Lipschitz graph. At the same time, Fav(En)/ Fav(Ln)→ 1, where Ln= [ 0 , 4 /n] × { 0 } . To see this, let Bn:= {θ∈ [ 0 , π) :πθis not injective on En} . Note that H1(Bn)→ 0, and at the same time for θ /∈Bn we have H1(πθ(En)) =H1(πθ(Ln)) . It follows easily that Fav(En)/ Fav(Ln)→1. 1A. Outline of the paper. A quick outline of the article is as follows: In Section 2 we introduce Crofton’s formula and prove that line segments maximize Favard length. In Section 3 we prove Theorem 1.1 using two main propositions, Proposition 3.3 and Proposition 3.11. The moral of these propositions is discussed at the beginning of Section 3. These two propositions are then proven in Section 4 and Section 5, respectively. Section 6 contains the counterexample mentioned above to the scale-invariant version of Theorem 1.1. Finally, in the Appendix we give an exact formula for the measure of lines spanned by two rectifiable curves — this is used in Section 5 but it might be of independent interest. 2. Measure-theoretic preliminaries 2A. Notation. For x∈Rd and r> 0, the notation B(x,r) stands for a closed ball of radius r centered at x . For A⊂Rd , we denote the cardinality of A by # A , and we write A(r):= {x∈Rd:dist(x,A)≤r} , where “ dist ” is Euclidean distance. For f,g≥ 0, we write f≲g if there exists an absolute constant C> 0 such that f≤Cg . The notation f≳g means the same as g≲f , and f∼g is shorthand for f≲g≲f . If the constant C>0 is allowed to depend on some parameter p, we signify this by writing f≲pg. 2B. Integralgeometry and Crofton’s formula. One of the main tools is Crofton’s formula for rectifiable sets, which states the following: if E⊂R2 is an H1 measurable 1-rectifiable set with H1(E) < ∞ , then H1(E)=1 2 Z π 0 Z R #(E∩π−1 θ{t})dt dθ. (2.1) Equation (2.1) is false without the rectifiability assumption, but the inequality “≥” remains valid in this case. This formula (and the inequality) is a special case of a more general relation between Hausdorff measure and integralgeometric measure for n -rectifiable sets in Rd ; see [Federer 1947, Theorem 9.7; 1969, Theorem 3.2.26]. We next rephrase the formula (2.1) in slightly more abstract terms. We define the following measure ηon the family A:= A(2,1)of all affine lines in R2: η(L)= Z π 0 H1({t∈R:π−1 θ{t} ∈ L})dθ, L⊂A.
1476 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA With this notation, the Crofton formula (2.1) can be rewritten as H1(E)=1 2 Z L(E) #(E∩ℓ) dη(ℓ), (2.2) where L(E):= {ℓ∈A:E∩ℓ= ∅}. Lemma 2.3 (the line segment maximizes Favard length). If E⊂R2 is H1 measurable ,H1(E) < ∞, and L⊂R2is a line segment with H1(E)=H1(L),then Fav(E)≤Fav(L)(2.4) and Fav(L)−Fav(E)≥ Z L(E) (#(E∩ℓ) −1)dη(ℓ). (2.5) If E is rectifiable,then equality holds in (2.5). Proof. Suppose E⊂R2 is H1 measurable, H1(E) < ∞ , and L⊂R2 is a line segment with H1(E)=H1(L) . Then Fav(E)=η(L(E)) = Z L(E) 1dη(ℓ) ≤ Z L(E) #(E∩ℓ) dη(ℓ) ≤2H1(E). (2.6) If we replace E with the line segment L , then equality holds in both inequalities above. Thus, Fav(L)= 2H1(L)=2H1(E), which combined with (2.6) (for E) proves (2.5). Next, (2.4) follows from the fact that the right-hand side of (2.5) is nonnegative. Finally, if E is rectifiable, then the second inequality in (2.6) becomes an equality, which implies that equality holds in (2.5).□ 2C. Coarea formula. We now record another tool in the proof of Theorem 1.1. It is closely related to Crofton’s formula, but only considers the intersections with lines in a fixed direction. The price to pay is that the tangent of the rectifiable set enters the formula. It is a generalization of the following standard fact: if f: [a,b] → Ris α-Lipschitz, then H1{(t,f(t)) :t∈ [a,b]}= Z b ap1+f′(t)2dt ≤p1+α2(b−a). Lemma 2.7 (coarea formula). Let α > 0. Let E⊂R2 be a countable union of α -Lipschitz graphs over the x-axis. Then, H1(A)≤p1+α2 Z R #(A∩π−1 0{t})dt (2.8) for all H1measurable subsets A ⊂E. (Recall that π0:R2→Ris the projection onto the x-axis.) Proof. This follows from the coarea formula for rectifiable sets. (See, e.g., [Federer 1969, Theorem 3.2.22] or [Krantz and Parks 2008, Theorem 5.4.9].) □
