INCIDENCE ESTIMATES FOR α -DIMENSIONAL TUBES AND

. We prove essentially sharp incidence estimates for a collection of δ -tubes and δ -balls in the plane, where the δ -tubes satisfy an α -dimensional spacing condition and the δ -balls satisfy a β -dimensional spacing condition. Our approach combines a combinatorial argument for small α,β and a Fourier analytic argument for large α,β . As an application, we prove a new lower bound for the size of a ( u,v ) -Furstenberg set when v ≥ 1 ,u + v 2 ≥ 1 , which is sharp when u + v ≥ 2 . We also show a new lower bound for the discretized sum-product problem.


Introduction
Let 0 < δ ≤ 1 be a small parameter.We will work with δ-tubes and δ-balls in the plane R 2 .A δ-ball is a ball of radius δ.A δ-tube is a δ × 1 rectangle.The direction of a rectangle is the vector pointing in the direction of its longest side.(This vector is only determined up to ±1.) Definition 1.1.Let P be a set of δ-balls and T be a set of δ-tubes.The number of incidences I(P, T) is the number of pairs (p, t) of δ-balls p ∈ P and δ-tubes t ∈ T such that p intersects t: p ∩ t ̸ = ∅.
The basic problem we will consider is the following: Given a set of δ-balls P and a set of δ-tubes T contained in the square [0, 1] 2 , what is the maximum number of incidences I(P, T)?
We will impose a spacing condition on the set of δ-balls and the set of δ-tubes.The spacing condition is standard, see e.g.[HSY21].
Definition 1.2.For 0 ≤ β ≤ 2 and K ≥ 1, we call a set of δ-balls P contained in [0, 1] 2 a (δ, β, K)-set of balls if for every w ∈ [δ, 1] and every ball B w of radius w, In the definition of (δ, β, K)-set, K may depend on δ.If K is constant, then we drop K from the notation.By taking w = 3δ, any δ-ball in a (δ, β, K)-set of balls may intersect up to ≤ 9K many other δ-balls in the set.
We will impose an analogous condition on the set of δ-tubes.
We can rephrase the problem as follows: given a (δ, β, K β )-set of balls P and a (δ, α, K α )-set of tubes T, what is the maximum number of incidences I(P, T)?
In [GSW19], incidence problems for δ-tubes with some spacing conditions were considered.They fix a parameter 1 ≤ W ≤ δ −1 and choose T to be a collection of W 2 well-spaced δ-tubes: each W −1 × 1 rectangle in R 2 contains at most one δ-tube in T. They also consider another spacing condition, where each W −1 × 1 rectangle contains ∼ N 1 many δ-tubes in each direction, for a fixed N 1 .Using a Fourier analytic approach, [GSW19] proved sharp incidence estimates for well-spaced δtubes.This Fourier analytic method is also used in [GWZ20, DGW20, GMW22, FGM21, FGM22] to derive incidence estimates, decoupling estimates, and square function estimates.

Note that the bound dim
As a quick corollary of Theorem 1.6, we obtain the following variant of Marstrand's slicing theorem [Mar54], which states that for all directions θ ∈ S 1 , then for a.e.line ℓ in direction θ, we have dim H (A ∩ ℓ) ≤ t − 1.In fact, we are able to bound the dimension of the exceptional set of lines for which dim H (A ∩ ℓ) > t − 1.
Corollary 1.7.Let A ⊂ R 2 be a set with dim H (A) = t > 1, and let L be a set of lines such that dim Our approach also allows us to obtain the following discretized sum-product estimate:
We conclude this introductory section by describing the organization of the paper.In Section 2, we will show the estimates in Theorem 1.4 are sharp up to C ϵ δ −ε , by constructing suitable examples.
We will then prove the upper bound of Theorem 1.4 by analyzing different cases for α, β.In Section 3, we will use a combinatorial argument (the L 2 argument as in [Cor77]) to resolve the case where α ≤ 1 or β ≤ 1.In Section 4, we will induct on scale δ to prove Theorem 1.5, the case where α, β ≥ 1.The starting point of this argument will be the Fourier-analytic Proposition 2.1 from [GSW19], which was inspired by ideas of Orponen [Orp18] and Vinh [Vin11].Finally, we derive Theorem 1.4, Theorem 1.6, and Corollary 1.8 in Section 5.
