A discretized Severi-type theorem with applications to harmonic analysis
AA discretized Severi-type theorem with applicationsto harmonic analysis
Joshua Zahl ∗ April 25, 2018
Abstract
In 1901, Severi proved that if Z is an irreducible hypersurface in P ( C )that contains a three dimensional set of lines, then Z is either a quadratichypersurface or a scroll of planes. We prove a discretized version of this resultfor hypersurfaces in R . As an application, we prove that at most δ − − ε direction-separated δ -tubes can be contained in the δ -neighborhood of a low-degree hypersurface in R .This result leads to improved bounds on the restriction and Kakeya prob-lems in R . Combined with previous work of Guth and the author, this resultimplies a Kakeya maximal function estimate at dimension 3 + 1 /
28, which isan improvement over the previous bound of 3 due to Wolff. As a consequence,we prove that every Besicovitch set in R must have Hausdorff dimension atleast 3 + 1 /
28. Recently, Demeter showed that any improvement over Wolff’sbound for the Kakeya maximal function yields new bounds on the restrictionproblem for the paraboloid in R . In [19], Severi classified projective hypersurfaces in P ( C ) that contain many lines. Theorem 1.1 (Severi) . Let Z ⊂ P ( C ) be an irreducible hypersurface. Let Σ be theset of lines contained in Z . Then • If dim(Σ) = 4 , then Z is a hyperplane. ∗ University of British Columbia, Vancouver BC. Supported by a NSERC Discovery Grant. a r X i v : . [ m a t h . C A ] A p r If dim(Σ) = 3 , then Z is either a quadratic hypersurface or a scroll of planes. This theorem was later generalized to higher dimensions by Segre [18]. See [17]for further discussion and [20, Appendix A] for a modern (and English) proof.Severi’s theorem allows us to control the set of directions of lines inside a hyper-surface.
Corollary 1.1.
Let Z ⊂ P ( C ) be an irreducible hypersurface. Then the lines con-tained in Z point in at most a two-dimensional set of directions. Algebraic varieties containing many lines have recently become a topic of interestwhen studying the Kakeya and restriction problems. In [12], Katz, (cid:32)Laba, and Taoobserved that the Heisenbeg group H = { ( z , z , z ) ∈ C : Im( z ) = Im( z ¯ z ) } is an “almost counter-example” to the Kakeya conjecture—it is a subset of C thatcontains many complex lines, few of which lie in a common plane. In four dimensions,Guth and the author showed in [11] that low-degree hypersurfaces containing manylines are the only possible obstruction to obtaining improved Kakeya estimates in R . Similar statements are implicit in the works of Guth [10] and Demeter [6].In this paper, we will prove a discretized version of Theorem 1.1. Our theoremwill classify the algebraic hypersurfaces in R whose δ -neighborhood, restricted tothe unit ball, contains many unit line segments. In contrast to the classical situationstudied by Severi, it is not true that if Z ( P ) is an irreducible hypersurface in R whose δ -neighborhood contains many unit line segments, then Z ( P ) must be a hyperplane,a quadric hypersurface, or a scroll of planes. For example, let P ( x ) = x x + δ .It is easy to verify that P is irreducible, and the δ -neighborhood of Z ( P ) containsmany unit line segments. Geometrically, Z ( P ) is a small perturbation of the variety { x = 0 } ∪ { x = 0 } , which is a union of two hyperplanes. In particular, largeregions of the δ -neighborhoods of Z ( P ) ∩ B (0 ,
1) and { x = 0 } ∪ { x = 0 } ∩ B (0 , N δ ( Z ) are contained in the δ –neighborhood of the hyperplane { x = 0 } , and “half”are contained in the δ –neighborhood of the hyperplane { x = 0 } .As a more extreme example, Z ( P ) might be a small perturbation of the variety Z ∪ Z ∪ Z ∪ Z , where Z is an arbitrary low-degree hypersurface containing few lines; Z is a scroll of planes; Z is a hyperplane; and Z is a quadratic hypersurface. Ourdiscretized version of Severi’s theorem follows this idea. Informally it says that if Z is a hypersurface, then the unit line segments contained in N δ ( Z ) can be partitionedinto four classes in the spirit of the above example.2s an application of our techniques, we prove a variant of Corollary 1.1, whichsays that the set of unit line segments in the unit ball lying near N δ ( Z ) can point inat most δ − − ε different δ -separated directions. As discussed in Section 10, this resultyields an improved bound for the Kakeya maximal function in R . In [5], Demeterproved that such an improvement for the Kakeya maximal function would yield newbounds on the restriction problem for the paraboloid in R . This will be discussedfurther in Section 1.2.Before stating the main result of this paper, we will introduce some notation. Definition . Let L be the set of lines in R . For each (cid:96) ∈ L , define Dir( (cid:96) ) to be aunit vector in R pointing in the same direction as (cid:96) . By convention, we will choosethe unit vector v = ( v , v , v , v ) with v ≥ v ≥ v = 0; v ≥ v = v = 0;and v = 1 if v = v = v = 0.If E is a set of unit vectors in R and δ >
0, we will write E δ ( E ) to denote the δ covering number of E . More generally if ( X, d ) is a metric space and E ⊂ X , E δ ( E )will denote the δ covering number of E .Our proof will refer to “rectangular prisms,” which are the discretized analoguesof lines, planes, and hyperplanes. These prisms will have “long directions,” whichhave length two, and “short directions,” which have length much smaller than two.Informally, we say a rectangular prism is k dimensional if it has k long directions (allsuch prisms will be contained in R ). We say that a line is covered by a rectangularprism if the intersection has length at least two.The following is a discretized version of Theorem 1.1. Theorem 1.2.
Let P ∈ R [ x , x , x , x ] be a polynomial of degree D , and let Z = Z ( P ) ∩ B (0 , . Let δ, κ, u, s ∈ (0 , be numbers satisfying < δ < u < s < and δ < κ < (if these conditions are not satisfied the theorem is still true, but it has nocontent).Define Σ = { (cid:96) ∈ L : | (cid:96) ∩ N δ ( Z ) | ≥ } . Then we can write
Σ = Σ ∪ Σ ∪ Σ ∪ Σ , where • There is a collection of O D (cid:0) | log δ | O (1) s − (cid:1) rectangular prisms of dimensions × s × s × s so that every line from Σ is covered by one of these prisms. • There is a collection of O D (cid:0) ( | log δ | /s ) O (1) u − (cid:1) rectangular prisms of dimen-sions × × u × u so that every line from Σ is covered by one of these prisms. • There is a collection of O D (cid:0) | log δ | O (1) (cid:1) rectangular prisms of dimensions × × × κ so that every line in Σ is covered by one of these prisms. There is a set Σ (cid:48) ⊂ Σ with E δ (Σ (cid:48) ) (cid:38) D (cid:0) usκ/ | log δ | (cid:1) O (1) E δ (Σ ) and a quadratic hypersurface Q so that for every line (cid:96) (cid:48) ∈ Σ (cid:48) , there is a line (cid:96) contained in Z ( Q ) with dist( (cid:96), (cid:96) (cid:48) ) (cid:46) D (cid:0) | log δ | / ( usκ ) (cid:1) O (1) δ. We will prove Theorem 1.2 (or actually a slightly more technical generalization)in Section 8 below. As a corollary of (the more technical version of) Theorem 1.2,we obtain the following discretized analogue of Corollary 1.1.
Corollary 1.2 (Few directions near a low-degree variety) . Let P ∈ R [ x , . . . , x ] bea polynomial of degree D and let Z = Z ( P ) ∩ B (0 , . Let < δ < and define Σ = { (cid:96) ∈ L : | (cid:96) ∩ N δ ( Z ) | ≥ } . Then for each ε > , E δ (cid:0) Dir(Σ) (cid:1) (cid:46)
D,ε δ − − ε . (1)In brief, Corollary 1.2 follows from Theorem 1.2 by choosing s = | log δ | − C , u = | log δ | − C , κ = | log δ | − C , where C , C , C are large absolute constants. Since thelines contained in a quadratic hypersurface in R can point in few directions, the setof directions of lines in Σ (cid:48) is small. Using a slightly more technical version of Theorem1.2, we can also guarantee that the set of directions of lines in Σ is small. The linesin Σ , Σ , and Σ are handled by re-scaling and induction on scales. Corollary 1.2will be proved in detail in Section 9. Remark . We could replace the condition | (cid:96) ∩ N δ ( Z ) | ≥ | (cid:96) ∩ N δ ( Z ) | ≥ c for any fixed constant c >
0. Then the implicit constant in (1)would also depend on c .In [10], Guth stated the following conjecture Conjecture 1.1.
Let Z ⊂ R d be a m -dimensional variety defined by polynomials ofdegree at most D , and let Σ be the set of lines in R d satisfying | (cid:96) ∩ N δ ( Z ) ∩ B (0 , | ≥ .Then for each δ > and ε > , the set of directions of lines in Σ can be covered by O d,D,ε ( δ − m ) balls of radius δ . When m = 2, Conjecture 1.1 is straightforward, and the result was used by Guthin [9] to obtain improved restriction estimates in R . Corollary 1.2 proves Conjecture1.1 in the case d = 4 , m = 3. For m ≥ Addendum added April 2018 : Conjecture 1.1 was recently proved in all dimen-sions by Katz and Rogers [13]. 4 .1 Progress on the Kakeya conjecture
Recall the Kakeya maximal function conjecture. In the statement below, a δ -tube isthe δ -neighborhood of a unit line segment. Conjecture 1.2.
Let T be a set of δ -tubes in R d that point in δ -separated directions.Then for each ε > , (cid:13)(cid:13)(cid:13) (cid:88) T ∈ T χ T (cid:13)(cid:13)(cid:13) p (cid:48) (cid:46) ε δ − d/p − ε , p = d. (2)Conjecture 1.2 has been proved when d = 2 by C´ordoba [4] and remains open for d ≥
3. If the exponent p = d in (2) is replaced by a number 1 ≤ p ≤ d , then (2) iscalled a Kakeya maximal function estimate in R d at dimension p .Using the results of Guth and the author from [11], Theorem 1.2 can be used toobtain a Kakeya maximal function estimate in R at dimension 3 + 1 / Theorem 1.3.
Let T be a set of δ -tubes in R that point in δ -separated directions.Then for each ε > , (cid:13)(cid:13)(cid:13) (cid:88) T ∈ T χ T (cid:13)(cid:13)(cid:13) p (cid:48) (cid:46) ε δ − /p − ε , p = 3 + 1 / . (3) Corollary 1.3.
