Polynomial Wolff axioms and Kakeya-type estimates in R4 Larry Guth∗ Joshua Zahl† January 26, 2017 7 1 0 2 n Abstract a J We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the 4 polynomial Wolff axioms. In brief, a collection of δ–tubes satisfies the Wolff axioms if not too 2 many tubes can be contained in the δ–neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the δ–neighborhood of ] a low degree algebraic variety. A First,weprovethatifasetofδ−3tubesinR4 satisfiesthepolynomialWolffaxioms,thenthe C unionofthetubesmusthavevolumeatleastδ1−1/28. Wealsoproveamoretechnicalstatement . h which is analogous to a maximal function estimate at dimension 3+1/28. Second, we prove t that if a collection of δ−3 tubes in R4 satisfies the polynomial Wolff axioms, and if most triples a m of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume atleastδ3/4. Again, we also provea slightly more technicalstatement which [ is analogous to a maximal function estimate at dimension 3+1/4. 1 We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are v unabletoprovethis. Ifourconjectureiscorrect,itimpliesaKakeyamaximalfunctionestimate 5 at dimension 3+1/28, and in particular this implies that every Kakeya set in R4 must have 4 0 Hausdorff dimension at least 3+1/28. This would be an improvement over the current best 7 bound of 3, which was established by Wolff in 1995. 0 . 1 1 Introduction 0 7 1 A Kakeya set in Rn is a compact subset of Rn that contains a unit line segment pointing in every : v direction. TheKakeyaconjectureassertsthateveryKakeyasetinndimensionsmusthaveHausdorff Xi dimension n. In R2, the conjecture was solved by Davies [5]. In three and higher dimensions the r conjectureremains open,though therehasbeenpartial progress. See[13,9]for asurveyof progress a on the Kakeya problem. A key step when proving Kakeya estimates is to firstdiscretize the problem. After this step has been performed, the Kakeya set is replaced by a finite set of “δ–tubes” (δ neighborhoods of line segments), which point in δ–separated directions; here δ > 0 is a small parameter. The Kakeya problem is transformed into the question of estimating the volume of the union of these tubes (or more precisely, the union of certain subsets of these tubes, which are known as “shadings” of the tubes). In [12], Wolff proved that every Kakeya set and every Nikodym set (a closely related object) in Rn must have Hausdorff dimension at least n+2. To handle both Kakeya sets and Nikodym sets 2 simultaneously, Wolff considered a more general type of set that satisfied the “Wolff axioms”; both Kakeya and Nikodym sets satisfy these axioms. A set of δ–tubes is said to satisfy the Wolff axioms ∗Massachusetts Instituteof Technology, Cambridge, MA, [email protected]. †University of British Columbia, Vancouver,BC, [email protected]. 1 if the tubes obey two of the key statistics possessed by collections of tubes coming from Kakeya sets: the cardinality of the set of δ tubes is δ1−n, and at most t/δ tubes can be contained in the intersection of the t–neighborhood of a line with the δ-neighborhood of a plane. In three dimensions, it is conjectured that the union of any set of tubes satisfying the Wolff axioms musthave volumeclose to1(for reference, if thetubesweredisjoint, thentheirunionwould have volume roughly 1). This is a deep conjecture which would imply the Kakeya conjecture in R3. Infourandhigherdimensions,however, theWolff axiomsarenotsufficienttoforcethetotalvolume tobeclose to 1. For instance, Wolff’s boundasserts thattheunionof any setof tubessatisfying the Wolff axioms in R4 must have volume at least δ, and this is in fact best