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1477 3. Proof of Theorem 1.1 in two main steps In this section we prove Theorem 1.1 using Propositions 3.3 and 3.11 introduced below. Proposition 3.3 says roughly the following: Assume a priori that E is a union of line segments (we reduce matters to something like this in Section 3A), fix a small angle α > 0, and let Eℓ,α be the union of those segments which make an angle ≤α with some given line ℓ⊂R2 . Evidently E can be expressed as the union of ∼ 1 /α sets of the form Eℓ,α .Proposition 3.3 says that if the parameter δ in our hypothesis Fav(E)≥Fav(L)−δ is sufficiently small, then each of the sets Eℓ,α can be (almost) covered by a single (∼α) -Lipschitz graph over ℓ . After this step, we know that E can be (almost) covered by a union of ∼ 1 /α Lipschitz graphs with constant ∼α . Thereafter, to complete the proof of Theorem 1.1, it remains to show that only one of these graphs can have a nontrivial intersection with E . This uses the hypothesis Fav(E)≥Fav(L)−δ once more, and is accomplished in Proposition 3.11 (and the discussion right below). 3A. Step 1 : first reductions. Let E⊂R2 be a Borel set with H1(E) < ∞ . We start with the following simple lemma: Lemma 3.1. It suffices to prove Theorem 1.1 under the additional assumption that E is a finite union of disjoint C1curves. Proof. We may assume that E⊂B( 1 ) is rectifiable, because by the Besicovitch projection theorem, the rectifiable part of E continues to satisfy all the assumptions of Theorem 1.1 (with the same constant δ > 0). By this assumption, H1 almost all of E can be covered by a countable union of C1 -curves. Decomposing the curves further, we may assume that they are disjoint, and for any given η > 0 we may write E= M1 S j=1 (γj∩E)∪S, where H1(S)≤η, and H1(E∩γj)≥(1−η)H1(γj). Now, the set E:= SM1 j=1γjsatisfies H1(E)≤(1−η)−1H1(E)and Fav(E)≥Fav(E)−η and is additionally a finite union of disjoint C1 -curves. If Theorem 1.1 is already known under this additional assumption, we may now infer that H1(E\0) ≤ϵ , where 0 is an ϵ -Lipschitz graph. But then also H1(E\0) ≤H1(E\E)+H1(E\0) ≤η+ϵ , and Theorem 1.1 follows for E by choosing the parameters ϵ, η appropriately. □ 3B. Step 2 : minigraphs and how to merge them. By Lemma 3.1, we may assume that E is a finite union of disjoint C1 -curves γ1, . . . , γM1 . We further chop up each curve γj into connected pieces whose tangent varies by less than α , where α is a small constant depending on ϵ fixed later on (see (3.5)). At this point, we have managed to write E as a finite union of disjoint α -Lipschitz graphs γ1, . . . , γM′ 1 , where M1≤M′ 1<+∞ . At this point we have no quantitative control on the constant M′ 1 . Each of the graphs γj will be called a minigraph, and their collection is denoted E . The main tasks in Theorem 1.1 are to combine the minigraphs into roughly 1 /α bigger graphs, and to show that nearly all of E lies on just one of these bigger graphs.
1478 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA To begin with, let M2= ⌈πα−1⌉ ∼ α−1 . We would like to divide the collection of minigraphs E into M2 subcollections E1, . . . EM2 , each of them containing the minigraphs with roughly the same direction. To do this, we consider M2vectors of the form vk:=(cos(kπ/M2), sin(kπ/M2)) for 1 ≤k≤M2∼α−1 . Observe that for each minigraph γ∈E there exists k∈ { 1 ,...,M2} such that γ is a 2 α -Lipschitz graph over the line span(vk) . The vector vk will be called the direction of the minigraph (if there are several suitable vectors for one minigraph, fix any one of them; we will only need to know that each minigraph is a 2 α -Lipschitz graph over the line spanned by its direction). Statements about the (relative) angles of minigraphs should always be interpreted as statements about the relative angles of the direction vectors vk . For k∈ { 1 ,...,M2} fixed, we define Ek⊂E as the collection of minigraphs with direction vk . We suggest that the reader visualize the minigraphs as line segments I with (I,span(vk)) ≤α . It seems likely that Theorem 1.1 could be reduced to the case where E is a finite union of line segments, but employing the minigraphs seems to spare us some unnecessary steps. We write Ek:= SEk. Thus E=E1∪···∪EM2.