Every Besicovitch set in R has Hausdorff dimension at least / . Theorem 1.3 will be proved in Section 10. It is an improvement over the previousbound p = 3, which was due to Wolff [22]. Previously, (cid:32)Laba and Tao [15] proved thatevery Besicovitch set in R must have upper Minkowski dimension at least 3 + ε forsome positive constant ε > . In a similar vein, Tao [21] proved that every Kakeyaset in F p must have cardinality at least cp / . This was later improved by Dvir [6]and then Dvir, Kopparty, Saraf, and Sudan [7], who proved nearly sharp bounds onthe size of Kakeya sets in F np for every n . Let f : [ − , d − → C . For each x = ( x, x d ) ∈ R d , define the extension operator Ef by Ef ( x ) = (cid:90) [ − , d − f ( ξ ) e ξ · x + | ξ | x n dξ. The restriction conjecture for the paraboloid relates the size of f and Ef . Without attempting to optimize their arguments, (cid:32)Laba and Tao obtained the estimate ε ≥ − . A careful analysis of their methods would likely yield a larger value of ε . onjecture 1.3. For each q > dd − and each f : [ − , d → C , we have (cid:107) Ef (cid:107) q (cid:46) q,d (cid:107) f (cid:107) ∞ . (4)Conjecture 1.3 has been proved when d = 2 by Fefferman and Zygmund [8, 25].For d ≥ d = 4, (4) is known for q > /
5. In [6], Demeter proved that improvementsto the Kakeya maximal function conjecture in R would lead to progress on therestriction conjecture. Theorem 1.4 ( [6], Theorem 1.4) . Let d = 4 . If (2) holds for some p > , then (4) holds for some q < / . When Theorems 1.3 and 1.4 are combined, they yield an improved restrictionestimate for the paraboloid in R . Inserting the exponent p = 3+1 /
28 into Demeter’sargument yields the exponent q = − . In this section we will survey the main ideas in the proof of Theorem 1.2. Let P ∈ R [ x , . . . , x ] be a polynomial of degree at most D and let Z = Z ( P ) ∩ B (0 , | (cid:96) ∩ N δ ( Z ) | ≥ ∇ P ( z ) ∼ z ∈ Z andthat | P ( x ) | ≤ δ for all x ∈ N δ ( Z ). While these assumptions certainly need not holdin general, a reduction performed in Section 3 allows us to reduce to the case wherea similar (though slightly more technical) statement holds.Let (cid:96) be a line satisfying | (cid:96) ∩ N δ ( Z ) | ≥ x ∈ (cid:96) ∩ N δ ( Z ). Let (cid:96) ( t ) be a unitspeed parameterization of (cid:96) with (cid:96) (0) = x . Then P ( (cid:96) ( t )) is a univariate polynomialof degree at most D , and | P ( (cid:96) ( t )) | is small for many values of t . More precisely, wehave |{ t ∈ [ − ,
1] : | P ( (cid:96) ( t )) | (cid:46) δ }| (cid:38) . This means that all of the coefficients of P ( (cid:96) ( t )) have magnitude (cid:46) δ (the implicitconstant may depend on D , but we will suppress this dependence here). In partic-ular, if v is a unit vector pointing in the same direction as (cid:96) and if z ∈ Z satisfiesdist( x, z ) ≤ δ , then | v · ∇ P ( z ) | (cid:46) δ and | ( v · ∇ ) P ( z ) | (cid:46) δ. (5)For each z ∈ Z ( P ), the set of vectors { v ∈ S : v · ∇ P ( z ) = 0 , ( v · ∇ ) P ( z ) = 0 } is called the quadratic cone of Z ( P ) at z, and it is closely related to the secondfundamental form of Z ( P ) at z . 6e wish to understand the relationship between the set of vectors satisfying (5)and the set of vectors in the quadratic cone of Z ( P ) at z . It thus seems reasonableto ask: if z ∈ Z and if a unit vector v ∈ S satisfies (5), must it be the case that v is contained in the (cid:46) δ -neighborhood of the quadratic cone of Z ( P ) at z ?In short, the answer is no. As a first example, consider the polynomial P ( x , x ,x , x ) = x + δx , and let z = 0. Then the quadratic cone of Z ( P ) at z = 0 is { ( v , v , v , v ) ∈ S : v = v = 0 } . However, the set of vectors satisfying (5) is muchlarger; it is comparable to { ( v , v , v , v ) ∈ S : | v | (cid:46) δ } . In this example, the δ -neighborhood of Z ( P ) ∩ B (0 ,
1) is comparable to the δ -neighborhood of a hyperplane.In Section 4 we will expand on this observation. We will prove a technical variant ofthe following idea: if the coefficients of the second fundamental form of Z ( P ) are allvery small at a typical point, then large pieces of Z ( P ) ∩ B (0 ,
1) can be containedin the thin neighborhood of a hyperplane. The lines having large intersection withthese pieces will end up in the set Σ from the statement of the theorem.As a second example, consider the polynomial P ( x , x , x , x ) = x + x and let z = 0. Then the quadratic cone of Z ( P ) at z = 0 is again { ( v , v , v , v ) ∈ S : v = v = 0 } , while the set of vectors satisfying (5) is comparable to { ( v , v , v , v ) ∈ S : | v | (cid:46) δ, | v | (cid:46) δ / } . In this example, at least one coefficient of the secondfundamental form of Z ( P ) at z = 0 has large magnitude, and the quadratic cone of Z ( P ) at z = 0 is a plane. In Section 6, we will show that if at least one coefficient ofthe second fundamental form of Z ( P ) is large at a typical point, and if the quadraticcone is comparable to either a plane or a union of two planes, then most of thelines contained in N δ ( Z ( P ) ∩ B (0 , and Σ from the statementof the theorem.Now let z ∈ Z ( P ) ∩ B (0 ,
1) and suppose that at least one coefficient of the secondfundamental form of Z ( P ) at z is large and that the quadratic cone of Z ( P ) at z isnot comparable to either a plane or a union of two planes. Then the set of vectorssatisfying (5) is comparable to the δ -neighborhood of the quadratic cone of Z ( P ) at z . In Section 7, we will show that if this happens at a typical point, then morallyspeaking Z ( P ) must be a quadratic hypersurface. More precisely, many of the lineslying near N δ ( Z ( P ) ∩ B (0 , from the statement of the theorem. The author would like to thank Larry Guth and Ciprian Demeter for helpful discus-sions. 7
A primer on real algebraic geometry
Our proof will use several facts about semi-algebraic sets. Further background canbe found in [2]. A semi-algebraic set is a set S ⊂ R d of the form S = { x ∈ R d : P ( x ) = 0 , . . . , P k ( x ) = 0 , Q ( x ) > , . . . , Q (cid:96) ( x ) > } , (6)where P , . . . , P k and Q , . . . , Q (cid:96) are polynomials. We define the complexity of S tobe the minimum of deg( P ) + . . . + deg( P k ) + deg( Q ) + . . . + deg( Q (cid:96) ), where theminimum is taken over all representations of S of the form (6).If S, T ⊂ R d are semi-algebraic sets of complexity E and E respectively, then S ∪ T, S ∩ T, and S \ T are semi-algebraic sets of complexity O d,E ,E (1). If π : R d → R e is a projection, then π ( S ) is a semi-algebraic set of complexity O d,E (1).If S ⊂ R d , T ⊂ R e are semi-algebraic sets, we say that a function f : S → T issemi-algebraic of complexity E if the graph of f is a semi-algebraic set of complexity E . Clearly if f : S → T is a semi-algebraic bijection of complexity E , then f − : T → S is also a semi-algebraic bijection of complexity E .If S ⊂ R d is a semi-algebraic set, we define its dimension dim( S ) to be the largestinteger e so that there is a subset S (cid:48) ⊂ S that is homeomorphic to the open e -dimensional cube (0 , e . If S ⊂ R d has dimension e and complexity E , then there isan e -dimensional real algebraic variety S ⊂ Z ⊂ R d that is defined by polynomialsof degree O d,E (1). If S and T are semi-algebraic sets, and if there is a semi-algebraicbijection f : S → T , then S and T have the same dimension.Occasionally, we will refer to semi-algebraic subsets of the sphere S d or the affineGrassmannian Grass( d ; e ) of e -dimensional affine subspaces of R d . To do this, wewill identify S d or Grass( d ; e ) with a semi-algebraic set in R N for some N = O d (1).In the remainder of this section, we will list several standard results about realalgebraic sets that will be used throughout the proof. Lemma 2.1 (Milnor-Thom theorem) . Let S ⊂ R d be a semi-algebraic set of com-plexity at most E . Then S has O E,d (1) connected components.
This is a variant of the Milnor-Thom Theorem [16, 21]. See Barone-Basu [1] forthe above formulation.
Lemma 2.2 (Wongkew [23]) . Let Z ⊂ R d be a real algebraic variety of dimension e that is defined by polynomials of degree at most D . Then for each u > , we have | N u ( Z ∩ B (0 , | ≤ e (cid:88) j =0 C d,j ( Du ) d − e , where the numbers C d,j are constants depending only on d and j . orollary 2.1. Let S ⊂ B (0 , ⊂ R d be a semi-algebraic set of dimension e andcomplexity at most E . Then for each u > , we have | N u ( S ) | (cid:46) E,d u d − e . Since E u (cid:0) N w ( S ) (cid:1) (cid:46) u − d | N u + w ( S ) | , we obtain the following corollary. Lemma 2.3 (Covering number of neighborhoods of semi-algebraic sets) . Let S ⊂ B (0 , ⊂ R d be a semi-algebraic set of dimension e and complexity at most E . Thenfor each u > , we have E u ( N u ( S )) (cid:46) E,d u − e . If < u < w , then E u ( N w ( S )) (cid:46) E,d u − d w d − e = u − e ( w/u ) e − d . A similar result can be found in [24, Theorem 5.9]. The next lemma is a quan-titative version of the statement that every connected smooth semi-algebraic set ispath connected.
Lemma 2.4.
Let S ⊂ R d be a semi-algebraic set of complexity at most E anddiameter at most one. Suppose as well that S is a connected smooth manifold. Thenany two points in S can be connected by a smooth curve of length O d,E (1) . We will also need the following multi-dimensional Remez-type inequality. See,e.g. [3, Theorem 2].
Lemma 2.5.
Let P ∈ R [ x , . . . , x d ] be a polynomial of degree at most D . Let Ω ⊂ R d be an open convex set. Let m = sup x ∈ Ω | P ( x ) | . Then for each < λ < , |{ x ∈ Ω : | P ( x ) | ≤ λm }| ≤ d | Ω | λ /D . (7) Corollary 2.2.
Let P ∈ R [ x , . . . , x d ] be a homogeneous polynomial of degree D . Let m = sup | x | =1 | P ( x ) | . Then for each < λ < , µ σ (cid:16) { x ∈ S d − : | P ( x ) | ≤ λm } (cid:17) (cid:46) d λ /O D (1) , (8) where µ σ is the Haar probability measure on the sphere. Lemma 2.6 (Selecting a point from each fiber) . Let X ⊂ R d be a semi-algebraicset and let f : X → Y be a semi-algebraic map. Suppose that both X and f havecomplexity at most E . Then there is a semi-algebraic set X (cid:48) ⊂ X of complexity O E,d (1) so that f ( X (cid:48) ) = f ( X ) and f : X (cid:48) → Y is an injection. roof. Define a semi-algebraic ordering “ > ” on R d with the following properties.(A): If x, x (cid:48) ∈ R d then exactly one of the following holds: x > x (cid:48) , x = x (cid:48) , or x < x (cid:48) .(B): The set O = { ( x, x (cid:48) ) ∈ R d × R d : x < x (cid:48) } is semi-algebraic of complexity O d (1).An example of such an ordering is the lexicographic order on x = ( x , . . . , x d ).Observe that { ( x, x (cid:48) ) ∈ X × X : f ( x ) = f ( x (cid:48) ) } is semi-algebraic of complexity O E (1). Thus B = { ( x, x (cid:48) ) ∈ X × X : f ( x ) = f ( x (cid:48) ) , x < x (cid:48) } is semi-algebraic of complexity O E,d (1). Let π : X × X → X be the projection to thefirst coordinate. Then X (cid:48) = X \ π ( B ) = { x ∈ X : x ≥ x (cid:48) ∀ x (cid:48) ∈ X with f ( x ) = f ( x (cid:48) ) } is semi-algebraic of complexity O E,d (1). Note that f ( X (cid:48) ) = f ( X ) and that f : X (cid:48) → Y is an injection. Indeed, if x, x (cid:48) ∈ W with f ( x ) = f ( x (cid:48) ) then x ≥ x (cid:48) and x (cid:48) ≥ x ,which implies x = x (cid:48) . The map (cid:96) (cid:55)→
Dir( (cid:96) ) described in Definition 1.1 is badly behaved for lines lying in(or near) the hyperplane x = 0. To avoid this issue, we will only consider lines (cid:96) ∈ L that make a small angle with the e direction. Abusing notation slightly, wewill re-define L to be the set of lines in R that make an angle ≤ /
10 with the e direction. Definition . For Z ⊂ R and 0 < δ < c , defineΣ δ,c ( Z ) = { (cid:96) ∈ L : | (cid:96) ∩ N δ ( Z ) | ≥ c } . Define Σ δ ( Z ) = Σ δ, ( Z ).Let (cid:96) ∈ L . Define v ( (cid:96) ) to be the unit vector v ∈ R that points in the samedirection as (cid:96) and satisfies ∠ ( v, e ) ≤ / Definition . Let Σ ⊂ L be a set of lines. For each x ∈ R , defineΣ( x ) = { (cid:96) ∈ Σ : x ∈ (cid:96) } . efinition . Let Z ⊂ R and let Σ ⊂ L be a set of lines. DefineΓ( Z, Σ) = { ( x, (cid:96) ) ∈ Z × Σ : x ∈ (cid:96) } . If Γ ⊂ Γ( Z, Σ), then for each x ∈ Z defineΓ( x ) = { (cid:96) ∈ L : ( x, (cid:96) ) ∈ Γ } , and for each (cid:96) ∈ Σ, define Γ( (cid:96) ) = { x ∈ R : ( x, (cid:96) ) ∈ Γ } . P by a better-behaved polynomial In this section we will perform a convenient technical reduction. Informally speaking,this reduction lets us assume that the polynomial P from the statement of Theorem1.2 obeys the bounds |∇ P ( x ) | ∼ x ∈ Z ( P ) ∩ B (0 , , | P ( x ) | (cid:46) δ for all x ∈ N δ ( Z ( P )) ∩ B (0 , . (9)If (9) were true, it would yield a lot of useful information about the unit line segmentscontained in N δ ( Z ( P )). For example, if (cid:96) ∈ Σ δ ( Z ) and x ∈ Z ( P ) ∩ N δ ( (cid:96) ) ∩ B (0 , v ( (cid:96) ) is almost contained in the tangent plane T x ( Z ( P )). While (9) need notbe true, the following proposition will still allow us to recover some of the usefulconsequences of (9). Proposition 3.1.
Let P be a polynomial of degree at most D . Let Z = Z ( P ) ∩ B (0 , , let δ > , and let Σ ⊂ Σ δ ( Z ) be a semi-algebraic set of complexity at most E . Then there exist sets Σ , . . . , Σ b ; polynomials P , . . . , P b , and sets Γ , . . . , Γ b sothat the following holds.1. b (cid:46) D,E | log δ | .2. Each P j has degree at most D . Each set Σ j and Γ j is semi-algebraic of com-plexity O D,E (1) .3. Γ j ⊂ Γ( N δ ( Z j ) , Σ j ) , where Z j = { x ∈ Z ( P j ) ∩ B (0 ,
1) : 1 ≤ |∇ P j ( x ) | ≤ } . . Σ = (cid:83) bj =1 Σ j
5. For each (cid:96) ∈ Σ j , we have | Γ j ( (cid:96) ) | (cid:38) D | log δ | − .6. For each ( x, (cid:96) ) ∈ Γ j , each y ∈ Z j with dist( x, y ) ≤ δ , and each non-negativeinteger i , we have | ( v ( (cid:96) ) · ∇ ) i P j ( y ) | (cid:46) D,i | log δ | i δ. (10) Proof.