possible—the set of tubes lying near a quadric hypersurface in R4 satisy the Wolff axioms, but the union of these tubes has volume δ. The above example suggests that in four and higher dimensions, Wolff’s axioms should be extended to not only forbid many tubes from lying near a plane, but to also forbid many tubes from lying near a low degree algebraic variety. We define a polynomial version of the Wolff axioms in this spirit. The precise definition is given in Definition 1.3 below. We conjecture that if a set of δ-tubes in Rn obeys the polynomial Wolff axioms, then the union of the tubes has volume close to 1. In this paper, we study the 4-dimensional case, and we prove that a set of tubes obeying the polynomial Wolff axioms obeys stronger estimates than a set of tubes that only obeys the regular Wolff axioms. We prove that the union of any set of δ−3 tubes in R4 that satisfy the polynomial Wolff axioms must have volume at least δ1−1/28. A key ingredient is a new trilinear Kakeya-type bound in R4. We believe this bound may be of independent interest. To establish the trilinear bound, we use a “grains decomposition” lemma for Kakeya-type sets in Rn,which isrelated tothegrainsdecomposition from[7]. Thislemmasays thatiftheunionof aset of δ–tubes in Rn has small volume, then this arrangement of tubes must have algebraic structure. More precisely, the union of tubes can be covered by the δ–neighborhoods of pieces of algebraic varieties, which are called “grains.” To state our results precisely, we will first need several definitions. Throughout the paper, we will assume that all points, sets, etc. are contained in the ball centered at the origin of radius 2. Definition 1.1. A δ–tube is the δ–neighborhood of a unit line segment (recall that δ–tubes, like all objects in this proof, must be contained in B(0,2)). Definition 1.2. A semi-algebraic set is a set of the form S = {x ∈ Rn: P (x) = 0,...,P (x) = 0, Q (x) > 0,...,Q (x) > 0}, (1) 1 k 1 ℓ where P ,...,P ,Q ,...,Q are polynomials. We define the complexity of S to be min deg(P )+ 1 k 1 ℓ 1 ...+deg(P )+deg(Q )+...+deg(Q ) , where the minimum is taken over all represe(cid:0)ntations of k 1 ℓ S of the form (1) (cid:1) Definition 1.3. Let T be a set of δ–tubes. We say that T satisfies the polynomial Wolff axioms if for every semi-algebraic set S ⊂ Rn of complexity at most E, and every δ ≤ λ ≤ 1, T ∈ T : |T ∩S| ≥ λ|T| ≤K |S|δ1−nλ−n. (2) E (cid:12)(cid:8) (cid:9)(cid:12) (cid:12) (cid:12) Here |{...}| denotes the numbers of tubes (i.e. counting measure), while |S| denotes the Lebesgue measure of S. The following remark should give some intuition for what it means to satisfy the polynomial Wolff axioms. Remark 1.1. Let T be a set of δ–tubes in Rn that satisfy the polynomial Wolff axioms. Then the following properties hold: 2 • The tubes in T are essentially distinct: If T ∈ T, then at most C tubes from T are contained in the 10δ neighborhood of T (denoted N (T)), where C depends only on the constants 10δ K , E = 1,...,6fromDefinition1.3. ThereasonthatC onlydependsonK forE = 1,...,6 E E is that the set N (T) is a semi-algebraic set of complexity ≤ 6. Often, we will not worry 10δ about the exact complexity of the semi-algebraic sets we encounter, so we will replace the number 6 by O(1). • More generally, at most Ct ···t δ1−n tubes are contained in a rectangular prism of dimen- 1 n−1 sions 1×t ×...×t , where C depends only on the constants K , E = 1,...,O(1) from 1 n−1 E Definition 1.3. Any set of tubes T with the property that at most Ct ···t δ1−n tubes are 1 n−1 contained in a rectangular prism of dimensions 1×t ×...