Notation.We will use A ≳ B to represent A ≥ CB for a constant C, and A ≲ B to represent A ≤ CB.The constant C is independent of the scale δ and the dimension parameters α, β, K α , K β .We will use A ∼ B to represent A ≳ B and A ≲ B. Finally, we let A ≳ ε B to denote A ≥ CB for a constant C which depends on ε, and define A ≲ ε B, A ∼ ε B similarly.
For a finite set A, typically a set of δ-tubes or δ-balls, let #A or |A| denote its cardinality.For a subset A ⊂ R 2 , let |A| δ denote the least number of δ-balls needed to cover A.
For a set P of δ-balls and a subset A ⊂ R 2 , let P ∩ A := {p ∈ P | p ⊂ A}.
The angle between two δ-tubes s and t, or ∠(s, t), is the acute angle between their directions.
For two sets A and B in R 2 , we say A and B intersect if For a δ-ball p and S ≥ 1, define the S-thickening p S to be the Sδ-ball concentric with p.For a δ-tube t, let t S denote the Sδ-tube coaxial with t.Finally, for a set of δ-balls P (respectively set of δ-tubes T), let P S := {p S : p ∈ P } (respectively T S := {t S : t ∈ T}).
We say two δ-tubes s, t are essentially identical if they intersect and their angle is ≤ δ.Otherwise, they are essentially distinct, and we say a collection T of δ-tubes is essentially distinct if the tubes in T are pairwise essentially distinct.

Constructions
We start with the sharpness part of Theorem 1.4.We will construct (δ, α)-sets of tubes and (δ, β)-sets of balls such that the number of incidences is at least δ −f (α,β) , where f was defined as in Theorem 1.4.We divide the constructions into four cases.Construction 1 is the main construction that works for most α and β.Constructions 2, 3, 4 can be considered as auxiliary constructions which take care of exceptional values of α, β not covered in Construction 1.The constructions will all take place inside a 1 × 1 square.In the constructions, some of the δ-tubes may not be fully contained within the 1 × 1 square, but we will ignore this minor detail.For ease of notation, let D = δ −1 .
To describe the construction, we will need a few auxiliary variables.Recall a = min(α, 1), b = min(β, 1), and we will eventually choose γ, κ, λ as parameters in [0, 1].Refer to Figure 2. The left picture depicts a single bundle with D (1−γ)a many δ-tubes and ∼ D γb many δ-balls.The δ-tubes are rotates of a single central δ-tube t, and the angle spacing between δ-tubes is δ γ+(1−γ)a , so that the maximal angle of two δ-tubes in the bundle is δ γ .By trigonometry, the intersection of all the tubes contains a δ × (∼ δ 1−γ ) rectangle with the same center and direction as the central δ-tube t.We may thus place ∼ D γb many δ-balls in the rectangle, spaced a distance of δ 1−γ+γb apart; then each ball of the bundle will intersect each tube in the bundle.Furthermore, since the maximum angle between two δ-tubes in the bundle is δ γ , we see that the bundle fits inside a δ γ × 1 rectangle.
It might be helpful to observe that the configuration of δ-balls is "dual" to the configuration of δ-tubes in a bundle, in the sense that the δ-balls in a bundle are evenly spaced along the central axis, while the δ-tubes are evenly spaced in direction.
In the right picture, there are D κ bundles in [0, 1] 2 .The bundles are arranged in a D (1−λ)κ × D λκ grid, with the horizontal spacing δ (1−λ)κ and the vertical spacing δ λκ .The bundles in the same row are translates of each other; two adjacent bundles in the same column are δ λκ rotates of each other.
If T is the set of δ-tubes and P is the set of δ-balls in the configuration, then we see that |T| ∼ D (1−γ)a+κ and |P | ∼ D γb+κ .
Intuitively, we can regard λ as controlling the "aspect ratio" of the bundle configuration.If λ = 1, then all the bundles are rotated copies of each other, arranged vertically; if λ = 0, then all the bundles are translated copies of each other, arranged horizontally.For the right values of λ, our constructed T will be a (δ, α)-set of δ-tubes and P will be a (δ, β)-set of δ-balls.Now, we choose suitable values for our parameters γ, κ, λ.We first choose γ = (b) There exists 0 ≤ λ ≤ 1 such that T is a (δ, α)-set of δ-tubes and P is a (δ, β)-set of δ-balls.