First, we can assume without loss of generality that the largest coefficient of P has magnitude 1. If Z ( P ) ∩ B (0 ,
1) = ∅ then Z = ∅ and thus Σ = ∅ so the resultis trivial. Thus we can also assume that at least one non-constant coefficient of P has magnitude (cid:38) D z ∈ B (0 , m ( z ) = sup x ∈ B ( z,δ ) |∇ P ( x ) | . Observe that |∇ P ( x ) | is a polynomial, all of whose coefficients have magnitude (cid:46) D (cid:38) D
1. Thus δ O D (1) (cid:46) D m ( z ) (cid:46) D z ∈ B (0 , . (11)For each z ∈ N δ ( Z ) ∩ B (0 , P ( B ( z, δ )) ⊂ (cid:2) − δm ( z ) , δm ( z ) (cid:3) . By Lemma 2.5, there is a constant c (cid:38) D z ∈ B (0 , (cid:12)(cid:12)(cid:8) x ∈ B ( z, δ ) } : |∇ P ( x ) | ≥ cm ( z ) (cid:9)(cid:12)(cid:12) ≥ | B ( x, δ ) | . (12)This implies that P ( B ( z, δ )) contains an interval of length (cid:38) D δm ( z ). Let m , . . . , m b ,b (cid:46) D | log δ | be a geometric sequence of non-negative numbers with m = δ O D (1) , m b = O D (1) and m j +1 = 2 m j . If we select m and b appropriately, then for each z ∈ N δ ( Z ) ∩ B (0 ,
1) there exists an index j so that m j ≤ m ( z ) < m j +1 . For each index j , let w j, , . . . , w j,h , h = O D (1) be real numbers in [ − δm ( z ) , δm ( z )] so that for every z ∈ N δ ( Z ) ∩ B (0 ,
1) satisfying m j ≤ m ( z ) < m j +1 , we have { w j, , . . . , w j,h } ∩ P ( B ( z, δ )) (cid:54) = ∅ , i.e. at least one of the values w j, , . . . , w j,h is contained in P ( B ( z, δ )). This canalways be achieved since P ( B ( z, δ )) must contain an interval of length (cid:38) D δm ( z ).12e can select h = O D (1) to be independent of j . Observe that there are O D ( | log δ | )pairs of indices j, h , which establishes Item 1 in the statement of the lemma (laterin the proof we will re-index the pairs ( j, h ) to use a single indexing variable).For each j = 1 , . . . , b and each k = 1 , . . . , h , define X j,k = { z ∈ N δ ( Z ) ∩ B (0 ,
1) : m j ≤ m ( z ) < m j +1 , and w j,k ∈ P (cid:0) B ( z, δ ) (cid:1) } . Then each set X j,k is semi-algebraic of complexity O D (1), and N δ ( Z ) ∩ B (0 ,
1) = b (cid:91) j =1 h (cid:91) k =1 X j,k . For each (cid:96) ∈ Σ, there exists indices j, k so that | (cid:96) ∩ X j,k | ≥ c / | log δ | , where c (cid:38) D
1. Since the set (cid:96) ∩ X j,k is semi-algebraic of complexity O D (1), it contains aninterval of length ≥ c / | log δ | , where c (cid:38) D
1. DefineΣ j,k = { (cid:96) ∈ Σ : (cid:96) ∩ X j,k contains an interval of length ≥ c / | log δ |} . With this definition, each set Σ j,k is semi-algebraic of complexity O D,E (1) and Σ = (cid:83) Σ j,k , so Item 4 in the statement of the lemma is satisfied.DefineΓ j,k = { ( x, (cid:96) ) : (cid:96) ∈ Σ j,k , x is contained in an interval in (cid:96) ∩ X j,k of length ≥ c / | log δ |} . Then | Γ j,k ( (cid:96) ) | (cid:38) D | log δ | − for each (cid:96) ∈ Σ j,k , so Item 5 is satisfied. Define P j,k ( z ) = m − j ( P ( z ) − h j,k );each polynomial P j,k ( z ) has degree ≤ D , so Item 2 is satisfied. If ( x, (cid:96) ) ∈ Γ j,k , then | P j,k ( y ) | ≤ δ for all y ∈ B ( x, δ ) . (13)This is because m ( x ) < m j , so |∇ P j,k ( y ) | ≤ y ∈ B ( x, δ ). (13) then followsfrom the fact that B ( x, δ ) ∩ N δ ( Z ( P j,k )) (cid:54) = ∅ .We have that if ( x, (cid:96) ) ∈ Γ j,k , then x ∈ N δ ( Z j,k ), where Z j,k = { z ∈ Z ( P j,k ) ∩ B (0 ,
1) : 1 ≤ |∇ P j,k ( z ) | ≤ } . Thus Item 3 is satisfied.It remains to verify Item 6. Fix indices j, k and let Γ = Γ j,k . Let ( x, (cid:96) ) ∈ Γ andlet y ∈ Z j,k with dist( x, y ) ≤ δ . Then there is a line segment I ⊂ Γ( (cid:96) ) of length13 yx L I Figure 1: The points x and y (black circles); the lines (cid:96) and L (thin black lines); theline segment I (thick black line) and the region N δ ( I ) (grey rectangle). Observe thatif dist( x, y ) ≤ δ , then | L ∩ N δ ( I ) | (cid:38) | I | (cid:38) D | log δ | − containing x . By (13) we have that | P j,k ( z ) | ≤ δ for all z ∈ N δ ( I ). Let L be a line containing y with | L ∩ N δ ( I ) | (cid:38) D | log δ | − (see Figure 1). Let L ( t ) : R → L be a unit speed parameterization of L , with L (0) = y (i.e. L ( t ) = y + tv ( L ) ). Then P j,k ( L ( t )) is a univariate polynomial of degree ≤ D that satisfies | P j,k ( L ( t )) | ≤ δ for all t in an interval J ⊂ [0 ,
1] of length (cid:38) D | log δ | − . This implies that (cid:12)(cid:12)(cid:12) d i dt i P j,k ( L ( t )) | t =0 (cid:12)(cid:12)(cid:12) (cid:46) D,i | log δ | − i δ, which gives us (10).The sets { Σ j,k } and { Γ j,k } , and the polynomials { P j,k } satisfy the conclusions ofLemma 3.1. All that remains is to re-index the indices j, k to use a single indexingvariable. In this section, we will consider the region where Z has small second fundamentalform. We will show that lines lying near this region must be contained in a thinneighborhood of a hyperplane; this will be the set of lines Σ from the statement ofTheorem 1.2. This result will be proved in Proposition 4.1, which is the main resultof this section. 14 .1 A primer on the second fundamental form Define the functions ϕ i : R → R , i = 0 , , , ϕ ( x , x , x , x ) = ( x , x , x , x ) ,ϕ ( x , x , x , x ) = ( − x , x , − x , x ) ,ϕ ( x , x , x , x ) = ( − x , x , x , − x ) ,ϕ ( x , x , x , x ) = ( − x , − x , x , x ) . Note that for each x ∈ R , ϕ ( x ) , ϕ ( x ) , ϕ ( x ) , and ϕ ( x ) have the same magnitudeand are orthogonal.Let P ∈ R [ x , . . . , x ] and let x ∈ Z ( P ). Suppose that ∇ P ( x ) (cid:54) = 0 and that Z ( P ) is a smooth manifold in a neighborhood of x . Define N ( x ) = ∇ P ( x ) |∇ P ( x ) | . For each i, j ∈ { , , } , define a ij ( x ) = (cid:16) ϕ i ( ∇ P ( x )) · ∇ y (cid:17) (cid:16) ϕ j ( ∇ P ( x )) · ∇ y (cid:17) P ( y ) (cid:12)(cid:12)(cid:12) y = x . To untangle the above definition: ∇ P ( x ) is a vector in R ; ϕ i ( ∇ P ( x )) is a vectorin R ; (cid:0) ϕ i ( ∇ P ( x )) · ∇ y (cid:1) is an operator acting on functions f : R → R . Similarly, (cid:0) ϕ j ( ∇ P ( x )) · ∇ y (cid:1) is an operator acting on functions f : R → R . We apply theseoperators to the function P ( y ), and then evaluate the resulting function at the point y = x .Note that for each i, j ∈ { , , } , a ij ∈ R [ x , . . . , x ] is a polynomial of degree O (deg P ). Define II ( x ) = 1 |∇ P ( x ) | a ( x ) a ( x ) a ( x ) a ( x ) a ( x ) a ( x ) a ( x ) a ( x ) a ( x ) . Then II ( x ) is the second fundamental form of Z ( P ) at x , written in the basis ϕ ( x ) | ϕ ( x ) | , ϕ ( x ) | ϕ ( x ) | , ϕ ( x ) | ϕ ( x ) | (this is a basis for the tangent space T x ( Z ( P )) ). Note thatif the polynomial P ( x ) is replaced by tP ( x ) for t (cid:54) = 0, then the matrix II ( x ) isunchanged. For each x ∈ R , define (cid:107) II ( x ) (cid:107) ∞ to be the (cid:96) ∞ norm of the entries of II ( x ) (so (cid:107) II ( x ) (cid:107) ∞ is a function from R to R ). Observe that if P ∈ R [ x , . . . , x ] isa polynomial of degree at most D , then for each κ >
0, the set { x ∈ Z ( P ) : (cid:107) II ( x ) (cid:107) ∞ > κ } is semi-algebraic of complexity O D (1). 15ote that if 0 ∈ Z ( P ) and if N (0) = (0 , , , Z ( P ) as the graph x = f ( x , x , x ), with f (0 , ,
0) = 0 and ∇ f (0) = (0 , , II (0) = ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) ∂ x x f (0) . Lemma 4.1.
Let P ∈ R [ x , . . . , x ] and let M ⊂ Z ( P ) be a smooth manifold. Sup-pose that ∇ P ( x ) (cid:54) = 0 for all x ∈ M , (cid:107) II ( x ) (cid:107) ∞ ≤ κ for all x ∈ M , and that everypair of points in M can be connected by a smooth curve of arclength ≤ t . Then theimage of the Gauss map N ( M ) can be contained in a ball of diameter tκ .Proof. Fix a point x ∈ M . For each x ∈ M , let γ ( s ) be a unit-speed paramateriza-tion of a smooth curve with γ (0) = x and γ ( s ) = x . By hypothesis, we can selectsuch a curve with s ≤ t . Then | dds N ( γ ( s )) | ≤ (cid:107) II ( γ ( s )) (cid:107) ∞ ≤ κ ; here N ( γ ( s )) is amap from R to the unit sphere S ⊂ R , and | · | denotes the Euclidean norm of thefour-dimensional vector dds N ( γ ( s )) . We conclude that N ( x ) is contained in the ball(in S ) centered at x of radius tκ . Lemma 4.2 (Hypersurfaces with small second fundamental form lie near a hyper-plane) . Let S ⊂ R be a semi-algebraic set contained in B (0 , of complexity atmost E . Suppose that S is a connected smooth three-dimensional manifold and that (cid:107) II( x ) (cid:107) ∞ ≤ κ for all x ∈ S . Then S can be contained in the O E ( κ ) –neighborhood ofa hyperplane.Proof. After applying a rigid transformation, we can assume that 0 ∈ S and N (0) =(0 , , , x ∈ S . By Lemma 2.4, we can find a smooth curve γ ⊂ S of length O E (1) whose endpoints are 0 and x . Let γ ( s ) be a unit-speed parameterization ofthis curve, so γ (0) = 0 and γ ( s ) = x , for some s = O E (1).Since dds γ ( s ) ∈ T γ ( s ) S , we must have | dds γ ( s ) · (0 , , , | (cid:46) s κ . In particular, the x coordinate of γ ( s ) must have magnitude (cid:46) s sκ = O E ( κ ). Thus after applyinga rigid transformation, S is contained in the O E ( κ )-neighborhood of the hyperplane { x = 0 } . Lemma 4.3 (Hypersurfaces with large second fundamental form escape every hy-perplane) . Let P ∈ R [ x , . . . , x ] be a polynomial of degree at most D and let Z ⊂ Z ( P ) ∩ B (0 , be a semi-algebraic set of complexity at most E . Suppose that ≤|∇ P ( x ) | ≤ and (cid:107) II ( x ) (cid:107) ∞ ≥ κ at every point x ∈ Z . Then for every hyperplane H and every < a ≤ b ≤ , we have | N a ( Z ) ∩ N b ( H ) | (cid:46) D,E a ( b/κ ) / . (14)16 roof. Let ρ (cid:38) D Z into O ρ,D,E (1) = O D,E (1) connected semi-algebraic sets so that on each set Z (cid:48) , each of ∂ x i P ( x ) , i ∈ { , , , } vary by at most an additive factor of ρ . It suffices to establish(14) for each of these sets individually. Fix one of these sets Z (cid:48) . After applying arotation, we can assume that for all x ∈ Z (cid:48) we have 1 − ρ ≤ | ∂ x P ( x ) | ≤ ρ and | ∂ x i P ( x ) | ≤ ρ, i = 1 , ,
3. In particular, each of the vectors ϕ i ( N ( x )) , i = 0 , , , ρ for all x ∈ Z (cid:48) .Since (cid:107) II ( x ) (cid:107) ∞ ≥ κ for all x ∈ Z (cid:48) , and 1 ≤ |∇ P ( x ) | ≤
2, there is a unit vector v ∈ T x ( Z (cid:48) ) with | ( v · ∇ ) P ( x ) | ≥ κ . Phrased differently, there is a unit vector( v , v , v ) ∈ R so that (cid:12)(cid:12)(cid:0) ( v ϕ ( N ( x )) + v ϕ ( N ( x )) + v ϕ ( N ( x ))) · ∇ (cid:1) P ( x ) (cid:12)(cid:12) ≥ κ. Since the map( v (cid:48) , v (cid:48) , v (cid:48) ) (cid:55)→ (cid:12)(cid:12)(cid:0) ( v (cid:48) ϕ ( N ( x )) + v (cid:48) ϕ ( N ( x )) + v (cid:48) ϕ ( N ( x ))) · ∇ (cid:1) P ( x ) (cid:12)(cid:12) is homogeneous of degree four, there is a constant c > v (cid:48) = ( v (cid:48) , v (cid:48) , v (cid:48) )with ∠ ( v, v (cid:48) ) ≤ c , we have (cid:12)(cid:12)(cid:0) ( v (cid:48) ϕ ( N ( x )) + v (cid:48) ϕ ( N ( x )) + v (cid:48) ϕ ( N ( x ))) · ∇ (cid:1) P ( x ) (cid:12)(cid:12) ≥ κ. Thus after further partitioning Z (cid:48) into O D (1) connected semi-algebraic sets, wehave that for each such set Z (cid:48)(cid:48) , there is a unit vector v so that (cid:12)(cid:12)(cid:0) ( v (cid:48) ϕ ( N ( x )) + v (cid:48) ϕ ( N ( x )) + v (cid:48) ϕ ( N ( x ))) · ∇ (cid:1) P ( x ) (cid:12)(cid:12) ≥ κ for all x ∈ Z (cid:48)(cid:48) and all vectors v (cid:48) = ( v (cid:48) , v (cid:48) , v (cid:48) ) ∈ S ⊂ R with ∠ ( v, v (cid:48) ) ≤ c . It sufficesto establish (14) for each of these sets Z (cid:48)(cid:48) individually. Fix such a set.Fix a point x ∈ Z (cid:48)(cid:48) and apply a rigid transformation so that N ( x ) = (0 , , , v = (1 , , ϕ ( N ( x )) = (0 , , − , ρ (cid:38) D ∠ (cid:0) ϕ ( N ( x )) , ϕ ( N ( x )) (cid:1) ≤ c/ x ∈ Z (cid:48)(cid:48) . This means thatfor all x ∈ Z (cid:48)(cid:48) , if v (cid:48) ∈ R is a unit vector in T x ( Z (cid:48)(cid:48) ) with ∠ ( v (cid:48) , (0 , , − , ≤ c/ | ( v (cid:48) · ∇ ) P ( x ) | ≥ κ .This means that we can write Z (cid:48)(cid:48) as the graph of a function f ( x , x , x ); moreprecisely, for each ( x , x , x , x ) ∈ Z (cid:48)(cid:48) , we can write x = f ( x , x , x ), and | ∂ x f ( x , x , x ) | (cid:38) D κ. (15)17et π ( x , x , x , x ) = ( x , x , x ), so Z (cid:48)(cid:48) is the graph of f above π ( Z (cid:48)(cid:48) ). By (15),we have that if I ⊂ π ( Z (cid:48)(cid:48) ) is a line segment pointing in the direction (0 , ,
1) and if | f ( x , x , x ) | ≤ b for all ( x , x , x ) ∈ I , then we must have | I | (cid:46) D ( b/κ ) / . Next,let L be a line in R pointing in the direction (0 , , (cid:12)(cid:12)(cid:12) { ( x , x , x ) ∈ L : | f ( x , x , x ) | ≤ b } (cid:12)(cid:12)(cid:12) (cid:46) D ( b/κ ) / , where | · | denotes one dimensional Lebesgue measure. Thus by Fubini’s theorem, (cid:12)(cid:12)(cid:12) { ( x , x , x ) ∈ π ( Z (cid:48)(cid:48) ) : | f ( x , x , x ) | ≤ b } (cid:12)(cid:12)(cid:12) (cid:46) D ( b/κ ) / , where | · | denotes three dimensional Lebesgue measure. Thus |{ ( x , x , x , x ) ∈ Z (cid:48)(cid:48) : | x | ≤ b } (cid:12)(cid:12)(cid:12) (cid:46) D ( b/κ ) / , where again | · | denotes three dimensional Lebesgue measure. We conclude that | N a ( Z (cid:48)(cid:48) ) ∩ N b ( H ) | (cid:46) D,E a (cid:12)(cid:12) Z (cid:48)(cid:48) ∩ N a + b ( H ) (cid:12)(cid:12) (cid:46) D,E a ( b/κ ) / . (16)Since (16) holds for each of the O D,E (1) connected sets Z (cid:48)(cid:48) , we obtain (14). Proposition 4.1.