×t is said to satisfy the linear 1 n−1 Wolff axioms1. • At most Cδ2−n tubes are contained in the Mδ–neighborhood of an algebraic hypersurface Z(P), where C depends only on M and the constant K from (2) with E = 2degP. E • More generally, if B is a ball of radius r then at most C(δ/r)δ2−n tubes satisfy T ∩ B ⊂ N (Z(P)). Cδ • If Z is a ℓ-dimensional algebraic variety, then at most Cδ1−ℓ tubes are contained in the Mδ neighborhood of Z, where again C depends only on M and and the constants K from (2), E with E = 1,...,O (1). deg(Z) • More generally, if B is a ball of radius r and Z is an ℓ-dimensional variety, then at most C(δ/r)ℓ−nδ1−ℓ tubes satisfy T ∩B ⊂ N (Z). Mδ • Let δ < ρ ≤ 1. If we only consider those tubes lying in the ρ–neighborhood of a line segment (we will call this set a cylinder), and if we re-scale this cylinder to have dimensions 1 × 1×...×1, then the re-scaled tubes will satisfy the polynomial Wolff axioms at scale δ/ρ 2. Furthermore, if {K′ } are the constants for the (rescaled) set of tubes, then K′ . K for E E E each E, where the implicit constant is independent of E. Remark 1.2. The various properties of T discussed in Remark 1.1 essentially characterize what it means for a set of tubes to satisfy the Wolff axioms. Definition 1.4. We say A . B if A ≤ CB. Here and throughout the paper, C will denote a constant (independent of δ) that is allowed to change from line to line. Numbered constants C ,C , 0 1 etc. will have specific meanings, and won’t be allowed to change. While C can be any constant, we will think of it as being large. We will use c to denote a small (positive) constant, which is also allowed to change from line to line. We say A / B if for every ǫ > 0, there exists a constant C (independent of δ) so that ǫ A≤ C δ−ǫB. ǫ Conjecture 1.1. (Kakeya conjecture for the polynomial Wolff axioms) For every dimension n, there is a complexity E so that the following holds. If T is a set of δ-tubes in Rn obeying the polynomial Wolff axioms for semi-algebraic sets of complexity at most E, then 1Note that Wolff’s original axioms only required that at most t/δ tubes are contained in a rectangular prism of dimensions 1×t1×δ×...×δ. Thus the linear Wolff axioms are more restrictive than the original Wolff axioms. Unliketheoriginal Wolff axioms, however, the linear Wolff axioms are preserved byre-scalings. 2Technically this is a lie, since the re-scaled tubes won’t actually be δ/ρ tubes. However, the rescaled tubes are contained in Cδ/ρ tubes, so theissue is easily fixable. 3 n χ / 1, for p = . T (cid:13)(cid:13)TX∈T (cid:13)(cid:13)p n−1 (cid:13) (cid:13) This would imply that T ' 1. (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:12) (cid:12) Before this paper, the only known result about the polynomial Wolff axioms is Wolff’s original result from [12]: if T obeys the original Wolff axioms then n+2 χ / 1, for p = . T (cid:13)(cid:13)TX∈T (cid:13)(cid:13)p n (cid:13) (cid:13) This maximal function boundimplies that Kakeya and Nikodym sets have Hausdorff dimension at least n+2. In this paper, we prove some stronger estimates in the 4-dimensional case. 2 1.1 Linear Kakeya-type bounds in R4 Our first result is a maximal function estimate for sets of tubes that satisfy the polynomial Wolff axioms. Theorem 1.1. Let T be a set of δ–tubes in R4 that satisfy the polynomial Wolff axioms. Then χ / δ−27/85. (3) T (cid:13)(cid:13)TX∈T (cid:13)(cid:13)85/57 (cid:13) (cid:13) See Proposition 2.1 for a slightly messier and more technical version of Theorem 1.1 that de- scribes the implicit constant in (3) in greater detail. In particular, Proposition 2.1 explains how the implicit constant in (3) depends on