(3.2) It turns out that, except for a small error, each set Ek is covered by a single Lipschitz graph with constant ∼α over span(vk) . Indeed, note that Lemma 2.3 and (1.2) together imply RL(E) # (E∩ℓ)− 1 dη(ℓ) ≤δ. Since for each k∈ { 1 ,...,M2} we have Ek⊂E , one sees immediately that L(Ek)⊂L(E) and # (Ek∩ℓ) ≤ # (E∩ℓ) , so that we also get RL(Ek) # (Ek∩ℓ) − 1 dη(ℓ) ≤δ. Then, the desired Lipschitz graph 0 covering most of Ekis constructed in the following proposition, whose proof will be carried out in Section 4: Proposition 3.3. There exist absolute constants C0, α0∈( 0 , 1 ) and Clip > 1such that the following holds : Let δ, ϵ ∈( 0 , 1 ) and α∈( 0 , α0) be such that δ≤C0α3ϵ2 . Let E⊂B( 1 ) be a set with H1(E) < ∞ of the form E=S γ∈E γ, where Eis a finite collection of disjoint α-Lipschitz graphs over a fixed line L ⊂R2 . Assume further that E satisfies Z L(E) (#(E∩ℓ) −1)dη(ℓ) ≤δ. (3.4) Then,there exists a Lipschitz graph 0over L,with Lipschitz constant at most Clip ·α,such that H1(E\0) ≤ϵ. We remark that the absolute constants α0 and Clip are such that α0≤C−1 lip . In particular, the Clipα-Lipschitz graph 0from above has a Lipschitz constant bounded by 1. The proof of Proposition 3.3 recycles most of the ideas from Besicovitch’s original proof of the Besicovitch projection theorem [1939]. Indeed, we first use (in Lemma 4.1) the assumption (3.4) to show that E must have arbitrarily low conical density in arbitrarily wide cones centered at most points x∈E , whose axis is perpendicular to the line L . The quantifications of arbitrarily low and arbitrarily wide can
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1479 be made stronger by reducing the value of the constants α and δ . After this step, we use Besicovitch’s two cones argument (quantified in Lemma 4.18) to show that most of E can be contained on a Lipschitz graph over L. 3C. Step 3 : there can only be one graph. In Proposition 3.3 we managed to pack a majority of each set Ej (as defined in (3.2)) to a Lipschitz graph of constant ∼α , up to errors which tend to zero as δ→ 0 in the main assumption (1.2). However, at this point there might be up to ∼α−1 distinct Lipschitz graphs, and to prove Theorem 1.1, we would (roughly speaking) like to reduce their number to one. That this should be possible is not hard to believe: if E consists of several distinct Lipschitz graphs of substantial measure, which nevertheless cannot be fit into a single Lipschitz graph, then Fav(E) cannot possibly be maximal. We turn to the details. We recall the given constant ϵ > 0 from the statement of Theorem 1.1, and we set δ:= ϵ70 Cthm for a sufficiently large absolute constant Cthm >1. We define also α:= ϵ Calp 10 (3.5) for some universal Calp > 1. The universal constant Cthm will depend on Calp , whereas Calp depends only on Clip and another constant Csep , which is introduced below. The additional constant Calp will make it easier for us to ensure that the Lipschitz graph 0 obtained from the application of Proposition 3.3 has Lipschitz constant smaller than ϵ; see the discussion around (3.8). We record that α7=C−70 alp ϵ70 =CthmC−70 alp ·δ. (3.6) Recall, once more, the decompositions E=E0∪···∪EM2 and E=E0∪···∪EM2 from the previous subsection: this decomposition depends on the parameter α fixed above. In addition to the decomposition E=E0∪· · ·∪ EM2 , we will also need another, coarser, decomposition of E in this section. Write κ:= 1 10 , fix M3∼α−κ, and decompose E=F0∪···∪FM3in such a way that •each Fkis a union of finitely many consecutive families Ej, and •Fk contains those minigraphs whose direction makes an angle no larger than ακ with wk= (cos(kπ/M3), sin(kπ/M3)) for 0 ≤k≤M3. We write Fk:= SFk,0≤k≤M3∼α−κ. At this point, we consider two distinct cases. Let Csep be a large constant depending only on the absolute constant Clip appearing in Proposition 3.3 (the letters sep stand for separation). Thus, the constant Csep is also absolute, and we may (and will) assume that Calp is large relative to Csep.