Let < δ < κ and let c > . Let P ∈ R [ x , . . . , x ] be a polynomialof degree at most D . Define Z = { x ∈ Z ( P ) ∩ B (0 ,
1) : ∇ P ( x ) (cid:54) = 0 , (cid:107) II ( x ) (cid:107) ∞ ≤ κ } . Then there is a set of O D ( c − O (1) ) rectangular prisms of dimensions × × × κ sothat every line in Σ δ,c ( Z ) is contained on one of these prisms.Proof. Let Z , . . . , Z p , p = O D (1) be the connected components of Z (without loss ofgenerality, we can assume that each of these components is a smooth 3-dimensionalmanifold.) Each line (cid:96) ∈ Σ δ,c ( Z ) satisfies | (cid:96) ∩ N δ ( Z j ) | ≥ c/p for some index j . ByLemma 4.2, each connected component Z j can be contained in the κ (cid:48) = O D ( κ )–neighborhood of a hyperplane; call this hyperplane H j .Finally, for each index j , we can select O D ( c − O (1) ) rectangular prisms of dimen-sions 2 × × × κ so that every line (cid:96) satisfying | (cid:96) ∩ N κ (cid:48) ( H j ) | ≥ c/p must must becovered by one of these prisms. 18 Multilinearity and quantitative broadness
In this section we will explore the notion of “broadness,” which was introduced byGuth in [10] to study the restriction problem. Throughout this section, we will oftenrefer to the following “standard setup.”
Definition . Let d be a positive integer. Let Z ⊂ R d and letΦ ⊂ Z × S d − be semi-algebraic sets of complexity at most E . Let π Z : Φ → Z and π S : Φ → S d − be the projection of Φ to Z and S d − , respectively. For each z ∈ Z ,define Φ( z ) = π − Z ( z ) . ( m, A ) -broadness Definition . Let d, Φ , and Z be defined as in the standard setup from Definition5.1. For each positive integer 0 ≤ m ≤ d −
1, let S m be the set of m dimensionalunit spheres contained in S d − (recall that a zero dimensional unit sphere in S d − isjust a pair of antipodal points). We will identify S m with a semi-algebraic subset of R N for some N = O d (1). Let A ≥ u ≥
0. Define( m, A ) -Narrow u (Φ) = (cid:8) z ∈ Z : ∃ S , . . . , S A ∈ S m s . t . ∀ v ∈ π S (Φ( z )) , ∃ i ∈ { , . . . , A } s.t. ∠ ( v, S i } ≤ u (cid:9) . In words, z ∈ ( m, A ) -Narrow u (Φ) if and only if there is a set of A m -dimensionalunit spheres S , . . . , S A so that every vector v ∈ π S (Φ( z )) makes an angle at most u with one of these spheres. Observe that if m ≤ m (cid:48) , A ≤ A (cid:48) , and u ≤ u (cid:48) , then( m, A ) -Narrow u (Φ) ⊂ ( m (cid:48) , A (cid:48) ) -Narrow u (cid:48) (Φ).Define ( m, A ) -Broad u (Φ) = Z \ ( m, A ) -Narrow u (Φ) . If ( m, A ) -Narrow u (Φ) = Z , we say that Φ is ( m, A )-narrow at width u . If ( m, A ) -Broad u (Φ) = Z , we say that Φ is ( m, A )-broad at width u .The sets ( m, A ) -Broad u (Φ) and ( m, A ) -Narrow u (Φ) have complexity O d,E (1). Inpractice we will have d = 4 so the complexity is O E (1). Lemma 5.1.
Let d, Φ , and Z be defined as in the standard setup from Definition5.1. Let u ≤ s . Let m ≥ . Suppose that Φ is ( m, -narrow at width u and ( m − , -broad at width s . Define A = { ( z, S ) ∈ Z × S m : ∠ ( v, S ) ≤ u ∀ v ∈ π S (Φ( z )) } . f we identify Z × S m with a subset of R d × R N , N = O d (1) , then E u ( A ) (cid:46) d,E u − dim( Z ) s − codim( Z ) , (17) where codim( Z ) = d + N − dim( Z ) = O d (1) .Proof. Since the constants d and E are fixed, all implicit constants may depend onthese quantities. First, since Φ is ( m, u , i.e. ( m,
1) -Narrow u (Φ) = Z , we have that π : A → Z is onto. Apply Lemma 2.6 to select a semi-algebraic set A (cid:48) ⊂ A of complexity O (1) so that π : A (cid:48) → Z is a bijection. In particular, we havedim( A (cid:48) ) = dim( Z ) . We claim that if ( z, S ) , ( z, S (cid:48) ) ∈ A , then dist( S, S (cid:48) ) (cid:46) u/s, where dist( S, S (cid:48) )denotes the Euclidean distance in R N between the points in R N identified with S and S (cid:48) . Indeed, note that π S (Φ( z )) ⊂ N u ( S ) ∩ N u ( S (cid:48) ) . Since z ∈ ( m − ,
1) -Broad s (Φ) we have that π S (Φ( z )) ⊂ N u ( S ) ∩ N u ( S (cid:48) ) cannotbe contained in the s –neighborhood of a ( m − N u ( S ) ∩ N u ( S (cid:48) ) cannot be contained in the s –neighborhood of a ( m − S, S (cid:48) ) (cid:46) u/s .By Lemma 2.3, E u ( A ) (cid:46) E u (cid:0) N u/s ( A (cid:48) ) (cid:1) (cid:46) u dim( A (cid:48) ) s − codim( A (cid:48) ) = u dim( Z ) s − codim( Z ) . In practice, we will use Lemma 5.1 in the special case Z ⊂ R , Φ ⊂ Z × S ⊂ R ,and m = 2. The lemma will help us analyze the situation where a hypersurface Z ( P ) ⊂ R is ruled by planes. If V ⊂ S d − is semi-algebraic of complexity at most E , then by Lemma 2.1 there isa number K d,E so that V is a union of at most K d,E connected components. Thismeans that for each s >
0, either V can be contained in the s –neighborhood of aunion of K d,E vectors, or V contains a connected component of diameter at least s . Definition . Let d, Φ , and Z be defined as in the standard setup from Definition5.1. Let E = O d,E be an integer so that π S (Φ( z )) has complexity at most E foreach z ∈ Z . 20efine 1 -SBroad s (Φ) = (1 , K d,E ) -Broad s (Φ) . The “S” stands for “Strong.” If 1 -SBroad s (Φ) = Z , we say that Φ is strongly 1-broadat width s .Note that if z ∈ s (Φ), then π S (Φ( z )) contains a connected component ofdiameter ≥ s . Conversely, if π S (Φ( z )) contains a connected component of diameterat least K d,E s , then z ∈ s (Φ). It would be preferable to just directly define1 -SBroad s (Φ) to be the set of points z ∈ Z so that π S (Φ( z )) contains a connectedcomponent of diameter at least s , but it is not clear whether 1 -SBroad s (Φ) wouldbe semi-algebraic with this definition. Under Definition 5.3, 1 -SBroad s (Φ) ⊂ Z issemi-algebraic of complexity O d,E (1). Remark . It would be straightforward to define a notion of strong m -broadnessfor each m ≥
1, but this definition is not particularly useful if m >
1. One of thekey properties of strong 1-broadness is that if z ∈ s (Φ), then E s (cid:0) π S (Φ( z )) (cid:1) (cid:38) d,E s − . Unfortunately, the analogous statement for strong m -broadness (with s − replacedby s − m ) need not be true. Lemma 5.2.
Let d, Φ , and Z be defined as in the standard setup from Definition 5.1and let s > . Suppose that s (Φ) = ∅ . Then E s (Φ) (cid:46) d,E s − dim( Z ) . Proof.
Since d and E are fixed, all implicit constants may depend on these quantities.For each k = 1 , . . . , O (1), define A k = { ( z, v , . . . , v k ) : z ∈ Z, v , . . . , v k ∈ π S (Φ( z )) , ∠ ( v i , v j ) ≥ s if i (cid:54) = j } . Let π k : A k → Z be the projection to Z , and let Z k = π k ( A k ). Since 1 -SBroad s (Φ) = ∅ , we have that O (1) (cid:91) k =1 Z k = π Z (Φ) . (18)Apply Lemma 2.6 to each map π k : A k → Z k to obtain sets A (cid:48) k ⊂ A k so that theprojection map π k : A (cid:48) k → Z k is a bijection. Define Z (cid:48) k = Z k \ (cid:83) j>k Z j . Note thateach set Z (cid:48) k has dimension ≤ dim( Z ) and is semi-algebraic of complexity O (1).21efine Φ k = { ( z, v ) ∈ Φ : z ∈ Z (cid:48) k } . Then by (18), we have Φ = O (1) (cid:71) k =1 Φ k . (19)Note that if z ∈ Z (cid:48) k and if ( z, v , . . . , v k ) = π − k ( z ) (here π k : A (cid:48) k → Z (cid:48) k is abijection, so it has a well-defined inverse), then every vector in π S (Φ( z )) must beclose to one of the vectors v , . . . , v k ; more precisely, π S (Φ( z )) ⊂ N s ( { v , . . . , v k } ) . (20)Indeed, if (20) did not hold, then there exists v k +1 ∈ π S (Φ( z )) \ N s ( { v , . . . , v k } ) . But then ( z, v , . . . , v k , v k +1 ) ∈ A k +1 , so z ∈ Z k +1 , which contradicts the assumptionthat z ∈ Z (cid:48) k .For each index k and each j = 1 , . . . , k , define the projections π k,j : A (cid:48) k → Z (cid:48) k × S d − by ( z, v , . . . , v k ) (cid:55)→ ( z, v j ). Define W k = (cid:83) kj =1 π k,j ( A (cid:48) k ). Thus W k ⊂ Z (cid:48) k × S d − ,and for each z ∈ Z (cid:48) k , we have that W k ∩ ( { z } × S d − ) is the set { ( z, v ) , . . . , ( z, v k ) } ,where ( z, v , . . . , v k ) = π − k ( z ).Observe that W k is a semi-algebraic set of dimension ≤ dim( Z ) and complexity O (1), and thus by Lemma 2.3, E s ( W k ) (cid:46) s − dim( Z ) . (21)On the other hand, by (20) we have that E s (Φ k ) (cid:46) E s ( W k ) . (22)The lemma now follows from (19), (20), (21), and (22). In this section we will relate the objects Z, Σ and Γ from Section 2.1 with the standardsetup from Definition 5.1.
Definition . Let δ > P ∈ R [ x , . . . , x ] be a polynomial of degree atmost D . Let Z ⊂ Z ( P ) ∩ B (0 ,
1) and Γ ⊂ Γ( N δ ( Z ) , L ) be semi-algebraic sets ofcomplexity at most E . For each x ∈ N δ ( Z ), define f Z ( x ) to be the point z ∈ Z that minimizes dist( x, z ). If more than one such point exists, select the one thatis minimal under the lexicographic order (any semi-algebraic total order would be22qually good). Then the set { ( x, z ) ∈ N δ ( Z ) × Z : z = f Z ( x ) } is semi-algebraic ofcomplexity O E,D (1). Define the set Φ ⊂ Z × S to be the set of ordered pairsΦ = { ( f Z ( x ) , v ( (cid:96) )) : ( x, (cid:96) ) ∈ Γ } . Then Φ ⊂ Z × S is semi-algebraic of complexity O D,E (1). We will say that set Φ isassociated to Γ (the set Z and the parameter δ will be obvious from context).By construction, the map Γ → Φ , ( x, (cid:96) ) (cid:55)→ ( f Z ( x ) , v ( (cid:96) )) is onto. If Φ (cid:48) ⊂ Φ, wewill define Γ (cid:48) to be the pre-image of Φ (cid:48) under this map.