The proof is computational, and we defer it to the Appendix.Now with this choice of parameters, we find |T| ∼ D The prototypical example is α = 1, β = 1, in which γ = κ = 0.5.In this case, the possible values for λ are 0 ≤ λ ≤ 1 2 .If we choose λ = 0, then we get a series of D 0.5 horizontally spaced, parallel bundles, as in Figure 3.
Refer to Figure 4.In each bundle, there are ∼ D many δ-tubes, each separated by angle δ.Thus, we can fit the bundle inside a 1 4 × 1 rectangle.We arrange ∼ D α−1 bundles as in the right figure, separated by distance ∼ δ α−1 , such that the centers of the bundles lie within a segment of length 1 2 centered at the unit square's center.Then, we place D β many δ-balls at some of the centers of the bundles, such that the δ-balls are δ β -separated.Thus, there are D α many δ-tubes and D β many δ-balls in the configuration.
Let T be the set of δ-tubes and P be the set of δ-balls.We will show that T is a (δ, α)-set of tubes and P is a (δ, β)-set of balls.
Fix w ∈ [δ, 1] and a w × 2 rectangle R w ; we will count how many δ-tubes in T are in R w .There are two main contributions.
• R w can contain tubes from ≲ ⌈ w δ α−1 ⌉ bundles of |T|.• For each bundle, R w can contain ≲ w δ δ-tubes.Thus, R w contains at most N δ-tubes in T, where (using w ∈ [δ, 1] and α ≥ 1): This means T is a (δ, α)-set of tubes.Now, we verify that P is a (δ, β)-set of balls.Fix w ∈ [δ, 1] and a ball B w of radius w; we will count how many δ-balls in P are in B w .Note that B w can intersect at most N many δ-balls, where (using 2.3.Construction 3.For this construction, we will assume β ≥ α + 1.Our goal is to obtain δ −(α+1) incidences.
Refer to Figure 5.There are D β−1 columns of D many δ-balls each.On D α of the columns, there is a δ-tube.The δ-tube-containing columns are separated by distance δ α .Thus, there are D β δ-balls and D α δ-tubes.Note that Construction 3 is "dual" to Construction 2, in the sense that a bundle of direction-separated δ-tubes is replaced by a bundle of evenly-spaced δ-balls.
The δ-tubes are a (δ, α)-set of tubes and the δ-balls are a (δ, β)-set of balls by a similar argument to Construction 2. Finally, each δ-tube contains D many δ-balls, so 2.4.Construction 4. For this construction, we will assume α + β ≥ 3. Our goal is to obtain δ −(α+β−1) incidences.0.5 Refer to Figure 6.The bundles of δ-tubes are the same as Construction 2. We then arrange ∼ D β many δ-balls in a D β/2 × D β/2 grid, such that adjacent δ-balls are separated by distance ∼ δ β/2 .We confine the δ-balls to a 1 2 × 1 2 square S concentric with the large 1 × 1 square.Let T be the set of δ-tubes and P be the set of δ-balls in this configuration.
From Construction 2, T is a (δ, α)-set of tubes.We now show that P is a (δ, β)-set of balls.
Fix w ∈ [δ, 1] and a ball B w of radius w; we will count how many δ-balls in P are in B w .Note that B w can intersect at most N many δ-balls in P , where Also, the δ-balls in P are essentially distinct, so P is a (δ, β)-set of balls.Finally, we will count the number of incidences.For a bundle centered at some point O ∈ S, the δ-tubes in the bundle cover a double cone with apex O and angle 1 4 .This double cone intersects square S in a polygonal region with positive area, so it contains a positive fraction of the balls in P .Hence, the number of incidences between a given bundle and P is ≳ D β .There are D α−1 bundles in T, so

Combinatorial upper bound
We will first prove the upper bound for α ≤ 1 or β ≤ 1.We further casework on whether α < β or α ≥ β, which are handled by Theorems 3.1 and 3.2 below.