1486 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA It follows that Z L(E) #(E∩ℓ) −1dη(ℓ) ≥ Z J(α′) Z #(E∩π−1 θ{t})−1dt dθ (4.17) ≳ Z J(α′) Z πθ(Eθ) #(Eθ∩π−1 θ{t})dt dθ (4.14) ≳ Z J(α′)P T∈Tθ Z πθ(Eθ∩T) #(Eθ∩π−1 θ{t})dt dθ (4.7) ≳α Z J(α′)P T∈Tθ H1(Eθ∩T)dθ (4.16) ≥α 2 Z J(α′)P T∈Tθ H1(E∩T)dθ (4.14) ≥α Z J(α′) H1(R′ θ)dθ (4.12) ≥α H·H1(R2). Recalling once again from (3.4) that the left-hand side above is ≤δ, we deduce that H1(R2)≲δH α∼δ εα2, which is (4.8) for R2. The proof of Lemma 4.1 is complete. □ Next, repeating the classical two cones argument of Besicovitch (e.g., [Mattila 1995, Lemma 15.14]), we show that we can pack most of points of low conical density into a single Lipschitz graph: Lemma 4.18 (most low conical density points fit into a Lipschitz graph). Let E⊂B( 1 )⊂R2 and let ε∈(0,1), β ∈0,1 2. Then,there exists a 2β-Lipschitz graph 0⊂R2over the x-axis such that H1({x∈E:2∗ E,β (x)≤ε} \ 0) ≲ε/β. Proof. Let G= {x∈E:2∗ E,β (x)≤ε} . Our task is to find a subset 0⊂G with H1(G\0) ≲ε/β and the property C2β(x)∩0= {x}for all x∈0. Then 0extends to a 2β-Lipschitz graph, as desired. Let B be the set of points x∈G with the “bad” property that there exists a point y∈G∩C2β(x) with y= x . The goal is to show that H1(B)≲ε/β . For each x∈B , let r(x)=sup{|x−y| : y∈G∩C2β(x)} , so B∩C2β(x)⊂B(x,r(x)), x∈B.(4.19) See Figure 2 for an illustration. Let Txbe the tube around the vertical line passing through xwith w(Tx):= 1 10 βr(x). Then Tx\Bx,1 2βr(x)⊂C1(x)⊂C2β(x)⊂Cβ(x). (4.20)
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1487 Txy(x) Bx,1 2βr(x) β x Figure 2. Containing the tube Tx in the union of the cones Cβ(x) and Cβ(y(x)) . The dotted cone illustrates C2β(x)∋y(x). (Recall that 2 β≤ 1.) In particular, (4.20) implies Tx\B(x,r(x)) ⊂C2β(x) . Using this, we observe that B∩Tx⊂B(x,r(x)) ∪ [(B∩Tx)\B(x,r(x))] =B(x,r(x)) ∪ [B∩(Tx\B(x,r(x)))] ⊂ B(x,r(x)) ∪ [B∩C2β(x)](4.19) ⊂B(x,r(x)). (4.21) Choose a point y(x)∈G∩C2β(x) such that |x−y(x)| ≥ 9 10r(x) . A slightly more delicate geometric fact is that Tx⊂Cβ(x)∪Cβ(y(x)). This is an exercise in elementary geometry; see Figure 2 (or the proof in [Mattila 1995, Lemma 15.14]for a more formal argument): the disc Bx,1 2βr(x) , and in particular the intersection Tx∩Bx,1 2βr(x) , is contained in the cone Cβ(y(x)) , whereas the rest of Tx is contained in Cβ(x) , as already noted in (4.20). Consequently, using (4.21), the trivial inclusion B(x,r(x)) ⊂B(y(x), 2 r(x)) , and x,y(x)∈G , we have H1(B∩Tx)≤H1(Cβ(y(x), 2r(x)) ∩E)+H1(Cβ(x,r(x)) ∩E)≤2εr(x)+εr(x)≤30(ε/β) ·w(Tx). We have now shown that every point x∈B is contained on the central line of a vertical tube Tx satisfying the estimate above. By the Besicovitch covering theorem, as in the proof of Lemma 4.1, we may then find a countable, boundedly overlapping subfamily T of these tubes which still cover B . All the tubes intersect B(1)⊃B, so PT∈Tw(T)≲1. It follows that H1(B)≤X T∈T H1(B∩T)≤30ε βX T∈T w(T)≲ε β. This completes the proof of Lemma 4.18.□ We are then ready to prove Proposition 3.3: Proof of Proposition 3.3.Fix ϵ > 0 as in the statement of the proposition, and set α′=Clipα/ 2. Define ϵ1:= αϵ/C for a suitable absolute constant C> 0. By Lemma 4.1 applied to ε=ϵ1 , we know that the set R⊂Eof bad points x∈Ewith 2∗ Eα,α′(x)≥ϵ1 satisfies H1(R)≲δ·ϵ−1 1α−2=Cδ·ϵ−1α−3 .
1488 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA Since δ≤C0ϵ2α3 , taking C0=C−2gives H1(R)≤ϵ/2 (assuming that C>0 was large enough). The set G:= E\R satisfies the hypotheses of Lemma 4.18 (with β=α′=Clipα/ 2 and ε=ϵ1 ), so there exists a Clipα -Lipschitz graph 0⊂R over the x -axis such that H1(G\0) ≲ϵ1/α =ϵ/C . If the constant C>0 was chosen large enough, we see that H1(E\0) ≤H1(R)+H1(G\0) ≤1 2ϵ+1 2ϵ=ϵ. This concludes the proof of Proposition 3.3.□ 5. Proof of Proposition 3.11 In this section we prove Proposition 3.11. Recall that we are assuming to be in Case 2; that is, E cannot be exhausted, up to measure ϵ , by a constant number of consecutive sets Fk,Fk+1,...,Fk+Csep (recall this notation from Section 3C). More precisely, this means that H1(E\(Fk∪···∪Fk+Csep )) ≤ϵ(5.1) fails for every k; thus we find an index pair k,l∈ {0,...,M3}with |k−l| ≥ Csep such that H1(Fk)≥α2κand H1(Fl)≥α2κ.(5.2) Recall that all the minigraphs in Fkmake an angle ≤ακwith Lk:=span(wk)=span(cos(kπ/M3), sin(kπ/M3)), and similarly all the minigraphs in Flmake an angle ≤ακwith Ll=span(wl). The existence of Fk and Fl will imply a configuration such as the one depicted in Figure 3. A more precise definition is given in the lemma below. Lemma 5.3. If the inequalities in (5.2) hold , then there exists an absolute constant C∼Clip ( the constant from Proposition 3.3)such that the following objects exist: (1) affine lines ℓkand ℓlwith (ℓk,Lk)≤ακand (ℓl,Ll)≤ακ, (2) tubes T ′ k:= ℓk(Cα) and Tk:= ℓk(α1/2), (3) tubes T ′ l:= ℓl(Cα) and Tl:= ℓl(α1/2), (4) Clipα-Lipschitz graphs γk, γlover the lines ℓk, ℓl,respectively such that γk∩B(1)⊂T′ kand γl∩B(1)⊂T′ l, (5) compact subsets Gk⊂(E∩γk)\Tl⊂B(1)and Gl⊂(E∩γl)\Tk⊂B(1)(5.4) of measure H1(Gk)≥α3/C and H1(Gl)≥α3/C. Once the objects in Lemma 5.3 are found, it follows from a relatively simple geometric argument, presented below, that positively many lines intersect E twice (the lines in question are depicted in red in Figure 3):