the constants {K } appearing in Definition 1.3. E Note that 85/57 is the dual exponent to 3+1/28. Thus Theorem 1.1 should be thought of as a maximal function bound of dimension 3+1/28. In particular, Theorem 1.1 gives us a lower bound on the volume of unions of tubes satisfying the polynomial Wolff axioms. Corollary 1.1. Let T be a set of δ−3 δ–tubes in R4 that satisfy the polynomial Wolff axioms. Then T ' δ1−1/28. (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:12) (cid:12) Theorem 1.1 does not tell us anything about the Kakeya conjecture, because we do not know whether every direction-separated set of tubes satisfies the polynomial Wolff axioms. However, we conjecture that this should be the case. Conjecture 1.2. Everysetof tubespointing inδ–separated directions satisfies the polynomial Wolff axioms. More precisely, if T is a set of tubes pointing in δ–separated directions, then T satisfies (2) with constants {K } that are independent of δ. E If Conjecture1.2 is true, then Theorem 1.1 would implya Kakeya maximal functionestimate at dimension 3+1/28. In particular, it would mean that every Kakeya set in R4 must have Hausdorff dimension at least 3+1/28. This would be a slight improvement over the previous best bound of 3 due to Wolff [12], and the related bound of L aba-Tao [10] that every Kakeya set in R4 must have upper Minkowski dimension at least 3+ǫ , where ǫ > 0 is a small absolute constant. 0 0 Remark 1.3. Theorem 1.1 does not actually require the full strength of the polynomial Wolff axioms. The exact conditions needed for Theorem 1.1 will be discussed further in Section 6 below. 4 1.2 Trilinear Kakeya-type bounds in R4 In [1], Bennett, Carbery and Tao proved that if T ,...,T are sets of δ–tubes in Rn, and if each 1 n tube in T makes a small angle with the e direction, then for all q > n , j j n−1 n n χ . δn/q|T | . (4) (cid:13) T (cid:13) j (cid:13)(cid:13)jY=1(cid:16)TX∈TJ (cid:17)(cid:13)(cid:13)Lq/n(Rn) jY=1(cid:0) (cid:1) (cid:13) (cid:13) The endpoint q = n was later established by the first author in [6]. n−1 Heuristically, (4)says thatifTis asetofδ1−n δ-tubes inRn, andifmostn-tuplesofintersecting tubes point in n (quantitatively) linearly independent directions, then T has volume close to T∈T 1. In particular, if T is a collection of δ1−n tubes for which T hSas small volume, then most T∈T of the tubes passing through a typical point in T mustSlie close to a hyperplane. T∈T Inequality (4) deals with the situation whereSthe number of families of tubes is the same as the dimension of the ambient Euclidean space ((4) is an inequality in Rn, and there are n families of tubes). However, itcan beeasily extended tothecase wherethereareℓ ≤ nfamilies oftubesinRn. Heuristically, this says that if T is a set of δ1−n δ-tubes in Rn, and if most ℓ-tuples of intersecting tubes point in ℓ quantitatively linearly independent directions, then T has volume at least T∈T δn−ℓ. This result is sharp: if δ1−n δ tubes are placed at random intSo the δ–neighborhood of an ℓ-dimensional flat in Rn, then the union of these tubes has volume ≤ δn−ℓ, and most ℓ–tuples of intersecting tubes will point in ℓ (quantitatively) linearly independent directions. Theorem 1.2 below is a stronger inequality for three collections of tubes T ,T ,T in R4, if 1 2 3 the tubes in each of T ,T , and T satisfy the polynomial Wolff axioms. Heuristically, Theorem 1 2 3 1.2 says that if T is a set of δ−3 δ-tubes in R4 that satisfy the polynomial Wolff axioms, and if most triples of intersecting tubes pointin three quantitatively linearly independentdirections, then T has volume at least δ3/4. T∈T S Theorem 1.2. Let