Notation.We first relate J to I. By Hölder's inequality, we have Next, we estimate J(P, T).For a given δ-tube t ∈ T, let Then, we have The main claim is the following: To prove Lemma 3.3, we introduce some notation.Let T δ (t) = {s ∈ T | s ∩ t ̸ = ∅, ∠(s, t) ≤ 2δ} and for w ≥ 2δ, We will now prove two lemmas involving T w (t).
Proof.Let R be the 200w × 2 rectangle with the same center as t such that the length-2 side of R is parallel to the length-1 side of t.By trigonometry, we observe that any δ-tube s ⊂ [0, 1] 2 with s ∩ t ̸ = ∅ and ∠(s, t) ≤ 2w must be contained in R. Since T is a (δ, α, K α )-set of tubes, there are at most Proof.We use a double counting argument.The left-hand side counts the number of pairs (p, s) × P × T w (t) with p ∼ s and p ∼ t.For each s ∈ T w (t), s ∩ t is contained in a δ × δ w rectangle R w .To upper-bound the number of δ-balls of P in R w , we split into cases: • If β ≥ 1, then cover R w with ∼ 1 w many 10δ-balls q i , such that any δ-ball that intersects R w must lie in some q i .By dimension, q i contains at most Thus, for each s ∈ T w (t), there are at most ≲ K β 1 w b many δ-balls p ∈ P with p ∼ s and p ∼ t, which proves the Lemma.
. Now, we perform the following calculation: This is the desired result.□

Fourier analytic upper bound
We will now prove an upper bound for I(P, T) when P is a (δ, α, K α )-set and T is a (δ, β, K β )-set of tubes, for α ≥ 1 and β ≥ 1.The proof method is using the high-low method in Fourier analysis.

4.1.
A Fourier Analytic result.We will need a variant of Proposition 2.1 from [GSW19].The version presented here is a modest refinement of [Bra23, Proposition 2.1].First, we review some notation.We say two δ-tubes s, t are essentially identical if they intersect and their angle is ≤ δ.Otherwise, they are essentially distinct, and we say a collection T of δ-tubes is essentially distinct if the tubes in T are pairwise essentially distinct.
For a δ-ball p and S ≥ 1, define the S-thickening p S to be the Sδ-ball concentric with p.For a δ-tube t, let t S denote the Sδ-tube coaxial with t.Finally, for a set of δ-balls P (respectively set of δ-tubes T), let P S := {p S : p ∈ P } (respectively T S := {t S : t ∈ T}).
Proposition 4.1.Fix a small ε > 0, and δ ≤ 1, D = δ −1 .There exists a constant C ε with the following property: Suppose that P is a set of δ-balls and T is a set of δ-tubes contained in [0, 1] 2 such that every p ∈ P intersects at most K β many δ-balls of P (including p itself ) and every t ∈ T is essentially identical to at most K α many δ-tubes of T. Let S = D ε/20 .Then we have the incidence estimate (4) then the δ-tubes in T are essentially distinct and the δ-balls in P are pairwise non-intersecting.Thus, we can directly apply [Bra23, Proposition 2.1] with choice of parameters α = ε 2 40 and weight function w ≡ 1.Now, we will tackle the general case.To do so, we will partition P into K β groups P 1 , P 2 , • • • , P K β such that all the balls in P i are disjoint.Consider a graph on the set of δ-balls of P , with two balls connected by an edge if they intersect.Then each ball has maximum degree K β − 1 by assumption.To construct the desired partition of P , we employ the following well-known lemma from graph theory, which follows from for example Brook's theorem in [Lov75]: Lemma 4.2.Any graph with maximum degree n admits a coloring of the vertices with n + 1 colors such that no two adjacent vertices share the same color.
In other words, we may partition of P into K β many sets P 1 , P 2 , . . ., P K β , such that any two intersecting δ-balls in P must belong in different sets of the partition, so the δ-balls in each P i are disjoint.
Similarly, we may partition T into K α groups T 1 , T 2 , • • • , T Kα such that the δtubes in each T i are essentially distinct.Finally, by applying our K α = K β = 1 incidence result to each P i and T j , we have The last line followed from Cauchy-Schwarz and The upper bound for I(P, T).Proposition 4.1 hints at an inductive approach to upper bound I(P, T).If the first term in (4) dominates, we get our desired upper bound.If the second term dominates, then we need to estimate I(P S , T S ), where P S is formed by thickening the δ-balls in P to Sδ-balls, and likewise T S is formed by thickening the δ-tubes in T to Sδ-tubes.(Here, S = D ε/20 .) We thus obtain an incidence problem at scale Sδ, so we can apply induction.The key idea is that if P is a (δ, β, K β )-set of balls, then P S is a (Sδ, β, S β K β )-set, and similarly for tubes T S .We now prove Theorem 1.5.