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1489 TkTl γl γk T′ l T′ k Figure 3. A configuration where positively many lines hit Etwice. Lemma 5.5. There exists a set of lines L(Gk,Gl) of measure η(L(Gk,Gl)) ≳α7 such that ℓ∩Gk= ∅ and ℓ∩Gl= ∅for all ℓ∈L(Gk,Gl). In particular,since Gk,Gl⊂E are disjoint, Z L(E) (#(E∩ℓ) −1)dη(ℓ) ≳η(L(Gk,Gl)) ≳α7 .(5.6) Proposition 3.11 follows immediately by Lemma 5.5. We will next derive Lemma 5.5 from Lemma 5.3. (See Remark 5.10 and the Appendix for an alternative proof of Lemma 5.5.) Proof. The key geometric observation is the following: if ℓ⊂R2is any line with Gk∩ℓ= ∅= Gl∩ℓ, then ℓ must make an angle ≳α1/2 with both ℓk and ℓl ; see Figure 3: indeed, if for example (ℓ, ℓl)≪α1/2 and ℓ∩Gl= ∅ , then ℓ∩B( 1 )⊂Tl , and hence ℓ∩Gk=∅ by (5.4). It follows that both ℓk, ℓl are Cα−1/2 - graphs over ℓ⊥ , for any line ℓ connecting Gk and Gl . But since γk, γl were by definition Clipα -Lipschitz graphs over ℓk, ℓl , it follows that also γk, γl are Cα−1/2 -Lipschitz graphs over ℓ⊥ (assuming that α > 0 is small enough). To prove the lower bound (5.6), start by fixing x∈Gl⊂γl , recall that ℓx,θ := π−1 θ{πθ(x)} , and consider the set of directions 2(x,Gk):= {θ∈ [0, π) :ℓx,θ ∩Gk= ∅}. With this notation, we claim that H1(2(x,Gk)) ≳α1/2H1(Gk), x∈Gl.(5.7) Indeed, if {B(θj,rj)}j∈N is an arbitrary cover of 2(x,Gk) , then the tubes ℓx,θj(Cr j) cover Gk , where C> 0 is an absolute constant. This is because Gk is covered by the cones Cj:= S{ℓx,θ :θ∈B(θj,rj)} by definition, and each intersection Gk∩Cj⊂B( 1 )∩Cj is further covered by a tube of the form ℓx,θj(Cr j) . Now recall that γk⊃Gkis an α−1/2-Lipschitz graph over each line ℓ⊥ x,θj: this gives α−1/2P j∈N rj≳P j∈N H1(Gk∩ℓx,θj(rj)) ≥H1(Gk), which implies (5.7).
1490 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA We now infer from (5.7) and Fubini’s theorem that Zπ 0 H1({x∈Gl:θ∈2(x,Gk)})dθ=ZGl H1(2(x,Gk)) dH1(x)≳α1/2H1(Gk)H1(Gl). (5.8) To proceed, write Gl(θ) := {x∈Gl:θ∈2(x,Gk)}. We claim that H1(Gl(θ)) = 0=⇒ H1πθ(Gl(θ))≳α1/2H1(Gl(θ)), θ ∈ [0, π). (5.9) This will complete the proof of the corollary, because (5.8) then implies Zπ 0 H1(πθ(Gl(θ)) dθ (5.8) ≳αH1(Gk)H1(Gl) Lem. 5.3 ≳α7 , and the left-hand side above is a lower bound for η(L(Gk,Gl)). Finally, let us prove (5.9). If H1(Gl(θ)) = 0, then θ∈2(x, γk) for at least one x∈Gl , which means that ℓx,θ =π−1 θ{πθ(x)} intersects both Gk and Gl . Thus, γl is a Cα−1/2 -Lipschitz graph over the line ℓ⊥ x,θ . Consequently, the relation H1(πθ(H)) ≳α1/2H1(H) holds for all H1 measurable subsets H⊂γl , in particular for H:= Gl(θ).□ Remark 5.10. In fact, we have an exact expression for η(L(Gk,Gl)): η(L(Gk,Gl)) =ZZGk×Gl |πθ(xk,xl)(τk(xk))||πθ(xk,xl)(τl(xl))| |xk−xl|d(H1×H1)(xk,xl). (5.11) In (5.11), τk(x) denotes the unit tangent vector to γk at x∈γk , and τl(x) is defined similarly. For distinct x,x′∈R2 ,θ(x,x′)denotes the angle θsuch that πθ(x)=πθ(x′). Now we show how (5.11) implies Lemma 5.5. By the key geometric observation in the first paragraph of the proof of Lemma 5.5 and the fact that Gk,Gl⊂B( 1 ) , the integrand in (5.11) is ≳α1/2α1/2/ 1 =α . Thus, η(L(Gk,Gl)) ≳αH1(Gk)H1(Gl)≳α7 . We state and prove a more general form of (5.11) in the Appendix. The remainder of this section is devoted to constructing the objects listed in Lemma 5.3. This is based on the assumption (3.9), that is, H1(Fk)≥α2κ and H1(Fl)≥α2κ . Recall also that Fk,Fl were the unions of the minigraphs in Fk and Fl . The minigraphs in Fk make an angle ≤ακ with Lk , while the minigraphs in Fl make an angle ≤ακ with Ll . Furthermore, (Lk,Ll)≥Csepακ , so the minigraphs from Fk and Fl point in quantitatively different directions. We also recall that Fk (respectively Fl) can be expressed as a union of certain consecutive families Ei: Fk=Es∪Es+1∪···∪Es+mand Fl=Et∪···∪Et+m.(5.12) Some of these families may be empty, but not all, according to (5.2). Of course m≲α−1 ,(5.13) since there were no more than α−1of the families Ejaltogether.