T be a set of δ-tubes in R4 that satisfy the polynomial Wolff axioms. Then 13/27 χ χ χ |v ∧v ∧v |12/13 / δ−1/3, (5) Z (cid:16)T1,TX2,T3∈T T1 T2 T3 1 2 3 (cid:17) where in the above expression v is the direction of the tube T . i i See Proposition 3.1 for a slightly messier and more technical version of Theorem 1.1 that de- scribes the implicit constant in (20) in greater detail. In particular, Proposition 3.1 explains how the constant depends on the constants {K } appearing in Definition 1.3. E 2 Preliminaries 2.1 Shadings, the two-ends condition, and dyadic pigeonholing Definition 2.1 (Two-ends condition). Let T ⊂ Rn be a δ–tube. We call a set Y(T) ⊂ T a shading of T. We will often use the variable λ to denote the quantity |Y(T)|/|T|. If Y(T) satisfies the bound |Y(T)∩B(x,r)| ≤ αrǫ|Y(T)| (6) for all x ∈ B(0,2) and all δ ≤ r ≤ 1, then we say Y(T) satisfies the two-ends condition with exponent ǫ and error α. 0 5 Definition 2.2. Let T be a set of δ–tubes, and for each T ∈ T let Y(T) be a shading of T. A refinement of Y is a set T′ ⊂ T, and for each T ∈ T′ a set Y′(T) ⊂ Y(T) so that |Y′(T)| & T∈T′ |logδ|−C |Y(T)|, where C is an absolute constant (to be pedantic, we shouPld call this a C– T∈T refinemenPt, but in practice the constant C will always be at most 3). We will sometimes abuse notation and use the same symbols T,Y to denote the refinement of T and Y. For example, if (T,Y) is a set of tubes and their associated shadings with |T| ≤ δ−C, then the function χ (x) can take integer values between 0 and |T|. However, there exists a set B, T∈T Y(T) a numberPµ, and a refinement T′ = T, Y′(T)⊂ Y(T) so that χ (x) ∼ µχ pointwise. T∈T′ Y′(T) B Similarly, if (T,Y) is a set of tubes and their associatedPshadings with δC ≤ Y(T) ≤ |T| for each T ∈ T, then there exists a number λ, a refinement T′ ⊂ T and Y′(T) = Y(T) so that λ ≤ |Y′(T)|/|T| ≤ 2λ for all T ∈T′. 2.2 The two-ends reduction First, we will state a slightly more precise (and uglier) version of Theorem 1.1 that explicitly describes the different constants involved in the bound. Proposition 2.1 (Messy version of Theorem 1.1). For all ǫ > 0, there exist constants c > 0 and ǫ d(ǫ) so that the following holds. Let T be a set of δ–tubes in R4 that satisfy the polynomial Wolff axioms. For each T ∈ T, let Y(T) ⊂ T with λ ≤ |Y(T)|/|T| ≤2λ. Then Y(T) ≥ c λ3+1/28K−1δ1−1/28+ǫ δ3|T| , (7) ǫ (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:0) (cid:1) (cid:12) (cid:12) where K = Kd(ǫ),T = sup1≤E≤d(ǫ)KE, and the constants {KE} are from Definition 1.3. Remark 2.1. Many statements throughout the proof will begin with the phrase “for all ǫ > 0, there exist constants c > 0 and d(ǫ) so that....” The reader should think of the constant c as differing ǫ ǫ between statements, but the function d(ǫ) is universal. Proposition 2.2 (Two-ends version of Proposition 2.1). For all ǫ > 0, ǫ > 0 there exist constants 0 c > 0,C′ , and d(ǫ) so that the following holds. Let T be a set of δ–tubes in R4 that satisfy the ǫ,ǫ0 ǫ,ǫ0 polynomial Wolff axioms. For each T ∈ T, let Y(T) ⊂ T with λ ≤ |Y(T)|/|T| ≤ 2λ. Suppose that each tube T ∈ T satisfies the two-ends condition with exponent ǫ and error α. Then 0 Y(T) ≥ cǫ,ǫ0α−Cǫ′,ǫ0λ3+1/28K−1δ1−1/28+ǫ δ3|T| , (8) (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:0) (cid:1) (cid:12) (cid:12) where K = Kd(ǫ),T = sup1≤E≤d(ǫ)KE, and the constants {KE} are from Definition 1.3. Proof of Proposition 2.1 using Proposition 2.2. The reduction from Proposition 2.1 to Proposition 2.2 is a standard application of the “two-ends reduction” argument. The one new feature is that instead of pointing in δ–separated directions, the tubes