First, we can assume ε < 1 2 .The proof will be by induction on n = ⌊− log 2 δ⌋.Let C 1 (ε) ≥ 1 be a constant to be chosen later, and The base case will be δ ≥ 2 −N .Then since P is a (δ, β, K β )-set, we have This gives the desired bound (5) since 2 3N ≤ C ε .
We first take care of the case when |P ||T| is small.
Thus, we may assume |P ||T| ≥ D • K α K β .Because P is a (δ, β, K β )-set, each p ∈ P intersects ≤ 9K β many δ-balls of P (see brief remarks after Definition 1.2).Likewise, since T is a (δ, α, K α )-set, each t ∈ T is essentially identical with ≲ K α many δ-tubes of T. Thus, we may apply Proposition 4.1 to obtain, for some constant To prove (5), we will show each term is bounded above by For the second term, observe that P S is a (Sδ, β, S β K β )-set of balls and T S is a (Sδ, α, S α K α )-set of tubes.Thus, by the inductive hypothesis and c(α + β − 1) ≤ 1, we have Recall that δ ≤ 2 −N , and by definition of N and S, we get We have showed that each term of (6) is bounded above by We restate Theorem 1.4 here: Theorem 5.1.Suppose α, β satisfy 0 ≤ α, β ≤ 2, and let K α , K β ≥ 1.For every ε > 0, there exists C = C ε K α K β with the following property: for every (δ, β, K β )set of balls P and (δ, α, K α )-set of tubes T, the following bound holds: where f (α, β) is defined as in Figure 1.These bounds are sharp up to C • δ −ε .
Proof.The sharpness of these bounds was proved in Section 2, with the constructed examples.We turn to showing the desired upper bounds.In this proof, let by dimension property (take w = 2).We will split into cases.
• Suppose β ≥ α + 1.By the short remark after Definition 1.2, each δ-ball in P intersects ≤ 9K β other δ-balls in P .Thus, using Lemma 4.2, we can partition P into P 1 , P 2 , • • • , P 9K β such that the δ-balls in each P i are disjoint.Using this disjointness property, each δ-tube in T can only intersect ≲ D many δ-balls of any P i .Thus, we get then using a similar partitioning argument as in the previous bullet point, we get I(P, T) ≲ K α K β D β+1 .Combining these results proves Theorem 1.4.□ Now we move to the proof of Theorem 1.6 and Corollary 1.8.We will deduce them from the following incidence estimate: Theorem 5.2.Fix ε > 0. Suppose T is an (δ, α, K α )-set of tubes contained in [0, 1] 2 .For every t ∈ T, let P t be a (δ, β, K β )-set of balls contained in [0, 1] 2 such that p ∩ t ̸ = ∅ for each p ∈ P t .If c = max(α + β, 2) −1 and P = t∈T P t , then Remark.The LHS of ( 7) is less than I(P, T): we only count incidences between t and p ∈ P t , and discard "stray" incidences between t and p ∈ P \ P t .5.1.Sharpness of Theorem 1.6 and idea for Theorem 5.2.We first establish the sharpness part of Theorem 1.6.Let C be a Cantor set in [0, 1] with Hausdorff dimension β, and consider the product set A = C × [0, 1] ⊂ R 2 .Then A is a (β, v)-Furstenberg set for any 0 ≤ v ≤ 2, since lines with angle ≤ 1 100 from vertical intersect A in an affine copy of C, which has dimension β.Also, dim H (A) = β + 1.
Moving onto the proof of Theorem 5.2, it would be nice to assume the set P is a (δ, β ′ , K β )-set for some β ′ < 2, so that an application of Theorem 4.3 would be stronger.Unfortunately, a priori P may contain some over-concentrated pockets.