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1491 Fl Fk Gk Gl Tl Tk Figure 4. Finding the graphs and tubes claimed by Lemma 5.3. 5A. Sketch of the proof. We now explain the proof strategy with a picture. In Figure 4, we have depicted the sets Fk and Fl , which are roughly speaking ακ -Lipschitz graphs over the lines Lk,Ll by Proposition 3.3 (details will follow). Both Fk and Fl are, moreover, tiled by ≲α−1 of the sets Ej . Most of sets Ej are (individually) contained on α -Lipschitz graphs γj , by another application of Proposition 3.3. The red sets shown in Figure 4 illustrate sets of the form Gj=Ej∩γj∩Bj, where Bj is some ball of radius α with the property that H1(Gj)∼αH1(Ej) . Each Gj is contained in a tube Tj of width α1/2 (or even a tube of width α , which was also required in Lemma 5.3). So, picking Gk⊂Fk and Gl⊂Fl arbitrarily, we would satisfy all the points (1)-(5) in Lemma 5.3, except for the inclusions (5.4). The problem is that if we pick Gk⊂Fk and Gl⊂Fl arbitrarily, the tube Tk associated with Gk might intersect Gl , or vice versa, violating (5.4). To satisfy (5.4), we need to pick Gk,Gl in such a way that the Gk -tube avoids Gl and the Gl -tube avoids Gk . To achieve this, we roughly choose three well-separated sets Gl 1,Gl 2,Gl 3⊂Fl, and two further well-separated sets Gk 1,Gk 2⊂Fk. Then, we use the transversality of the graphs Fk,Fl to deduce the following: each Gk i -tube can intersect at most one of the sets Gl j , and vice versa. At this point, we may deduce from the pigeonhole principle that there must exists a pair (Gk i,Gl j) such that the Gk i -tube does not intersect Gl j , and the Gl j -tube does not intersect Gk i. Indeed, there are six pairs (Gk i,Gl j), but only five tubes. This will complete the proof. 5B. Proof. We turn to the details. First, we apply Proposition 3.3 to the sets Fk,Fl , each of which can be written as a finite union of ακ -Lipschitz minigraphs over the lines Lk,Ll , respectively. It follows from the choice of constants δ=ϵ70/Cthm and α=(ϵ/Calp)10 made in Section 3C that δ≪α5κ , assuming that Cthm is chosen sufficiently small compared to the absolute constant Calp . Writing α5κ=(ακ)3α2κ , this means that the main hypothesis of Proposition 3.3 is valid with constants ακ and 1 2α2κ in place of α and ϵ . It follows that there exist Clipακ -Lipschitz graphs 0k, 0l over Lk,Ll , respectively, which cover
1492 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA most of Fkand Flin the sense H1(Fk\0k)≤1 2α2κ(3.9) ≤1 2H1(Fk)and H1(Fl\0l)≤1 2H1(Fl). We write F′ k:= Fk∩0kand F′ l:= Fl∩0l. Next, recall from (5.12) that Fk=Es∪···∪Es+mand Fl=Et∪···∪Et+m, and each Ej is a finite union of α -Lipschitz minigraphs Ej over a certain line (which makes an angle ≤ακ with Lk ). Applying Proposition 3.3 again, for each Ej with either j∈ {s,...,s+m} or j∈ {t,...,t+m} , we find Lipschitz graphs γjwith constant ≤Clipαand the property H1(Ej\γj)≲α2 ,s≤j≤s+mor t≤j≤t+m. For this application of Proposition 3.3 to be legitimate, we need δ≪α3(α2)2=α7 , which also follows from our choice of constants recalled above, taking Cthm ≫C70 alp . We write E′ j:= Ej∩γj . With these choices, a major part of F′ k is covered by the union of the graphs γj : indeed since F′ k⊂Fk⊂(Es∪ · · · ∪ Es+m) , we have H1F′ k\ m S j=1 E′ s+j≤ m P j=1 H1(Es+j\γs+j)≲ m P j=1 α2(5.13) ≲α. Since H1(F′ k)≳H1(Fk)≥α2κ , and κ=1 10 , we infer that at least half of F′ k is covered by the (subsets of) α -Lipschitz graphs E′ j with s≤j≤s+m . The same conclusion mutatis mutandis holds for F′ l and the sets E′ jwith t≤j≤t+m. We finally redefine Fk:= F′ k∩ m S j=1 E′ s+jand Fl:= F′ l∩ m S j=1 E′ t+j. This should cause no confusion, since the original sets Fk,Fl will no longer be used. We list all the properties of Fk,Flwe will need in the sequel: •Fk,Fl⊂Eand H1(Fk)≳α2κand H1(Fl)≳α2κ(compare with (3.9)). •Fkis covered by the