In this section, we will consider the region where Z either fails to be robustly 1-broad,or fails to be (2 , and Σ from the statement of Theorem 1.2. Themain results of this section are Proposition 6.1, which describes what happens when Z fails to be robustly 1-broad, and Proposition 6.2, which describes what happenswhen Z has large second fundamental form and fails to be (2 , -Narrow varieties are ruled by lines Proposition 6.1.
Let < δ < s and c > δ . Let P ∈ R [ x , . . . , x ] be a polynomial ofdegree at most D ; let Z = Z ( P ) ∩ B (0 , ; let Σ ⊂ Σ δ,c ( Z ) be a semi-algebraic set ofcomplexity at most E . Let Γ ⊂ Γ( N δ ( Z ) , Σ) be a semi-algebraic set of complexity atmost E . Let Φ ⊂ Z × S be associated to Γ , in the sense of Definition 5.4. Supposethat s (Φ) = ∅ , (23) | Γ( (cid:96) ) | ≥ c for every (cid:96) ∈ Σ . (24) Then there is a set of O D,E ( c − s − ) rectangular prisms of dimensions × s × s × s ,so that every line from Σ is covered by one of the prisms.Proof. By Lemma 5.2, E s (Φ) (cid:46) D,E s − . (25)23et R max be a maximal set of essentially disjoint rectangular prisms of dimensions2 × s/ × s/ × s/ B (0 , R ∈ R max , define v ( R ) to be thedirection of the long axis of R , and define R ∗ = { ( x, v ) ∈ R × S : x ∈ R, ∠ ( v, v ( R )) ≤ s/ } . Observe that every line intersecting B (0 ,
1) is covered by some rectangular prismfrom R max , and that for each C ≥
1, the C –fold dilates of the sets { CR ∗ } R ∈R max are O C (1)–overlapping.Let ( x, (cid:96) ) ∈ Γ and let ( f Z ( x ) , v ( (cid:96) )) be the corresponding element of Φ. Note thatif (cid:96) × { v ( (cid:96) ) } ∩ R ∗ (cid:54) = ∅ then (cid:96) ∩ R (cid:54) = ∅ and ∠ ( v ( (cid:96) ) , v ( R )) ≤ s/
4, and thus (cid:96) is coveredby the 4-fold dilate of R . Furthermore, if this happens then Φ ∩ R ∗ (cid:54) = ∅ . Thus toprove the lemma, it suffices to show that |{ R ∈ R max : R ∗ ∩ Φ (cid:54) = ∅}| (cid:46) D,E c − s − . (26)Let R ∈ R max and suppose R ∗ ∩ Φ (cid:54) = ∅ . We will show that E s (Φ ∩ R ∗ ) ≥ c/s. (27)To see this, let ( z, v ) ∈ R ∗ ∩ Φ. Then there exists a point ( x, (cid:96) ) ∈ Γ with dist( x, z ) < δ and v ( (cid:96) ) = v . Thus ∠ ( v ( (cid:96) ) , v ( R )) ≤ s/ . Thus (cid:96) ∩ B (0 , ⊂ (cid:96) ∩ S . Since (cid:96) ∈ Σ , we have E s (Γ( (cid:96) )) ≥ c/s and thus there exist ≥ c/s s -separated points x (cid:48) ∈ Γ( (cid:96) ) with ( x (cid:48) , v ( (cid:96) )) ∈ S ∗ . Of course for each of thesepoints x (cid:48) ∈ Γ( (cid:96) ) there exists a point z (cid:48) ∈ B ( x (cid:48) , δ ) with ( z (cid:48) , v ( (cid:96) )) ∈ Φ ∩ R ∗ , whichestablishes (27). Since the sets { R ∗ } R max are O (1) overlapping, by combining (25)and (27) we have (26). -narrow varieties are ruled by planes Lemma 6.1.
Let δ, u, s, κ, c, D, E be parameters with < δ < u < s < c and δ < κ .Then there exists a number w (cid:38) D,E ( usκc ) O (1) so that the following holds.Let P ∈ R [ x , . . . , x ] be a polynomial of degree at most D and let Z ⊂ { z ∈ Z ( P ) ∩ B (0 ,
1) : 1 ≤ |∇ P ( z ) | ≤ , (cid:107) II ( z ) (cid:107) ∞ ≥ κ } . (28) Let Σ ⊂ Σ δ,c ( Z ) , and let Γ ⊂ Γ( N δ ( Z ) , Σ) . Suppose that Z, Σ , and Γ are semi-algebraic sets of complexity at most E . Let Φ be associated to Γ , in the sense of efinition 5.4. Suppose that Z ⊂ (2 ,
1) -Narrow w (Φ) , (29) | Γ( (cid:96) ) | ≥ c for all (cid:96) ∈ Σ . (30) Then we can write
Σ = Σ (cid:48) ∪ Σ (cid:48)(cid:48) , where the lines in Σ (cid:48) can be covered by O D,E ( c − O (1) s − ) rectangular prisms of dimensions × s × s × s , and the lines in Σ (cid:48)(cid:48) can be coveredby O D,E (( cs ) − O (1) u − ) rectangular prisms of dimensions × × u × u .Proof. Since D and E are fixed, whenever we write A (cid:46) B , the implicit constantmay depend on these quantities. LetΦ = { ( z, v ) ∈ Φ : z ∈ s (Φ) } , and let Φ (cid:48) = Φ \ Φ . Define Γ to be the pre-image of Φ under the map Γ → Φ, anddefine Γ (cid:48) = Γ \ Γ . Note that Γ (cid:48) is the pre-image of Φ (cid:48) . Γ (cid:48) is a set of complexity E = O D,E (1); in particular, the constant E can be chosen to be independent of s and δ .Let Σ = { (cid:96) ∈ Σ : | Γ ( x ) | > c/ } , Σ (cid:48) = { (cid:96) ∈ Σ : | Γ (cid:48) ( x ) | > c/ } . We will put the lines in Σ (cid:48) into Σ (cid:48) ; by Proposition 6.1, these lines can be coveredby O ( s − c − O (1) ) rectangular prisms of dimensions 2 × s × s × s .Define Z = { z ∈ Z : Φ ( z ) (cid:54) = ∅} ⊂ s (Φ) . Let w > ( usκc ) O (1) be a number that will be determined below. Define A = { ( z, S ) ∈ Z × S : ∠ ( v, S ) ≤ w ∀ v ∈ π S (Φ( z )) } . (31)Since Z ⊂ s (Φ), we have that if ( z, S ) , ( z, S (cid:48) ) ∈ A , then ∠ ( S, S (cid:48) ) ≤ w/s. (32)This is because there exists two vectors v, v (cid:48) ∈ π S (Φ( z )) with ∠ ( v, v (cid:48) ) ≥ s , and thesevectors satisfy ∠ ( v, S ) ≤ w ; ∠ ( v, S (cid:48) ) ≤ w ; ∠ ( v (cid:48) , S ) ≤ w ; and ∠ ( v (cid:48) , S (cid:48) ) ≤ w . Observethat A obeys the hypotheses of Lemma 5.1.Define π : A → Z to be the projection ( z, S ) (cid:55)→ z . Then by (29) π ( A ) = Z .Use Lemma 2.6 to select a set A (cid:48) ⊂ A so that π : A (cid:48) → Z is a bijection. By (32),we have that A ⊂ N w/s ( A (cid:48) ) . (33)25or each z ∈ Z , define S ( z ) to be the (unique) great circle S so that ( z, S ) ∈ A (cid:48) .The function S ( z ) is semi-algebraic of complexity O (1). If ( x, (cid:96) ) ∈ Γ , then f Z ( x ) ∈ Z . Thus for each (cid:96) ∈ Σ, the set S ◦ f Z ◦ Γ ( (cid:96) ) = { S ( f Z ( x )) : x ∈ Γ ( (cid:96) ) } ⊂ S is a union of O (1) connected components in S (recall that S is the parameter spaceof one-dimensional great circles in S ).For each S ∈ S , define span( S ) to be the two-dimensional vector space in R that contains the great circle S , i.e. span( S ) = { rv : r ∈ R , v ∈ S } . For each z ∈ Z ,Define Π( z ) = z + Span( S ( z )); this is an affine 2-plane containing z .Since S ◦ f Z ◦ Γ ( (cid:96) ) is a union of O (1) connected components, heuristically, thismeans that either (A): { Π ◦ f Z ( z ) : z ∈ Γ ( (cid:96) ) } can be covered by the thickenedneighborhoods of a small number of planes containing (cid:96) , or (B): the union of theplanes in { Π ◦ f Z ( z ) : z ∈ Γ ( (cid:96) ) } fill out a large fraction of N δ ( Z ). We will make thisheuristic precise in the arguments below.let h (cid:38) c be a constant that will be determined below. Define Y = (cid:8) ( (cid:96), S, x ) ∈ Σ × S × R : x ∈ Γ ( (cid:96) ) , ∀ x (cid:48) ∈ (cid:96) ∩ B ( x, h ) , we have x (cid:48) ∈ Γ ( (cid:96) ) and ∠ ( S ◦ f Z ( x (cid:48) ) , S ) < u (cid:9) . Let π L ( (cid:96), S, x ) = (cid:96) and define Σ = π L ( Y ), Σ = Σ \ Σ . In words, Σ is the setof lines (cid:96) ∈ Σ so that there exists a line segment I ⊂ Γ ( (cid:96) ) of length 2 h with theproperty that the great circle S ◦ f Z ( x (cid:48) ) does not change much as x (cid:48) moves along I . Remark . Heuristically, if the variety Z ( P ) was ruled by planes, and if every linewas contained in one of these planes, then Σ = Σ . We will show that if this is thecase, then Σ can be partitioned into disjoint pieces that do not interact with eachother (geometrically, if Z ( P ) is ruled by planes and if every line lies in one of theseplanes, then we can write Z ( P ) as a disjoint union of planes and consider each ofthese planes individually).On the other hand, Σ is the set of lines (cid:96) ∈ Σ so that for every interval I ⊂ Γ ( (cid:96) )of length 2 h , S ◦ f Z ( I ) has diameter ≥ u . Since S ◦ f Z ◦ Γ ( (cid:96) ) is a union of O (1)connected components, if h (cid:38) c is selected sufficiently small, then there must existan interval I ⊂ Γ ( (cid:96) ) of length 2 h so that S ◦ f Z ( I ) is connected. Remark . Heuristically, if the variety Z ( P ) is a small perturbation of a hyperplane,then it could be the case that Σ = Σ . We will show that if this is the case, thenmost of Z ( P ) can be contained in a thin neighborhood of a hyperplane, and this willcontradict the assumption that (cid:107) II ( z ) (cid:107) ∞ is large on Z .26 nderstanding lines in Σ Apply Lemma 2.6 to the surjection π L : Y → Σ to obtain a set Y (cid:48) ⊂ Y so that π L : Y (cid:48) → Σ is a bijection. Define S ( (cid:96) ) to be the circle in S containing the vector v ( (cid:96) ) that makes the smallest angle with the circle π S ◦ π − L ( (cid:96) ). Observe that if (cid:96) ∈ Σ and if ( (cid:96), S, x ) = π − L ( (cid:96) ), then ( x, v ( (cid:96) )) ∈ Γ and thus ∠ ( v ( (cid:96) ) , S ) < w ≤ u , so ∠ ( S, S ( (cid:96) )) < u .For each (cid:96) ∈ Σ , define Π( (cid:96) ) = (cid:96) + span( S ( (cid:96) )); this is an affine plane containing (cid:96) that points in the directions spanned by S ( (cid:96) ).Define Γ = { ( x, (cid:96) ) ∈ Γ : ∠ ( S ( (cid:96) ) , S ( x )) < u } . Then for each (cid:96) ∈ Σ , we have | Γ ( (cid:96) ) | ≥ h . This is because (cid:96) ∈ Σ implies that thereexists a point ( (cid:96), S, x ) = π − L ( x ) ∈ Y, and thus there exists an interval I ⊂ Γ ( (cid:96) ) oflength 2 h containing x so that for every point x (cid:48) ∈ I , we have ∠ ( S ( (cid:96) ) , S ( x (cid:48) )) ≤ ∠ ( S ( (cid:96) ) , S ) + ∠ ( S ( x (cid:48) ) , S ) < u + u = 2 u. Let Φ be the set associated to Γ , in the sense of Definition 5.4. DefineΦ = { ( z, v ) ∈ Φ : z ∈ s (Φ ) } . Define Γ ⊂ Γ to be the pre-image of Φ and define Γ (cid:48) = Γ \ Γ (this is the pre-imageof Φ (cid:48) = Φ \ Φ ). Define Σ (cid:48) = { (cid:96) ∈ Σ : | Γ ( (cid:96) ) | ≥ h } , Σ (cid:48)(cid:48) = { (cid:96) ∈ Σ : | Γ (cid:48) ( (cid:96) ) | ≥ h } . Γ (cid:48) and Σ (cid:48)(cid:48) are semi-algebraic sets of complexity O (1); the lines in Σ (cid:48)(cid:48) will be addedto Σ (cid:48) ; by Proposition 6.1, these lines can be covered by O ( s − h − O (1) ) = O ( s − c − O (1) )rectangular prisms of dimensions 2 × s × s × s .Define Σ (cid:48)(cid:48) = Σ (cid:48) (recall that Σ (cid:48) and Σ (cid:48)(cid:48) are the output of the lemma). We will showthat these lines can be covered by O ( u − ( sc ) − O (1) ) rectangular prisms of dimensions2 × × u × u .Define Z = { z ∈ Z : Φ ( z ) (cid:54) = ∅} . Define A = { ( z, S ) ∈ A : z ∈ Z } . z, S ) so that z ∈ Z and all of the vectors from π S (Φ( z ))(and thus all of the vectors from π S (Φ ( z ))) are contained in the 2 u –neighborhoodof S . In practice, it will be more convenient to work with the set˜ A = { ( z, z + span( S ) : ( z, S ) ∈ A } ;this is the set of pairs ( z, Π), where Π is an affine 2-plane containing z that is spannedby the vectors in S ( z ).Let R max be a maximal set of essentially disjoint rectangular prisms of dimensions2 × × u × u that intersect B (0 , R ∈ R max , define Π( R ) to be the affineplane concentric with the long axes of R , and define R ∗ = { ( x, Π) ∈ R × Grass(4; 2) : dist(Π , Π( R )) ≤ u } . The expression dist(Π , Π( R )) should be interpreted as follows: Select a semi-algebraicembedding of the affine Grassmannian Grass(4; 2) into R