To remedy this, we can replace these over-concentrated pockets with a discretized copy of A (from the sharpness of Theorem 1.6).Then we decrease the number of balls in P , but increase the number of δ-balls of P intersecting a given tube t ∈ T. In the end, we obtain a (δ, β + 1, K β )-set P ′ with |P ′ | ≲ |P | but I(P ′ , T) ≳ t∈T |P t |, and then we apply Theorem 4.3 on P ′ and T to finish.

Proving Theorem 5.2 with extra assumptions.
It is convenient to make some assumptions about our setup.Fortunately, these extra assumptions are harmless, as we will show in Subsection 5.4.
Theorem 5.3.Fix ε > 0. Suppose T is an (δ, α, K α )-set of tubes contained in [0, 1] 2 .For every t ∈ T, let P t be a (δ, β, K β )-set of balls contained in [0, 1] 2 such that p ∩ t ̸ = ∅ for each p ∈ P t .Let P = t∈T P t .Suppose we have the additional simplifying assumptions: (S1) δ = 2 −n for some n ≥ 1, and K α , K β ≥ 1 are integers.(S2) All the δ-tubes of T have angle [ π 4 , π 4 + π 100 ] with the y-axis.(S3) All the δ-balls in P are centered in the lattice (δ(2Z + 1)) We prove Theorem 5.3 in the remainder of this subsection.As stated in the last subsection, the main idea is to replace P with a (δ, β + 1, K β )-set P ′ with |P ′ | ≲ |P | but I(P ′ , T) ≳ t∈T |P t |.A priori, P may contain some over-concentrated pockets, or balls B w that contain > K β • ( w δ ) β+1 many δ-balls in P .We would like to locally replace the portion of P in each over-concentrated pocket with a smaller set of δ-balls (to be constructed later) with cardinality ≤ K β • ( w δ ) β+1 ; then the resulting set P ′ will not have over-concentrated pockets and thus will be a (δ, β + 1, K β )-set.Unfortunately, this argument does not work because some of the over-concentrated pockets may overlap.Instead, we will find a set of disjoint over-concentrated pockets such that if we fix them, then the new set P ′ will be a (δ, β + 1, CK β )-set for some absolute constant C > 0. The disjoint over-concentrated pockets will turn out to be dyadic squares, which we define next.
We will also adopt the convenient shorthand: Notation.For a set of δ-balls P and a subset The next well-known lemma roughly says that fixing the over-concentrated dyadic squares is sufficient to ensure P ′ is a (δ, β + 1, CK β )-set.
Proof.Pick a r-ball B r with r ∈ [δ, 1]; we want to show |P ∩ B r | ≤ 64K • ( r δ ) β .Suppose r ≥ 1 4 .Let D 1 , D 2 , D 3 , D 4 be the four dyadic squares in D 1/2 ; their union is [0, 1] 2 .Furthermore, for each p ∈ P ∩ [0, 1] 2 , we know that the center of p lies in (δ(2Z + 1)) 2 , so p must lie inside some D i .By applying (9) to each D i , we have (since r ≥ 1 4 and β ≤ 2): Suppose δ ≤ r < 1 4 .Let w = 2 −n satisfy r < w ≤ 2r.Let B w be the w-ball concentric with B r , and let A be the point in (2wZ) 2 closest to the center of B w .There are (at most) four dyadic squares D 1 , D 2 , D 3 , D 4 in D 2w with A as a vertex.Using geometric intuition, we see the union ∪ 4 i=1 D i contains B w ∩ [0, 1] 2 , and hence B r ∩ [0, 1] 2 .Furthermore, since w δ = 2 N −n is an even integer, a δ-ball p ∈ P that lies inside ∪ 4 i=1 D i must lie inside some D i .By applying (9) to each D i with side length 2w ≤ 4r < 1, we have (since β ≤ 2): Hence, P is a (δ, β, 64K)-set by Definition 1.2.□ Thanks to Lemma 5.5, we shall only look at the set R of over-concentrated dyadic squares for P , or squares Q ∈ ∪ δ<w<1 dyadic D w satisfying |P ∩ Q| ≥ K β • ( w δ ) β .The squares in R are not necessarily disjoint because a larger dyadic square can contain a smaller dyadic square.However, if we partially order the set of dyadic squares by inclusion, then the set R ′ of maximal elements in R with respect to inclusion will be pairwise disjoint.Thus, the final question remaining is: how to fix the over-concentrated pockets in R ′ ?