in T instead satisfy the polynomial Wolff axioms. As a result, we must take greater care when performing the “rescaling” part of the argument. Fix ǫ > 0. Let ǫ = ǫ/(6+1/14). Replace each shading Y(T) by a “two-ends” piece Y(T)∩ 0 B(x,r), satisfying |Y(T)∩B(x,r)| ≥rǫ0λ|T|, and |B(x,r′)∩Y(T)| ≤ (r′)ǫ0|B(x,r)∩Y(T)| 6 for all r′ ≤ r (note that r and x depend on the choice of tube T). Let T′ be a refinement of T so that r ∼r and λ′ ≤ |B(x,r)∩Y(T)| ≤ 2λ′ for all T ∈ T′, where 0 λ′ ≥ δǫ0λ. Observe that each tube in T′ satisfies the two-ends condition with exponent ǫ and error 0 α = 1. Cover B(0,2) by boundedly overlapping balls of radius 10r , so that each ball of radius r is 0 0 entirely contained within one of the balls. Associate each T ∈ T′ to one of the balls. This gives us a partition T′ = T′ . B For each ballFB, re-scale the tubes in T′ to have dimensions 1×δ/r ×δ/r ×δ/r , and denote B 0 0 0 this new set by T˜′ . Observe that if {K˜ } are the Wolff constants associated to T˜′ , and if {K } B E B E aretheWolff constantsassociated toT,thenK˜ = r−3K foreach E 3. Toseethis, letS˜ ⊂B(0,2) E 0 E be a semi-algebraic set of complexity E, and let S be the pre-image of S under the re-scaling that sends B to the ball B(0,2). Then |{T˜ ∈ T˜: |T˜∩S˜| ≥ λ |T˜|}| ≤ |{T ∈ T: |T ∩S| ≥ r λ |T|}| 1 0 1 ≤ K |S|δ−3(r λ )−4 E 0 1 (9) = K |S˜|δ−3λ−4 E 1 = K r−3|S˜|(δ/r )−3λ−4. E 0 0 1 Furthermore, |Y˜(T˜)| ∼ (λ′/r ) for each T˜ ∈T˜. 0 Apply Proposition 2.2 to T˜′ . We conclude that B Y˜(T˜) ≥ c (λ′/r )3+1/28(r−3K)−1(δ/r)1−1/28+ǫ/2 (δ/r )3|T˜′ | (cid:12)(cid:12)T˜[∈T˜′ (cid:12)(cid:12) ǫ/2,ǫ0 0 0 (cid:0) 0 B (cid:1) (cid:12) B (cid:12) = r−4c (λ′)3+1/28K−1δ1−1/28+ǫ/2 δ3|T˜′ | . 0 ǫ/2,ǫ0 B (cid:0) (cid:1) and thus after undoing the scaling δ → δ/r , we have 0 Y(T) ≥ c (λ′)3+1/28K−1δ1−1/28+ǫ/2 δ3|T˜′ | (10) ǫ/2 B (cid:12)(cid:12)T∈T[:T˜∈T′ (cid:12)(cid:12) (cid:0) (cid:1) (cid:12) B (cid:12) Since |T| ≥ δǫ/2 |T˜′ |, and the sets in (10) are at most 106–fold overlapping for different balls B B B, we have P Y(T) ≥ c Y(T) (cid:12)(cid:12)T[∈T (cid:12)(cid:12) XB (cid:12)(cid:12)T∈T[:T˜∈T′ (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) B (cid:12) ≥ c ≥ c (λ′)3+1/28K−1δ1−1/28+ǫ/2 δ3|T˜′ | ǫ/2,ǫ0 B XB (cid:0) (cid:1) ≥ c c λ3+1/28δ(3+1/28)ǫ/(6+1/14)K−1δ1−1/28+ǫ/2 δ3|T| ǫ/2,ǫ0 (cid:0) (cid:1) ≥ c c λ3+1/28K−1δ1−1/28+ǫ δ3|T| . ǫ/2,ǫ0 (cid:0) (cid:1) 2.3 The robust transversality reduction Theorem 1.2 gives the strongest bounds when most triples of intersecting tubes point in three linearly independentdirections. As a starting point, one would hopethat most pairs of intersecting tubes point in two linearly independentdirections. In this section, we will show that we can always reduce to this situation. 3thisscalingfactorofr−3 isthesamescalingfactoronewouldexpectfortubespointinginδ–separateddirections: 0 ifwerestrictattentiontoaballofradiusr0,therewillbeatmostr0−3tubespointingineachδ/r0–separateddirection. 7 Definition 2.3. Let (T,Y) be a set of δ tubes and their associated shadings. We say that (T,Y) is s–robustly transverse (with error t) if for all x ∈ R4 and all vectors v, we have |{T ∈ T: x ∈Y(T), ∠(T,v) < s}|≤ t|{T ∈ T: x ∈ Y(T)}|. (11) We wish to reduce Proposition 2.2 to the case where T is robustly transverse. This reduction will involve an induction argument. The next proposition is identical to Proposition 2.2, except we have added the requirement that the tubes are robustly transverse. Proposition 2.3. For all ǫ > 0, ǫ > 0 there exist constants c > 0,C′ , and d(ǫ) so that the 0 ǫ,ǫ0 ǫ,ǫ0 following holds. Let T be a set of δ–tubes in R4 that satisfy the polynomial Wolff axioms. For each T ∈ T, let