N ; then dist(Π , Π( R )) is theEuclidean distance between the points in R N corresponding to the images of Π andΠ( R ).Observe that for each C , the C -fold dilates { CR ∗ } R ∈R max are O C (1)–fold overlap-ping. For each z ∈ Z , there is a prism R ∈ R max so that ( z, Π( z )) ∈ R ∗ . This valueof R satisfies R ∗ ∩ ˜ A (cid:54) = ∅ . (34)For this R , we also have Π( z ) ∩ B (0 , ⊂ R, (35)where 4 R is the four-fold dilate or R . Note as well that if ( x, (cid:96) ) ∈ Γ with x ∈ f − Z ( z ),then (cid:96) is covered by 4 R . This is because (cid:96) ∩ R (cid:54) = ∅ and ∠ ( (cid:96), Π( R )) ≤ ∠ (Π( (cid:96) ) , Π( z )) + ∠ (Π( z ) , Π( R )) ≤ u. Thus we must establish the bound |{ R ∈ R max : R ∗ ∩ ˜ A (cid:54) = ∅}| (cid:46) ( hs ) − O (1) u − . (36)By Lemma 5.1, E u ( ˜ A ) = E u ( A ) (cid:46) s − O (1) u − . (37)Since the sets { CR ∗ : R ∈ R max } are O C (1) overlapping, in order to prove (36),it suffices to prove that if R ∗ ∩ ˜ A (cid:54) = ∅ , then E u (8 R ∗ ∩ ˜ A ) (cid:38) ( hs ) O (1) u − . (38)28uppose that R ∗ ∩ ˜ A (cid:54) = ∅ and let ( z, Π) ∈ R ∗ ∩ ˜ A . Since z ∈ s (Φ ),there are ≥ s/u lines (cid:96) that point in pairwise ≥ u separated directions with ( x, (cid:96) ) ∈ Γ for some x ∈ f − Z ( z ). For each of these lines (cid:96) , we have ∠ (Π( (cid:96) ) , Π( R )) ≤ ∠ (Π( (cid:96) ) , Π( z )) + ∠ (Π( z ) , Π( R )) < u. On each of these lines, we have | Γ ( (cid:96) ) | ≥ h , so we can select ≥ h/ (2 u ) points thatare all pairwise u separated and have distance ≥ h/ z . For each such point x (cid:48) , we have ∠ (cid:0) Π ◦ f Z ( x (cid:48) ) , Π( R ) (cid:1) ≤ ∠ (cid:0) Π ◦ f Z ( x (cid:48) ) , Π( (cid:96) ) (cid:1) + ∠ (cid:0) Π( (cid:96) ) , Π( R ) (cid:1) ≤ u. This gives us a set of ≥ hs/ (2 u ) (cid:38) ( sc ) O (1) u − points f Z ( x (cid:48) ) ∈ Z that are pairwise (cid:38) u separated, are contained in Z ∩ N u (Π), and satisfy ∠ (Π( u ) , Π( R )) ≤ u . Thisestablishes (38). Understanding lines in Σ We will prove that if w ≥ ( cushκ ) O (1) is sufficiently small, then the lines in Σ canbe covered by O ( s − ) rectangular prisms of dimensions 2 × s × s × s .Observe that for each (cid:96) ∈ Σ , Γ ( (cid:96) ) has complexity O (1). Thus if we select h (cid:38) c sufficiently small, then there is an interval I ⊂ Γ ( (cid:96) ) of length ≥ h so that Π( I ) isconnected. Since (cid:96) ∈ Σ , Π( I ) must also have diameter ≥ u . We will now fix a valueof h so that this holds. DefineΓ = { ( x, (cid:96) ) ∈ Γ : (cid:96) ∈ Σ , ∃ an interval x ∈ I ⊂ Γ ( (cid:96) ) of length 2 h } . Observe that | Γ ( (cid:96) ) | ≥ h (cid:38) c for all (cid:96) ∈ Σ . For each (cid:96) ∈ Σ , let γ (cid:96) ⊂ Grass(4; 2)be a connected set of diameter ≥ u that is contained in Π ◦ f Z (Γ( (cid:96) )). Let Φ be theset associated to Γ , in the sense of Definition 5.4. DefineΦ (cid:48) = { ( z, v ) ∈ Φ : z ∈ s (Φ ) } , and let Φ (cid:48)(cid:48) = Φ \ Φ (cid:48) . Let Γ (cid:48) be the pre-image of Φ (cid:48) under the map Γ → Φ , anddefine Γ (cid:48)(cid:48) = Γ \ Γ (cid:48) . Define Σ = { (cid:96) ∈ Σ : | Γ (cid:48) ( (cid:96) ) | ≥ h } , Σ (cid:48) = { (cid:96) ∈ Σ : | Γ (cid:48)(cid:48) ( (cid:96) ) | ≥ h } . (cid:48) can be covered by O ( c − O (1) s − ) rectangularprisms of dimensions 2 × s × s × s . We must now repeat the above process one moretime. Define Γ = { ( x, (cid:96) ) ∈ Γ : (cid:96) ∈ Σ } . Let Φ be the set associated to Γ , in the sense of Definition 5.4. DefineΦ (cid:48) = { ( z, v ) ∈ Φ : z ∈ s (Φ ) } , and let Φ (cid:48)(cid:48) = Φ \ Φ (cid:48) . Let Γ (cid:48) be the pre-image of Φ (cid:48) under the map Γ → Φ , anddefine Γ (cid:48)(cid:48) = Γ \ Γ (cid:48) . Define Σ = { (cid:96) ∈ Σ : | Γ (cid:48) ( (cid:96) ) | ≥ h/ } , Σ (cid:48) = { (cid:96) ∈ Σ : | Γ (cid:48)(cid:48) ( (cid:96) ) | ≥ h/ } . Again, the lines in Σ (cid:48) can be covered by O ( c − O (1) s − ) rectangular prisms of dimen-sions 2 × s × s × s . We will show that if w ≥ ( csuκ ) O (1) is sufficiently small, thenΣ is empty. The basic idea is as follows: If Σ is not empty, then we will findlines (cid:96) and (cid:96) (cid:48) that are (quantitatively) skew so that there are many (i.e. about w − )points x ∈ (cid:96) where the plane Π ◦ f Z ( x ) intersects (cid:96) (cid:48) . This implies that the planeΠ ◦ f Z ( x ) is contained in the hyperplane H spanned by (cid:96) and (cid:96) (cid:48) . Each of these planesΠ ◦ f Z ( x ) contains many lines (almost) contained in Σ , which will imply that the w –neighborhood of these planes each intersect N w ( Z ) in a set of measure roughly w . Since there are roughly w − such planes, this implies that | N w ( H ) ∩ N w ( Z ) | hassize roughly w . This contradicts the fact that (cid:107) II ( x ) (cid:107) ∞ ≥ κ on Z , which impliesthat | N w ( H ) ∩ N w ( Z ) | has size at most κ − / w / .We will now make this argument precise. For each (cid:96) ∈ Σ , let T (cid:96) be the w -neighborhood of (cid:96) ∩ B (0 , Y ( T (cid:96) ) ⊂ T be the w -neighborhood of Γ ( (cid:96) ).Observe that | Y ( T (cid:96) ) | (cid:38) c | T (cid:96) | for each such (cid:96) ∈ Σ . Let T be a maximal set ofessentially distinct tubes from { T (cid:96) : (cid:96) ∈ Σ } .Similarly, for each (cid:96) ∈ Σ , let T (cid:96) be the w -neighborhood of (cid:96) ∩ B (0 , Y ( T (cid:96) ) ⊂ T be the w -neighborhood of Γ ( (cid:96) ). Again, we have | Y ( T (cid:96) ) | (cid:38) c | T (cid:96) | for eachsuch (cid:96) ∈ Σ . Let T be a maximal set of essentially distinct tubes from { T (cid:96) : (cid:96) ∈ Σ } . Remark . Observe that for every T ∈ T and every plane Π containing theline coaxial with T , there are (cid:38) ( sc ) O (1) w − tubes T (cid:48) ∈ T with T ∩ T (cid:48) (cid:54) = ∅ , ∠ ( v ( T ) , v ( T (cid:48) )) (cid:38) ( sch ) O (1) , and ∠ ( v ( T (cid:48) ) , Π) (cid:38) ( uch ) O (1) .Our next task is to establish the bound | T | (cid:46) c − O (1) w − . (39)30or each x ∈ Z , define T ( x ) = { T ∈ T : x ∈ Y ( T ) } . For each T ∈ T , define v ( T ) = v ( (cid:96) ), where (cid:96) is the line coaxial with T , and define T ∗ = { ( x, v ) ∈ R × S : x ∈ T, ∠ ( v, v ( T )) ≤ w } . Then the sets { T ∗ } T ∈ T are O (1) overlapping.Recall the set A from (31). By Lemma 5.1, E w ( A ) (cid:46) s − O (1) w − . Define G = { ( z, v ) ∈ Z × S : v ∈ S ( z ) } . By Lemma 2.3, we have E w ( G ) (cid:46) s − O (1) w − . (40)On the other hand, if T (cid:96) ∈ T , then there are (cid:38) cw w -separated points z ∈ T (cid:96) ∩ Z with ∠ ( v ( (cid:96) ) , Π( z )) ≤ w and thus ( z, v ( T )) ∈ G . Thus if T ∈ T , we have N w ( T ∗ ∩ G ) (cid:38) cw − . (41)Combining (40) and (41), we obtain (39).We will now show that Σ is empty. Suppose not. Let (cid:96) ∈ Σ . Let T be the w -neighborhood of (cid:96) ∩ B (0 , Y ( T ) be the w -neighborhood of Γ (cid:48) ( (cid:96) ). Let z , . . . , z p , p (cid:38) ( csu ) O (1) w − be points in f Z (Γ (cid:48) ( (cid:96) )) ⊂ Y ( T ) so that the planes Π( z i )point in w –separated directions. (Recall that each of these planes makes an angle ≤ w with v ( (cid:96) ) = v ( T ))For each i = 1 , . . . , p , since z i ∈ Y ( T ), we can select a set T ( i ) of (cid:38) ( cs ) O (1) w − tubes from T passing through z i , so that each of these tubes makes an angle (cid:38) ( cs ) O (1) with v ( T ). Since the set T ( i ) is contained in the s − O (1) w –neighborhood ofthe plane Π( z i ), and these planes point in w –separated directions, we can refine theset of indices 1 , . . . , p to a new indexing set 1 , . . . , p (cid:48) with p (cid:48) (cid:38) ( csu ) O (1) w − so thatthe corresponding sets of tubes { T ( i ) } p (cid:48) i =1 are disjoint. As discussed in Remark 6.3,for each index i and each T ∈ T ( i ) , there exist (cid:38) ( csu ) O (1) w − tubes from T thatintersect T , make an angle (cid:38) ( csu ) O (1) with v ( T ), and that make an angle (cid:38) ( csu ) O (1) with the plane spanned by v ( T ) and v ( T ) (and thus make an angle (cid:38) ( csu ) O (1) withthe plane Π( z i )). We can also require that each of these tubes intersect T in a pointthat has distance (cid:38) ( csu ) O (1) from T , i.e. each of these tubes is (cid:38) ( csu ) O (1) skew to T . 31hus if C = O (1) is chosen sufficiently large, there are (cid:38) ( csu ) O (1) w − pairs { ( T, T (cid:48) ) ∈ T × T : T ∩ T (cid:54) = ∅ , ∠ ( v ( T ) , v ( T )) (cid:38) ( csu ) C ,T ∩ T (cid:48) (cid:54) = ∅ , ∠ ( v ( T ) , v ( T (cid:48) )) (cid:38) ( csu ) C ,T and T (cid:48) are (cid:38) ( csu ) C skew } . (42)Since | T | (cid:46) c − O (1) w − , we can select a tube T (cid:48) ∈ T that is (cid:38) ( csu ) C skewto T , so that there are (cid:38) ( csu ) O (1) w − tubes T ∈ T with ( T, T (cid:48) ) ∈ (42) . Notethat at most (cid:46) ( csu ) − of these tubes T ∈ T can lie in the w –neighborhood of acommon plane containing v ( T ), since T (cid:48) is (cid:38) ( csu ) C skew to T . Thus we can select (cid:38) ( csu ) O (1) w − tubes T with ( T, T (cid:48) ) ∈ (42) so that the planes { span( v ( T ) , v ( T ) } point in w –separated directions.Let H be the hyperplane containing the lines coaxial with T and T (cid:48) . Observethat if ( T, T (cid:48) ) ∈ (42), and if x ∈ T ∩ T , then ∠ (Π( x ) , H ) (cid:46) ( csu ) − O (1) w . Re-indexingthe sets of tubes { T (1) , . . . , T ( p (cid:48) ) } again, we can select sets T (1) , . . . , T ( p (cid:48)(cid:48) ) , p (cid:48)(cid:48) (cid:38) ( csu ) O (1) w − so that every set of tubes T ( i ) contains a tube T with ( T, T (cid:48) ) ∈ (42).This means that for each index i , the tubes in T ( i ) are contained in the (cid:46) ( csu ) − O (1) w –neighborhood of the hyperplane H (see Figure 2). We have (cid:12)(cid:12)(cid:12) p (cid:48)(cid:48) (cid:91) i =1 (cid:91) T ∈ T ( i ) Y ( T ) (cid:12)(cid:12)(cid:12) (cid:38) ( csu ) O (1) w. Since Y ( T ) ⊂ N w ( Z ) for each tube T in the above union, we have | N w ( Z ) ∩ N (cid:46) ( csu ) − O (1) w ( H ) | (cid:38) ( csu ) O (1) w. But by Lemma 4.3, we have | N w ( Z ) ∩ N (cid:46) ( csu ) − O (1) w ( H ) | (cid:46) ( csu ) − O (1) κ − / w / , and thus ( csu ) O (1) w (cid:46) ( csu ) − O (1) κ − / w / . (43)If w (cid:38) ( cκus ) O (1) is chosen sufficiently small, then (43) is impossible, which contra-dicts the assumption that Σ (cid:54) = ∅ . We conclude that Σ = ∅ , which completes theproof of Lemma 6.1.Lemma 6.1 can be used to understand hypersurfaces that are doubly-ruled byplanes. 32 z z z p (cid:48)(cid:48) T ( p (cid:48)(cid:48) ) T (cid:48) ∩ Π( z ) T (cid:48) T T (cid:48) ∩ Π( z p (cid:48)(cid:48) )Figure 2: The tubes T and T (cid:48) ; the points z , . . . , z p (cid:48)(cid:48) ∈ T , the planesΠ( z ) , . . . , Π( z p (cid:48)(cid:48) ), and the sets of tubes T (1) , . . . , T ( p (cid:48)(cid:48) ) . This entire figure is ostensiblycontained in R , but in fact it is contained in the (cid:46) ( csu ) − O (1) w –neighborhood ofthe hyperplane spanned by the lines coaxial with T and T (cid:48) . Proposition 6.2.