Lipschitz graph 0kover Lkwith constant ≤Clipακ. •Flis covered by the Lipschitz graph 0lover Llwith constant ≤Clipακ. •Fk is covered by the union of ≲α−1 Lipschitz graphs γs, . . . , γs+m with constant ≤Clipα over certain lines ℓs+jmaking an angle ≤ακwith Lk. •Fl is covered by the union of ≲α−1 Lipschitz graphs γt, . . . , γt+m with constant ≤Clipα over certain lines ℓt+jmaking an angle ≤ακwith Ll. We have now defined carefully the objects Fk and Fl in Figure 4. In defining the objects Ek and El in the same picture, there is the technical problem that the initial sets Ej need not be localized, as the picture suggests. This will be easily fixed by intersecting the initial sets Ej with balls. First, using that H1(Fk)≳α2κ, we choose two special points x1,x2∈Fkwith the properties |x1−x2|≳α2κand H1(Fk∩B(xj, α)) ≥α2for j∈ {1,2}.(5.14)
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1493 This can be arranged, because the set of points x∈Fk with H1(Fk∩B(x, α)) ≤α2 has total length at most ≲α≪H1(Fk) . Thus, the admissible points for the second condition in (5.14) have total length ≥1 2H1(Fk)≳α2κ . Then, to finish the selection, it remains to pick two of these points with separation α2κ : this is possible because Fk lies on a Lipschitz graph with constant ≤ 1, so in particular H1(Fk∩B(x,r)) ≲r for all r>0. Next, we move attention from Fk to Fl . This time we pick three special points y1,y2,y3∈Fl with properties similar to those in (5.14): |yi−yj|≳α2κfor i= jand H1(Fl∩B(yj, α)) ≥α2for j∈ {1,2,3}.(5.15) The details of the selection are the same as we have seen above. Next, recall that both Fk and Fl can be written as a finite union of (subsets of) Clipα -Lipschitz graphs: the covering graphs for Fk were denoted γs, . . . , γs+m and the covering graphs for Fl were denoted γt, . . . , γt+m , where m≲α−1 . Since H1(Fk∩B(x1, α)) ≥α2 , at least one of the graphs γs, . . . , γs+m must have large intersection with Fk∩B(x1, α). We denote this graph by γk 1; then we have H1(Fk∩γk 1∩B(x1, α)) ≳α3 .(5.16) We find similarly a graph γk 2∈ {γs, . . . , γs+m} such that H1(Fk∩γk 2∩B(x2, α)) ≳α3 . Then, we also repeat the argument for the three balls B(yj, α) : we find three graphs γl 1, γ l 2, γ l 3∈ {γt, . . . , γt+m} with the property H1(Fl∩B(yj, α) ∩γl j)≳α3 ,1≤j≤3.(5.17) The sets Gk i:= Fk∩γk i∩B(xi, α), i=1,2,and Gl j:= Fl∩γl j∩B(yj, α), j=1,2,3,(5.18) are the ones we informally discussed below Figure 4. Next, we associate the lines and tubes (required by Lemma 5.3) to the sets Gk i,Gl j . We associate to each graph γk ior γl jan affine line ℓk ior ℓl jwith the following properties: •γk iis a Clipα-Lipschitz graph over ℓk ifor i∈ {1,2}. •γl jis a Clipα-Lipschitz graph over ℓl jfor j∈ {1,2,3}. •The lines are chosen so that Gk j⊂ℓk i(Cα) for i∈ {1,2}and Gl j⊂ℓl j(Cα) for j∈ {1,2,3}, where C∼Clip. We now define (Tk i)′:= ℓk i(Cα) and Tk i:= ℓk i(α1/2) for i∈ {1,2}, and similarly (Tl j)′:= ℓl j(Cα) and Tl j:= ℓl j(α1/2)
1494 ALAN CHANG, DAMIAN D ˛ABROWSKI, TUOMAS ORPONEN AND MICHELE VILLA 0l Ll Lk Tk i ℓk j Figure 5. Transversality of Tk iand 0l. The angle between ℓk jand Llis ≳Cακ. for j∈ { 1 , 2 , 3 } . Thus, Gk i⊂(Tk i)′⊂Tk i and Gl j⊂(Tl j)′⊂Tl j . Since moreover H1(Gk i)≳α3 and H1(Gl j)≳α3 by (5.16)–(5.17),any pair (Gk i,Gl j) (with associated lines and tubes) would now satisfy all the requirements of Lemma 5.3, except perhaps the inclusions (5.4). We will now use the pigeonhole principle to show that at least one of the pairs (Gk i,Gl j) also satisfies the inclusions (5.4). The main geometric observation is diam(Tk i∩0l)≲α1/2−κand diam(Tl j∩0k)≲α1/2−κ.