Y(T)⊂ T with λ ≤ |Y(T)|/|T| ≤ 2λ. Suppose that (T,Y) is s–robustly transverse (with error 1/100) and that each tube T ∈ T satisfies the two-ends condition with exponent ǫ and error 0 α. Then (cid:12)(cid:12)T[∈TY(T)(cid:12)(cid:12) ≥ csc′ǫ,ǫ0α−Cǫ′,ǫ0λ3+1/28K−1δ1−1/28+ǫ(cid:0)δ3|T|(cid:1), (12) (cid:12) (cid:12) where K = Kd(ǫ),T = sup1≤E≤d(ǫ)KE, and the constants {KE} are from Definition 1.3. We will show that Proposition 2.3 implies Theorem 1.1. Proof of Proposition 2.1 using Proposition 2.3. Suppose that Proposition 2.3 holds; we will prove Proposition 2.1 by induction on δ. Fix the value of ǫ from the statement of Proposition 2.1. We will assume that ǫ < 1/4. By making the constant c sufficiently small, we can assume that Proposition 2.1 holds for all ǫ δ > 0 satisfying |logδ| ≤ δ−ǫ/100. (13) Suppose Proposition 2.1 has been established for all δ ≥ δ , and let (T,Y) be a set of δ tubes 0 satisfying the hypotheses of Proposition 2.1. Let X ⊂ Y(T) be the set where (11) holds with T∈T s > 0 a small constant (depending only on ǫ and ǫ ) toSbe determined later. Either 0 (A): |Y(T)∩X| ≥ 1 |Y(T)|, or T∈T 2 T∈T P P (B): (A) fails Suppose (A) holds. Let Y′(T) = Y(T) ∩ X and let T′ = {T ∈ T: |Y′(T)| ≥ 1λ|T|}. Then 4 |T′| ≥ 1|T|. We can refine each shading Y′(T) slightly so that λ/4 ≤ |Y′(T)|/|T| ≤ λ/2. Each of 4 the tubes still satisfies the two-ends condition with exponent ǫ and error α/4. Apply Proposition 0 2.3 to T′ with the shading Y′(T) and with the value of ǫ specified above. We conclude that Y(T) ≥ Y′(T) (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1 ≥ csc′ǫ,ǫ0(4α)−Cǫ′,ǫ0(4λ)3+1/28K−1δ1−1/28+ǫ(δ3|T′|) ≥ 4−Cǫ′,ǫ0−4csc′ǫ,ǫ0α−Cǫ′,ǫ0λ3+1/28K−1δ1−1/28+ǫ(δ3|T|). Thus Theorem 1.1 holds as long as cǫ,ǫ0 ≤ c′ǫ,ǫ0cs4−Cǫ′,ǫ0−4. Now suppose (B) holds. Let A= R\X. Then for every x ∈ A, there is a vector v so that x 1 |{T ∈ T: x ∈ Y(T), ∠(T,v )< s}| ≥ |{T ∈ T: x ∈ Y(T)}|. (14) x 100 8 For each T ∈ T, let Y′(T) = Y(T)∩{x ∈ A: ∠(T,v ) < s}. Then x 1 |Y′(T)| ≥ λ(δ3|T|). (15) 200 TX∈T Let T′ = {T ∈ T: |Y′(T)| ≥ 1 λ|T|}. Then |T′| ≥ 1 |T|. We can refine each shading Y′(T) 400 400 slightlysothatλ/400 ≤ |Y′(T)|/|T| ≤λ/200. Eachofthetubesstillsatisfiesthetwo-endscondition with exponent ǫ and error 400α. 0 Cover the sphere S3 with ≤ 100-fold overlapping caps of radius 3s, so that every ball (in S3) of radius s is entirely contained in one of the caps. Note that if x ∈ A, then there is a cap τ so that every tube T ∈ T′ with x ∈ Y′(T) points in a direction lying in τ. For each cap τ, let T′(τ) ⊂ T′ be a set of tubes pointing in directions lying in τ, so that T′ = T′(τ) is a partition of T′. For each τ cap τ, partition B(0,2) into ≤ 100–fold overlapping cylinderFs of dimensions 1×10ρ×10ρ×10ρ; call this set of cylinders Cyl(τ). Note that if T ∈ T′(τ), then T is contained in at least one of these cylinders. For each such cylinder U, T′(τ,U) ⊂ T′(τ), so that T′ = T′(τ,U), G G τ U∈Cyl(τ) i.e. each tube from T is assigned to a cap τ and a 100ρ cylinder pointing in the direction τ. Observe that the sets { Y′(T)} are at most 104–fold overlapping. Thus T∈T(τ,U) τ,U S Y(T) ≥ 10−4 Y′(T) . (16) (cid:12)(cid:12)T[∈T′ (cid:12)(cid:12) Xτ U∈XCyl(τ)(cid:12)(cid:12)T∈T[′(τ,U) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Now, for each cap τ and each cylinder U ∈ Cyl(τ), let L bea line pointing in the same direction as U and distance 100s from U. Let f: R4 → R4 be the map that fixes L and dilates R4 by a factor of s−1 in all directions orthogonal to L. Then f(U) contains a ball of radius 1/1000 and is contained in a ball of radius 1000; if Y ⊂ U is a set, then 1 ρ−3|Y|≤ |f(Y)| ≤ 1000ρ−3|Y|; (17) 1000 if T ∈ T(τ,U), then f(T) is contained in a 1000ρ−1δ tube and contains a 1 ρ−1T. For each 1000 T ∈ T′(τ,U), let T˜ be a 1000ρ−1δ–tube that contains f(T), and let Y˜(T) = f(Y(T)). Then T˜ = {T˜: T ∈ T(τ,U)} satisfies the polynomial Wolff axioms (at scale ρ−1δ). By (17), the constant K associated to T˜ is at most 1000 times the constant K associated to T′ (which is also the constant associated to T) (K also depends on ǫ, but we have fixed a value of ǫ throughout this proof.) Apply Theorem 1.1 to (T˜,Y˜) (with the same value of ǫ as above). We conclude that Y˜(T) ≥ cǫ,ǫ0(400α)−Cǫ′,ǫ0(10−5λ)3+1/28(103K)−1(δ/s)1−1/28+ǫ (δ/s)3|T˜| , (cid:12)(cid:12)T˜[∈T˜ (cid:12)(cid:12) (cid:0) (cid:1) (cid:12) (cid:12) and thus by (17), Y′(T) ≥ cǫ,ǫ0(400α)−Cǫ′,ǫ0(10−5λ)3+1/28(103K)−1(δ/s)1−1/28+ǫ δ3|T(τ,U)| . (18) (cid:12)(cid:12)T∈T[(τ,U) (cid:12)(cid:12) (cid:0) (cid:1) (cid:12) (cid:12) Combining (16) and (18), we conclude Y(T) ≥ 10−20−3Cǫ′,ǫ0 cǫ,ǫ0α−Cǫ′,ǫ0λ3+1/28K−1(δ/s)1−1/28+ǫ δ3|T(τ,U)| (cid:12)(cid:12)T[∈T′ (cid:12)(cid:12) Xτ U∈XCyl(τ) (cid:0) (cid:1) (19) (cid:12) (cid:12) ≥ c0 cǫ,ǫ0α−Cǫ′,ǫ0λ3+1/28K−1(δ/s)1−1/28+ǫ δ3|T| , (cid:0) (cid:1) 9 Thus, provided we select s sufficiently small (depending only on C′ , which in turn depends only ǫ,ǫ0 on ǫ and ǫ0) so that s1−1/28−ǫ < s1/2 < 10−20−3Cǫ′,ǫ0, then Y(T) ≥ cǫ,ǫ0α−Cǫ′,ǫ0λ3+1/28K−1δ1−1/28+ǫ(δ3|T|), (cid:12)(cid:12)T[∈T (cid:12)(cid:12) (cid:12) (cid:12) which closes the induction and completes the proof of Proposition 2.1 . In Section 5 we will prove Proposition 2.3. 3 Trilinear Kakeya in R4 In this section, we will prove Theorem 1.2. First, we will state a slightly more technical version of the theorem Proposition 3.1. For all ǫ > 0, there exist constants C ,d(ǫ) so that the following holds. Let T be ǫ a set of δ tubes in R4 that satisfy the polynomial Wolff axioms. Then 13/27 χ χ χ |v ∧v ∧v |12/13 ≤ C δ−1/3−ǫK1/9(δ3|T|)4/3, (20) Z (cid:16)T1,TX2,T3∈T T1 T2 T3 1 2 3 (cid:17) ǫ where in the above expression vi is the direction of the tube Ti, and K = KT,d(ǫ) = sup1≤E≤d(ǫ)KE, where {K } are the constants from Definition 1.3 E Corollary 3.1. For all ǫ > 0, there exist constants c > 0,d(ǫ) so that the following holds. Let T ǫ be a set of δ–tubes in R4 that satisfy the polynomial Wolff axioms. For each T ∈ T, let Y(T) ⊂ T with |Y(T)| ∼ λ|T|. Suppose that (T,Y) is s–robustly transverse with error 1/100, and that for all x ∈ R4 and all 2–planes Π, we have 1 |{T ∈ T: x ∈ Y(T), ∠(T,Π) < θ}|≤ |{T ∈ T: x ∈ Y(T)}|. (21) 100 Then Y(T) ≥ c c λ3+1/4K−1/4θδ3/4+ǫ(δ3|T|)1/4, (22) ǫ s (cid:12)[ (cid:12) (cid:12) (cid:12) where K = sup K(cid:12) , and {(cid:12)K } are the constant from Definition 1.3 1≤E≤d(ǫ/9) E E Remark 3.1. Heuristically, Corollary 3.1 says that if T is a set of δ−3 essentially distinct δ–tubes that satisfy the polynomial Wolff axioms, and if most triples of tubes passing through a typical cube in R4 span three quantitatively linearly independent directions, the the union of the tubes has volume at least δ3/4; this corresponds to a Hausdorff dimension bound of 3+1/4. In contrast, the multilinear Kakeya theorem says that the union of these tubes has volume at least δ (this is a weaker statement that corresponds to a Hausdorff dimension bound of 3). Remark 3.2. If T satisfies the polynomial Wolff axioms, then since each tube in T is contained in B(0,2), and B(0,2) is a semi-algebraic set of measure ≤ 100 and complexity 2, we have (δ3|T|) ≤ 100C , where C is the constant from Definition 1.3. Thus (22) can be replaced by the (weaker) 2 2 bound Y(T) ≥ c c λ3+1/4K−1θδ3/4+ǫ(δ3|T|). (23) ǫ s (cid:12)[ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 10