Let δ, u, s, κ, c, D, E be parameters with < δ < u < s < c and δ < κ . Then there exists a number w (cid:38) D,E ( usκc ) O (1) so that the following holds.Let P ∈ R [ x , . . . , x ] be a polynomial of degree at most D and let Z ⊂ { z ∈ Z ( P ) ∩ B (0 ,
1) : 1 ≤ |∇ P ( z ) | ≤ , (cid:107) II ( z ) (cid:107) ∞ ≥ κ } . (44) Let Σ ⊂ Σ δ,c ( Z ) , and let Γ ⊂ Γ( N δ ( Z ) , Σ) . Suppose that Z, Σ , and Γ are semi-algebraic sets of complexity at most E . Let Φ be associated to Γ , in the sense ofDefinition 5.4. Suppose that Z ⊂ (2 ,
2) -Narrow w (Φ) , (45) | Γ( (cid:96) ) | ≥ c for all (cid:96) ∈ Σ . (46) Then we can write
Σ = Σ (cid:48) ∪ Σ (cid:48)(cid:48) , where the lines in Σ (cid:48) can be covered by O D,E ( c − O (1) s − ) rectangular prisms of dimensions × s × s × s , and the lines in Σ (cid:48)(cid:48) can be coveredby O D,E (( cs ) − O (1) u − ) rectangular prisms of dimensions × × u × u . roof. Let w (cid:38) D,E ( usκc ) O (1) be the constant from Lemma 6.1 associated to thevalues δ (cid:48) = δ, u (cid:48) = u, s (cid:48) = s, κ (cid:48) = κ, c (cid:48) = c/ , D (cid:48) = D, E (cid:48) = O D,E (1). Define A = { ( z, S , S ) ∈ Z × ( S ) : min (cid:0) ∠ ( v, S ) , ∠ ( v, S ) (cid:1) ≤ w ∀ v ∈ π S (Φ( z )) } . Since Z ⊂ (2 ,
2) -Narrow w (Φ), the projection A (cid:55)→ Z is onto. Use Lemma 2.6 to selecta set A (cid:48) ⊂ A so that the map π : A (cid:48) → Z is a bijection. Define the semi-algebraicfunctions S ( z ) and S ( z ) : Z → S so that for each z ∈ Z, ( z, S ( z ) , S ( z )) ∈ A (cid:48) .For i = 1 ,
2, define Γ i = { ( x, (cid:96) ) ∈ Γ : ∠ ( v ( (cid:96) ) , S i ◦ f Z ( x ) ≤ w } . By (45), for every ( x, (cid:96) ) ∈ Γ we have that ∠ (cid:0) v ( (cid:96) ) , S ◦ f Z ( x ) (cid:1) ≤ w and/or ∠ (cid:0) v ( (cid:96) ) , S ◦ f Z ( x ) (cid:1) ≤ w. Thus Γ = Γ ∪ Γ . For i = 1 , , defineΣ i = { (cid:96) ∈ Σ : | Γ i ( (cid:96) ) | ≥ c/ } . Then Σ = Σ ∪ Σ . We have that Z, Γ i , and Σ i , i = 1 ,
2, are semi-algebraic ofcomplexity E (cid:48) = O D,E (1). For i = 1 ,
2, apply Lemma 6.1 to the data
P, Z, Σ i , Γ i , with the parameters δ (cid:48) , u (cid:48) , s (cid:48) , κ (cid:48) , c (cid:48) , D (cid:48) , and E (cid:48) described above, and let Σ (cid:48) i and Σ (cid:48)(cid:48) i bethe output from the lemma. Define Σ (cid:48) = Σ (cid:48) ∪ Σ (cid:48) and define Σ (cid:48)(cid:48) = Σ (cid:48)(cid:48) ∪ Σ (cid:48)(cid:48) . In this section, we will consider the region where Z is robustly 1-broad, (2 , (cid:48) from the statement of Theorem 1.2. This result will beproved in Proposition 7.1, which is the main result of this section. Lemma 7.1.
Let Q ( x , x ) = a x + a x + a x x be a monic quadratic polyno-mial. Let S = S ∩ Z (cid:16)(cid:0) a + R [( a − a a ) / ] (cid:1) x + 2 a x (cid:17) ,S = S ∩ Z (cid:16)(cid:0) a − R [( a − a a ) / ] (cid:1) x + 2 a x (cid:17) , here R [ z ] is the real part of z . Then there is an absolute constant C so that foreach t > , { x ∈ S : | Q ( x ) | ≤ t } ⊂ N Ct / ( S ∪ S ) . (47) Remark . The requirement that Q be monic can be replaced by the conditionthat the largest coefficient of Q has magnitude A >
0. Then the constant C in (47)depends on A . Definition . Let Q ∈ R [ x , x , x ] be a homogeneous polynomial of degree 2. Wesay that Q is w -degenerate if there exist great circles S , S ⊂ S so that Z ( Q ) ∩ S ⊂ N w ( S ∪ S ). If Q is not w -degenerate, then we call it w -non-degenerate. Lemma 7.2.
Let Q ∈ R [ x , x , x ] be a homogeneous quadratic polynomial. Supposethat Q is w -non-degenerate. Then for every t < w and for every great circle S ⊂ S ,we have | S ∩ Z ( Q ) ∩ N t ( S ) | (cid:46) ( t/w ) / , where | · | denotes one-dimensional Lebesgue measure. Lemma 7.3.
Let Q ( x , x , x ) be a monic homogeneous polynomial of degree 2 andlet w > . Then there is an absolute constant c > so that at least one of thefollowing two things must hold1. There exist two great circles S , S ⊂ S so that { x ∈ S : | Q ( x ) | ≤ cw } ⊂ N w ( S ∪ S ) .
2. For each t > , we have { x ∈ S : | Q ( x ) | ≤ ct } ⊂ N t/w ( Z ( Q )) . Proof.
Let c > p ∈ Z ( Q ) ∩ S ⊂ R where the map x (cid:55)→ Q ( x ) hassmall derivative, i.e. | DQ ( p ) | ≤ c w . We will show that Item 1 must hold. After arotation, we can assume that p = (1 , , Q ( x , x , x ) = a x + a x + a x x + a x x + a x x , and DQ ( p ) = ( ∂ x Q (1 , , , ∂ x Q (1 , , a , a ) . Thus Q ( x , x , x ) = a x + a x + a x x + O ( c w )( x x + x x ) , a , a , a has magnitude ∼ c > c ), then { x ∈ S : | a x + a x + a x x | ≤ c w } ⊂ N w ( S ∪ S ) , where S = S ∩ Z (cid:16)(cid:0) a + R [( a − a a ) / ] (cid:1) x + 2 a x (cid:17) ,S = S ∩ Z (cid:16)(cid:0) a − R [( a − a a ) / ] (cid:1) x + 2 a x (cid:17) . Next, if c > c ), then { x ∈ S : | Q ( x ) | ≤ c w } ⊂ { x ∈ S : | a x + a x + a x x | ≤ c w } , which completes the analysis of Case (A).Now suppose we are in Case (B): | DQ ( p ) | ≥ c w for all p ∈ Z ( Q ) ∩ S . Since Q is quadratic, DQ : R → R is a linear map. If c > c ), then | Q ( x ) | ≤ c t implies dist( t, Z ( Q )) ≤ t/w . Thus Item 2 musthold.To complete the proof, choose c = min( c , c ). Lemma 7.4.
Let P ∈ R [ x , x , x , x ] be a polynomial of degree at most D . Let Z ⊂ { x ∈ Z ( P ) ∩ B (0 , , ≤ |∇ P ( x ) | ≤ , (cid:107) II ( x ) (cid:107) ∞ ≥ κ } . Let Γ ⊂ Γ( N δ ( Z ) , L ) and let Φ ⊂ Z × S be associated to Γ , in the sense of Definition5.4. Suppose that • Z ⊂ (2 ,
2) -Broad w (Φ) . • For each z ∈ Z and each v ∈ π S (Φ( z )) , we have | ( v · ∇ ) j P ( z ) | ≤ Kδ, j = 1 , . (48) Then for each z ∈ Z , the vectors in π S (Φ( z )) are contained in the δ (cid:0) K/ ( wκ ) (cid:1) O (1) -neighborhood of the set C z = { v ∈ S : ( v · ∇ ) P ( z ) = 0 , ( v · ∇ ) P ( z ) = 0 } . (49)36 roof. Let x ∈ Z . After a translation and rotation, we can assume that x = 0 and N (0) = (1 , , , P ( x , x , x , x ) = x + (cid:88) | I | =2yes x a I x I + (cid:88) | I | =2no x a I x I + (cid:88) | I | > a I x I , where the first sum is taken over all multi-indices I of length two that include atleast one x term, and the second sum includes all the other multi-indices of lengthtwo.Let v ∈ π S (Φ( x )); we can write v = ( v , v , v , v ). Since v satisfies (48), we have | v | ≤ Kδ . Define A = (cid:107) II (0) (cid:107) − ∞ a a a a a a a a a . Since v satisfies (48) and (cid:107) II (0) (cid:107) ∞ ≥ κ , we have (cid:12)(cid:12)(cid:12) [ v , v , v ] T A [ v , v , v ] (cid:12)(cid:12)(cid:12) (cid:46) κ − Kδ. (50)Now consider the function Q ( v , v , v ) = [ v , v , v ] T A [ v , v , v ] . Since 0 ∈ (2 ,
2) -Broad w (Φ) (remember, originally we had z ∈ (2 ,
2) -Broad w (Φ),but we applied a translation sending z to 0), the set of unit vectors ( v , v , v ) sat-isfying (50) cannot be contained in the w –neighborhood of the union of two greatcircles in S . Thus by Lemma 7.3, we have that( v , v , v ) ∈ S ∩ N cκ − Kδ/w ( Z ( Q )) , where c > v ⊂ N t ( C z ) , where t (cid:46) δK/ ( κw ) . Definition . In (49) above, we defined the set C z = { v ∈ S : ( v · ∇ ) P ( z ) = 0 , ( v · ∇ ) P ( z ) = 0 } . (51)We will call this the quadratic cone of Z ( P ) with vertex z . More generally, any setof the form (51) will be called a quadratic cone. Following Definition 7.1, we saythat the quadratic cone C z is w -degenerate if there exist great circles S , S ⊂ { v ∈ S : ( v · ∇ ) P ( z ) = 0 }
37o that C z ⊂ N w ( S ∪ S ). Otherwise we say C z is w -non-degenerate.Define ˜ C z = z + span( C z );this is a two-dimensional algebraic variety in R ; it is the union of all lines thatintersect z and also intersect the curve z + C z . We say that ˜ C z is w -non-degenerateif C z is w -non-degenerate. Observe that ˜ C z is a degree-two algebraic surface; it canbe defined as the common zero locus of a degree one and a degree two polynomial in R [ x , x , x , x ]. If C z is a quadratic cone, (cid:96) ∈ L , z ∈ (cid:96), and dist (cid:0) v ( (cid:96) ) , C z (cid:1) = t , then (cid:96) ∩ B (0 , ⊂ N t ( ˜ C z ). For the next lemma, we will introduce some standard notation from the Kakeyaproblem. This notation will be used throughout the remainder of this section. Let T be a set of essentially distinct δ -tubes, i.e. a set of δ -neighborhoods of unit linesegments so that no tube is contained in the two-fold dilate of any other. For eachtube T ∈ T , let Y ( T ) ⊂ T . For each x ∈ R , define T ( x ) = { T ∈ T : x ∈ Y ( T ) } . If the set Y is ambiguous, we will sometimes use the notation T Y ( x ) in place of T ( x ).For each T ∈ T , define v ( T ) to be the direction of the line coaxial with T . Thus forexample v ( T ( x )) = { v ( T ) : T ∈ T ( x ) } . For each T ∈ T , define H ( T ) = { T ∈ T : Y ( T ) ∩ Y ( T ) (cid:54) = ∅} . If the set Y is ambiguous, we will sometimes use the notation H Y ( T ) in place of H ( T ).The next lemma says that if T is a set of tubes, and if the tubes passing througha typical point lie near a non-degenerate cone, then the tubes in a typical hairbrushare mostly disjoint and thus their union has large volume. This is a variant of Wolff’s“hairbrush argument” from [22]. However, unlike in [22] we do not assume that thetubes point in different directions. Lemma 7.5.
Let δ, λ, t > . Let T be a set of essentially distinct δ -tubes. For each T ∈ T , let Y ( T ) ⊂ T with Y ( T ) ≥ λ | T | . Let T ∈ T . Suppose that | H ( T ) | ≥ δ − and that for every x ∈ Y ( T ) , the vectors v ( T ( x )) are contained in the Kδ –neighborhood of a w -non-degenerate cone C x . Then (cid:12)(cid:12)(cid:12) (cid:91) T ∈ H ( T ) Y ( T ) (cid:12)(cid:12)(cid:12) ≥ ( wλt/K ) O (1) δ. Proof.