Let w ∈ (δ, 1) be a dyadic number.We shall construct a set of δ-balls P w contained in [0, w] 2 with the following three properties (where the implicit constants are absolute): (P2) Let t be a δ-tube that forms angle π 4 ± π 100 with the y-axis.Suppose P t is a (δ, β, K β )-set satisfying p ∩ t ̸ = ∅ for all p ∈ P t , and each δ-ball in P t is centered in the lattice (δ(2Z + 1)) 2 .Then we have We defer the construction of P w to Section 5.3.Now, we will formalize our previous ideas and prove Theorem 5.3 assuming the existence of P w .
Proof of Theorem 5.3.Recall that for w = 2 −n , we define D w to be the set of dyadic squares of side length w contained in [0, 1] 2 .Let R be the set of squares in δ<w<1 dyadic D w that contain ≥ K β • w δ β+1 many δ-balls in Q.We call R the "over-concentrated" squares.Let R ′ be the maximal elements of R, i.e. the squares Q ∈ R such that no square of R properly contains Q.Since two elements of R are either disjoint or one lies inside the other, we see that the elements of R ′ are disjoint.
As a notational convenience in this proof, for any subset A ⊂ R 2 , we let P ∩ A := {p ∈ P | p ⊂ A} to be the set of δ-balls in P that lie in Q. Similarly define P \ A := {p ∈ P | p ̸ ⊂ A}.We will also define the set ∪R ′ ⊂ R 2 to be the union of the squares in R ′ .
Using this notation, we observe an important fact (already noticed in the proof of Lemma 5.5): Since the δ-balls in P are centered in (δ(2Z + 1)) 2 (by (S3)), and since the dyadic squares in R ′ have side length being multiples of 2δ, we have that any δ-ball in P is either contained in some Q ∈ R ′ or contained in [0, 1] 2 \ ∪R ′ .This fact will be used throughout the argument without further mention, but let us mention a particular example: we have P = (P \ ∪R ′ ) ∪ Q∈R ′ (P ∩ Q).
We now construct a new set of balls P ′ that has fewer δ-balls than P in the over-concentrated squares R (and equals P outside the set of over-concentrated squares), yet For each Q ∈ R ′ with side length w, we let P ′ (Q) be a superposition of K β copies of P w placed inside Q.Finally, define P ′ := (P \ ∪R ′ ) ∪ Q∈R ′ P ′ (Q), which replaces the δ-balls in P ∩ Q with P ′ (Q) for each Q ∈ R ′ .Then for each t ∈ T, we have (To get from the first to the second line, we used P \∪R ′ = P ′ \∪R ′ and p ∈ P t =⇒ p ∩ t ̸ = ∅ to lower bound the first term, and property (P2) with our assumptions (S2), (S3) to lower bound the summation term.)Summing over all t ∈ T gives (10) Similarly, by (P1) and the definition of R ′ , we have We now check that P ′ satisfies the conditions of Lemma 5.5 with β + 1 for β.
Then by estimate (P1) applied to P ′ (Q ′ ), we get then by definition of P ′ and R, we already know . In either case, we get By assumption, T is a (δ, α, K α )-set of tubes, and by Lemma 5.5, P ′ is a (δ, β + 1, CK β )-set of balls (here C > 0 is an absolute constant).Hence, we can apply Theorem 4.3, (10), and (11) to get (with c = 1 max(α+β,2) ): This completes the proof of Theorem 5.3.□ 5.3.Constructing the set P w .Let δ < w ≤ 1 2 such that w δ is an even integer.Recall that we want to construct a set P w contained in [0, w] 2 with the following properties (where the implicit constants are absolute): (P2) Let t be a δ-tube that forms angle π 4 ± π 100 with the y-axis.Suppose P t is a (δ, β, K β )-set satisfying p ∩ t ̸ = ∅ for all p ∈ P t , and each δ-ball in P t is centered in the lattice (δ(2Z + 1)) 2 .Then we have Let C be a Cantor set with Hausdorff dimension β that contains 0 and w, and let C δ := C (δ) ∩ δZ be a discretization of C at scale δ. (Recall that C (δ) is the δ-neighborhood of C.) Now let P w be the set of δ-balls centered at (mδ, nδ), for all m, n ∈ Z satisfying 1 ≤ m, n < δ −1 and at least one of mδ or nδ belong to C δ .