(5.19) The first inequality holds for i∈ {1,2}, the second for j∈ {1,2,3}. The proof of (5.19) is contained in Figure 5. Recall that Tk i is an α1/2 -tube around a certain line ℓk i with (ℓk i,Lk)≤ακ . On the other hand, (Lk,Ll)≥Csepακ , so also (ℓk i,Ll)≥(Csep − 1 )ακ . Finally, 0l is a Clipακ -Lipschitz graph over Ll , so every tangent of 0l makes an angle ≳Csepακ with ℓk i , since we chose Csep much larger than Clip in Section 3C. Thus 0lis an α−κ-Lipschitz graph over (ℓk j)⊥. It follows that diam(Tk i∩0l)≤H1(Tk i∩0l)≲α1/2−κ. Now that we have proved (5.19), recall from (5.15) the three balls B(yj, α) , all of which were centered at yj∈Fl⊂0l , and whose centers yj had pairwise separation ≳α2κ . Since κ=1 10 , we have α1/2−κ≪α2κ for α > 0 small enough (or in other words assuming that the constant Calp > 0 is chosen large enough), and therefore (5.19) implies that #{j∈ {1,2,3} : Tk i∩B(yj, α) = ∅} ≤ 1,i∈ {1,2}.(5.20) By a similar argument, #{i∈ {1,2} : Tl j∩B(xi, α) = ∅} ≤ 1,j∈ {1,2,3}.(5.21) We finally claim, as a consequence of (5.20)-(5.21) and the pigeonhole principle, that there exists a pair of balls (B(xi0, α), B(yj0, α)), for some i0∈ {1,2}and j0∈ {1,2,3}with the property Tk i0∩B(yj0, α) =∅and Tl j0∩B(xi0, α) =∅.(5.22) This, by definition, yields Gk i0 (5.18) ⊂B(xi0, α) \Tl j0and Gl j0 (5.18) ⊂B(yj0, α) \Tk i0, which (combined with (5.18)) completes the proof of the inclusions (5.4), and Lemma 5.3.
STRUCTURE OF SETS WITH NEARLY MAXIMAL FAVARD LENGTH 1495 To prove (5.22), consider the bipartite graph with 5 vertices {v1, v2}∪{w1, w2, w3} and the following edge set: •For i∈ {1,2}and j∈ {1,2,3}, the edge (vi, wj)is included if Tk i∩B(yj, α) = ∅. •For j∈ {1,2,3}and i∈ {1,2}, the edge (wj, vi)is included if Tl j∩B(xi, α) = ∅. Now, (5.20)–(5.21) can be restated as follows: for vi fixed, there can be at most one edge (vi, wj) , and for wi fixed, there can be at most one edge (wi, vj) . Thus, the edge set contains at most five edges. On the other hand, the product set {v1, v2}×{w1, w2, w3} contains six elements, so there must be a pair {vi, wj} so that neither (vi, wj) nor (wj, vi) lies in the edge set. This is equivalent to (5.22). This completes the proof of Lemma 5.3. 6. The grid example In this section we provide an example showing that Theorem 1.1 is optimal in the sense that the assumption Fav(E)≥Fav(L)−δcannot be relaxed to Fav(E)≥δ. Proposition 6.1. There exists an absolute constant δ > 0and a sequence of compact rectifiable sets En⊂ [0,1]2⊂R2such that (1) H1(En)=1, (2) Fav(En)≥δ, (3) for any α∈ [2n−2,1)and any curve 0with H1(0 ∩En)≥αwe have H1(0) ≳αn. In particular , property (3) implies that if M≥ 1 , then for any M -Lipschitz graph 0,H1(0 ∩En)≲Mn−1 . We begin the construction. Fix an integer n≥ 2, and let [n]:= { 1 ,...,n} . For any j=(k,l)∈ [n]2 set xj=k n+1,l n+1(6.2) and Bj=Bxj,1 2πn2. Note that Bj⊂ [0,1]2and if i,j∈ [n]2 ,i= j, then dist(Bi,Bj)≥1 n+1−2 2πn2≥1 2n.(6.3) Define Sj=∂Bj, and observe that H1(Sj)=n−2 . We define the set Enas En:=S j∈[n]2 Sj. Since H1(Sj)=n−2 , we have H1(En)= 1. This verifies property (1) for En . It is also clear that En is compact and rectifiable. Now we check property (3). We will use the following result:
ANALYSIS & PDE Volume 17 No. 4 2024 1127The singular strata of a free-boundary problem for harmonic measure SEAN MCCURDY 1175On complete embedded translating solitons of the mean curvature flow that are of finite genus GRAHAM SMITH 1237Hausdorff measure bounds for nodal sets of Steklov eigenfunctions STEFANO DECIO 1261On full asymptotics of real analytic torsions for compact locally symmetric orbifolds BINGXIAO LIU 1331The Landau equation as a gradient Flow JOSÉ A. CARRILLO, MATIAS G. DELGADINO, LAURENT DESVILLETTES and JEREMY S.-H. WU 1377Degenerating hyperbolic surfaces and spectral gaps for large genus YUNHUI WU, HAOHAO ZHANG and XUWEN ZHU 1397Plateau flow or the heat flow for half-harmonic maps MICHAEL STRUWE 1439Noncommutative maximal operators with rough kernels XUDONG LAI 1473Structure of sets with nearly maximal Favard length ALAN CHANG, DAMIAN D˛ABROWSKI, TUOMAS ORPONEN and MICHELE VILLA ANALYSIS & PDE Vol. 17, No. 4 2024