5.4.The simplifying assumptions are harmless.We will show how to use Theorem 5.3 to prove Theorem 5.2; it is largely an exercise in pigeonholing.
Proof of Theorem 5.2.First, partition the set of δ-tubes T into 100 groups T 1 , T 2 , • • • , T 100 , where T i consists of the tubes in T with angle in [ 2πi 100 , 2π(i+1) 100 ] with the y-axis.By the pigeonhole principle, there exists i with t∈Ti |P t | ≥ 1 100 t∈T |P t |.Henceforth, we work only with the tubes in T i .By rotating the configuration appropriately, we may assume the tubes in T i have angle in [ π 4 , π 4 + π 100 ] with the y-axis.Let w = 2 −n satisfy 4δ ≤ w < 8δ.We first show that every δ-ball in P is contained in a w-ball centered at some point in (wZ) 2 .Indeed, let p ∈ P , and let x be the point in (wZ) 2 closest to p. Then d(x, p) ≤ √ 2 2 w < 3 4 w and δ + d(x, p) < δ + 3 4 w < w, so p ⊂ B w (x).Now we replace the δ-balls in P = t∈T P t with w-balls centered in (wZ) 2 containing the respective δ-balls, forming P ′ = t∈T P ′ t .Likewise, we thicken the δ-tubes in T i to w-tubes, forming T ′ i .The resulting sets P ′ t and T ′ i will be (w, β, ⌈64K β ⌉) and (w, α, ⌈64K α ⌉)-sets respectively.
We would further like P ′ to have centers in ((2w + 1)Z) 2 .To ensure this, for a, b ∈ {0, 1}, let P ′ t (a, b) be the elements in P centered at some (mw, nw) with m ≡ a, n ≡ b (mod 2).Then by the pigeonhole principle, there exist a, b such that By translating the configuration appropriately, we may assume that a = b = 1.
Definition 5.6.For 0 < v ≤ 2 and 0 < u ≤ 1, we call a collection P of essentially for any ϵ > 0, the bound in Theorem 1.6 follows from the corresponding discretized version.
The proof is a very slight modification of Section 6.3 of [DOV22], using our Theorem 5.2.We provide full technical details below.Let X be a minimal (disjoint) covering of A + B by δ-balls, and let Y be a minimal covering of AC by δ-balls.Let X denote the set of centers of the δ-balls in X, and define Ỹ , Ã, B, C analogously.Finally, for x ∈ A + B, let X(x) be the center of the δ-ball in X containing x, and similarly for Ỹ (x).Let   Now, we turn to proving the facts.From Figure 7 and the conditions in equation (13), we can easily show Facts 1 and 2. Now, we will verify the δ-tubes are a (δ, α)set of tubes.Fix w ∈ [δ, 1] and a w × 2 rectangle R w ; we will count how many δ-tubes are in R w .Recall that the bundles in the construction are arranged in a rectangular grid, with bundles in the same row being translates of each other, and bundles in the same column being rotates of each other.We will estimate the number of bundles per row and column that R w intersects (where R w intersects a bundle if it contains a tube from that bundle), as well as the number of tubes R w can contain from each bundle.
• R w can intersect bundles of ≲ ⌈ w δ λκ ⌉ different rows.This is because the angle between bundles of adjacent rows is δ λκ , which is at least the angle δ γ of a single bundle (since γ ≥ λ ≥ λκ).
• B w can intersect bundles of ≲ ⌈ w δ λκ ⌉ different rows.This is because the vertical spacing between two adjacent rows is δ λκ , which is at least the height δ 1−γ of a single bundle (since 1 − γ ≥ λ ≥ λκ).
The first line uses the fact that the sets T w (t) cover T(p).The second line follows from Hölder's inequality.The third line follows from |T(p) ∩ T w (t)| ≤ |T w (t)| and b α ≥ 1.The fourth line follows from Lemma 3.5.The fifth line follows from Lemma 3.4.□ Using Equations (2), (3) and Lemma 3.3, we can finish the proof of Theorem 3.1.
completing the inductive step and thus the proof of Theorem 4.3.□ 5. Proof of Theorems 1.4, 1.6, and Corollary 1.8