By pigeonholing, we can select a set of ≥ tλ/ δ points x ∈ Y ( T ) that are δ separated and that satisfy | T ( x ) | ≥ tλδ − . The line coaxial with T passes through x and makes an angle ≤ Kδ with a line ˜ (cid:96) in the cone ˜ C x (see Figure 3). Let Π x bethe plane that is tangent to ˜ C x along ˜ (cid:96) . x T ˜ C x Figure 3: The tube T , the point x , and the cone ˜ C x . The plane Π x (not pictured)contains the line coaxial with T and is tangent to ˜ C x .Let p = wt λ K − , and let T ( x ) (cid:48) = { T ∈ T ( x ) : ∠ ( v ( T ) , Π( x )) ≥ p } . Since C x is w –non-degenerate, by Lemma 7.2, we have that |{ v ∈ S : ∠ ( v, Π x ) ≤ p, v ∈ N Kδ ( C x ) }| (cid:46) ( Kδ )( p/w ) / , where | · | denotes two-dimensional Haar measure on the sphere S ; this set cancontain at most K ( p/w ) / δ − δ -separated points on S , which implies that | T ( x ) \ T (cid:48) ( x ) | ≤ K ( p/w ) / δ − ≤ | T ( x ) | / , | T (cid:48) ( x ) | ≥ tλδ − for each of the values of x chosen above. Furthermore, forevery plane Π containing the line coaxial with T , we have that |{ T ∈ T (cid:48) ( x ) : T ⊂ N δ (Π) }| (cid:46) (cid:0) K/ ( wλt ) (cid:1) O (1) . Define H (cid:48) ( T ) = (cid:83) x T (cid:48) ( x ); all of these tubes intersect T . We have that | H (cid:48) ( T ) | ≥ (cid:0) wλt ) /K (cid:1) O (1) δ − . (52)Furthermore, for each plane Π containing the line coaxial with T ; for each point z ∈ T ; and for each δ ≤ ρ ≤
1, we have |{ T ∈ H (cid:48) ( T ) : T ⊂ N δ (Π) , dist( z, T ∩ T ) ≤ ρ }| (cid:46) (cid:0) K/ ( wλt ) (cid:1) O (1) ( ρ/δ ) . (53)Wolff’s hairbrush argument from [22] says that the union of any set of tubes inter-secting T that satisfy (52) and (53) must have volume (cid:38) (cid:0) wλt/K (cid:1) O (1) δ . Thus (cid:12)(cid:12)(cid:12) (cid:91) T ∈ H ( T ) Y ( T ) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) (cid:91) T ∈ H (cid:48) ( T ) Y ( T ) (cid:12)(cid:12)(cid:12) (cid:38) (cid:0) wλt/K (cid:1) O (1) δ. The following lemma gives sufficient conditions for a semi-algebraic subset of ahypersurface in R to be large (specifically, for it to have δ -covering number roughly δ − ). In short, if a semi-algebraic subset of a hypersurface contains at least one linewhose hairbrush contains many cones, then the set must be large. Lemma 7.6.
Let δ, c, s, w, κ be positive real numbers. Let P be a polynomial of degreeat most D . Let Z ⊂ Z ⊂ { x ∈ Z ( P ) ∩ B (0 , , ≤ |∇ P ( x ) | ≤ , (cid:107) II ( x ) (cid:107) ∞ ≥ κ } . (54) Let ∅ (cid:54) = Σ ⊂ Σ with Σ i ⊂ Σ δ,c ( Z i ) for i = 1 , . Let Γ ⊂ Γ with Γ i ⊂ Γ( N δ ( Z i ) , Σ i ) for i = 1 , . Suppose that the sets Z i , Σ i , and Γ i , i = 1 , are semi-algebraic ofcomplexity at most E . For each i = 1 , , let Φ i ⊂ Z i × S be associated to Γ i , in thesense of Definition 5.4. Suppose that Z ⊂ s (Φ ) ∩ (2 ,
2) -Broad w (Φ ) , (55) | Γ i ( (cid:96) ) | ≥ c for each (cid:96) ∈ Σ i , i = 1 , , (56) | ( v · ∇ ) j P ( z ) | ≤ Kδ, j = 1 , , for each z ∈ Z and each v ∈ π S (Φ ( z )) . (57) Then E δ ( Z ) (cid:38) D,E ( scwκ ) O (1) δ − . roof. For i = 1 ,
2, define T i to be a maximal δ -separated subset of Σ i and for each T ∈ T i , define Y i ( T ) to be the δ -neighborhood of Γ i ( T ). Since Σ is non-empty, thereexists a tube T ∈ T . By Lemma 7.4, the tube T and the pair ( T , Y ) satisfy thehypotheses of Lemma 7.5. Applying Lemma 7.5, we conclude that (cid:12)(cid:12)(cid:12) (cid:91) T ∈ T : Y ( T ) ∩ Y ( T ) (cid:54) = ∅ Y ( T ) (cid:12)(cid:12)(cid:12) (cid:38) ( scwκ ) O (1) δ. But since the above set is contained in N δ ( Z ), we have E δ ( Z ) (cid:38) ( scwκ ) O (1) δ − . We are now ready to state the main result of this section.
Proposition 7.1.
Let δ, s, w, κ, t be positive real numbers. Let P be a polynomial ofdegree at most D . Let Z ⊂ { x ∈ Z ( P ) ∩ B (0 , , ≤ |∇ P ( x ) | ≤ , (cid:107) II ( x ) (cid:107) ∞ ≥ κ } . (58) Let Σ ⊂ L with |E δ (Σ) | ≥ L − δ − and let Γ ⊂ Γ( N δ ( Z ) , Σ) . Suppose that Z, Σ , and Γ are semi-algebraic of complexity at most E . Let Φ ⊂ Z × S be associated to Γ , inthe sense of Definition 5.4. Suppose that Z ⊂ s (Φ) ∩ (2 ,
2) -Broad w (Φ) , (59) E δ ( Z ) ≥ tδ − , (60) | ( v · ∇ ) j P ( z ) | ≤ Kδ, j = 1 , for each z ∈ Z and each v ∈ π S (Φ( z )) . (61) Then there is a set Σ (cid:48) ⊂ Σ and a quadratic polynomial Q so that E δ (Σ (cid:48) ) (cid:38) D,E (cid:0) swκt/KL (cid:1) O (1) E δ (Σ) , (62) and for every (cid:96) (cid:48) ∈ Σ (cid:48) , there is a line (cid:96) ⊂ Z ( Q ) with dist( (cid:96), (cid:96) (cid:48) ) (cid:46) (cid:0) swκt/KL (cid:1) − O (1) δ .Proof. Since the constants D and E are fixed, all implicit constants may depend onthese quantities; i.e. we will write (cid:46) instead of (cid:46) D,E .For each (cid:96) ∈ Σ, let T (cid:96) = N δ ( (cid:96) ) ∩ B (0 ,
1) and define Y ( T ) = N δ (Γ( (cid:96) )). Let T be amaximal essentially distinct subset of { T (cid:96) : (cid:96) ∈ Σ } . Note that | T | ∼ E δ (Σ) ≥ L − δ − . (63)41y (59), we have that for each x ∈ (cid:83) T Y ( T ), | T ( x ) | (cid:38) sδ − . (64)By (71), (59), (61), and Lemma 7.4, we have that v ( T ( x )) is contained in the (cid:46) δ ( K/wκ ) O (1) -neighborhood of the quadratic cone C x of P at x . By (59), this cone is (cid:38) ws O (1) -non-degenerate. Thus there exists a constant A (cid:46) ( K/wκ ) O (1) so that forevery x ∈ Z , we have that v ( T ( x )) is contained in the Aδ -neighborhood of C x . Inparticular, | T ( x ) | (cid:46) A O (1) δ − . (65)Since tδ (cid:46) | N δ ( Z ) | (cid:46) δ (the lower bound comes from (60) and the upper boundcomes from the fact that | N δ ( Z ) | (cid:46) δ ), we have st (cid:46) (cid:90) N δ ( Z ) (cid:88) T ∈ T χ T ( x ) dx (cid:46) A O (1) , (66)and thus by (63) and (66), L − δ − (cid:46) | T | (cid:46) A O (1) δ − . (67)By pigeonholing, we can select a point x ∈ (cid:83) Y ( T ) with (cid:88) T ∈ T ( x ) | Y ( T ) | (cid:38) ( st ) O (1) δ . For each point x ∈ (cid:83) T ( x ) Y ( T ), define N ( x ) = (cid:12)(cid:12)(cid:12)(cid:8) T ( x ) ∩ (cid:91) T ∈ T ( x ) H ( T ) (cid:9)(cid:12)(cid:12)(cid:12) .N ( x ) is an integer satisfying 0 ≤ N ( x ) (cid:46) A O (1) δ − . For each T ∈ T \ T ( x ), define anew shading Y (cid:48) ( T ) = { x ∈ Y ( T ) : N ( x ) (cid:38) ( st ) C δ − } . If the constant C is chosen sufficiently large, then (cid:88) T ∈ T \ T ( x ) | Y (cid:48) ( T ) | ≥ (cid:88) T ∈ T | Y ( T ) | . Thus by pigeonholing, we can select a set T (cid:48) ⊂ T so that | T (cid:48) | (cid:38) ( st/L ) O (1) | T | and | Y (cid:48) ( T ) | (cid:38) ( st/L ) C | T | for all T ∈ T (cid:48) . Select a point x with dist( x , x ) (cid:38) ( st/L ) O (1) and | T (cid:48) ( x ) | (cid:38) ( st/L ) O (1) δ − . (68)42or this value of x , if we select the constant C (cid:46) (cid:38) ( st/L ) O (1) δ − tubes T ∈ T that satisfy | Y ( T ) | ≥ ( st/L ) C | T | and ∃ T ∈ T ( x ) , T ∈ T (cid:48) ( x ) : T ∩ T i (cid:54) = ∅ , i = 1 , , dist( T ∩ T , T ∩ T ) ≥ C − ( st/L ) C . Call this set of tubes T (cid:48)(cid:48) . Select a tube T ∈ T (cid:48)(cid:48) with | H ( T ) ∩ T (cid:48)(cid:48) | (cid:38) ( st/L ) O (1) δ − .Let C x be the quadratic cone associated to x and let ˜ C x = x + span( C x ). Define˜ C x similarly, with x in place of x .Equation (68) implies that | N Aδ ( ˜ C x ) ∩ N Aδ ( ˜ C x ) (cid:12)(cid:12) (cid:38) ( st/L ) O (1) δ . Since the cones ˜ C x are (cid:38) A non-degenerate and their vertices are (cid:38) ( sc ) C -separated,the set N Aδ ( ˜ C x ) ∩ N Aδ ( ˜ C x ) is contained in the (cid:46) ( AL/st ) O (1) δ –neighborhood of acurve. However, it need not be the case that the cones ˜ C x and ˜ C x themselvesintersect. To overcome this annoying technicality, we will replace ˜ C x by a differentcone that is comparable to ˜ C x but which does intersect ˜ C x in a curve. We will callthis cone ˜ C ∗ ; we will describe its construction in the next paragraph.By our choice of x , there exist three points p , p , p ∈ ˜ C x ∩ N Aδ ( ˜ C x ) so thatall 3 × p − x , p − x , p − x ] have magnitude (cid:38) ( st/AL ) O (1) . Let H be the hyperplane passing through x , p , p , p (our condition on the minors of[ p − x , p − x , p − x ] ensures that this hyperplane is “well conditioned” in thesense that a small perturbation to one of the points p , p , or p will only cause a smallchange in the choice of hyperplane). Since p , p , p ∈ N Aδ ( ˜ C x ), and ˜ C x ⊂ T x ( Z ),the condition on the minors of [ p − x , p − x , p − x ] implies that ∠ ( H, T x ( Z )) (cid:46) ( L/st ) O (1) Aδ. (69)Define C ∗ = H ∩ Z ( P ), where P is the homogeneous polynomial of degree 2 arisingfrom the Taylor expansion of P around x . Since H and Z ( P ) intersect ≥ κ trans-versely (i.e. the tangent plane of Z ( P ) and of H make an angle ≥ κ at every pointof intersection), (69) implies that ˜ C x and C ∗ are comparable in the sense that B (0 , ∩ ˜ C x ⊂ N ( AL/ ( stκ )) O (1) δ ( C ∗ ) , and B (0 , ∩ C ∗ ⊂ N ( AL/ ( stκ )) O (1) δ ( ˜ C x ) . We also have that ˜ C x ∩ C ∗ is a degree-two curve lying in the plane T x Z ∩ H .Let (cid:96) be a line with (cid:96) ∩ B (0 , ⊂ T so that (cid:96) intersects ˜ C x and C ∗ at pointsthat are (cid:38) ( stκ/AL ) O (1) separated. Observe that the cones ˜ C x and C ∗ intersect ina one-dimensional degree-two curve, and the line (cid:96) intersects each of ˜ C x and C ∗ atdistinct points that are not on this curve.43e can now use the “14 point” argument from [14] to find a monic polynomial Q that vanishes on ˜ C x , C ∗ , and (cid:96) . In brief, select 5 points p , . . . , p ∈ ˜ C x ∩ C ∗ ; anypolynomial of degree ≤ p , . . . , p must vanish on the degree-twoplane curve ˜ C x ∩ C ∗ . Let p , p be the points of intersection of ˜ C x ∩ (cid:96) and C ∗ ∩ (cid:96) ,and let p be another point on (cid:96) ; any polynomial of degree ≤ p , p , p must vanish on (cid:96) . Let p = x and let p = x . Let p and p be twopoints on ˜ C x , and let p and p be two points on C ∗ . See Figure 4. Let Q be apolynomial of degree ≤ p , . . . , p . We can choose Q so that itslargest coefficient has magnitude 1. Q will be the output from this proposition. Theremainder of the proof is devoted to finding the set Σ (cid:48) so that Q and Σ (cid:48) satisfy theconclusions of the proposition. p p p p p p p p p p p p p p Figure 4: The cones ˜ C x (left), C ∗ (right), the line (cid:96) , and the 14 points p , . . . , p .Let (cid:96) be the line passing through p and p ; this is a line in ˜ C x passing throughthe vertex p = x , so it intersects the curve ˜ C x ∩ C ∗ at some point x . Since Q vanishes at the three collinear points p , p , and x , Q must vanish on the entire line (cid:96) . Similarly, Q vanishes on the line (cid:96) passing through p and p , and the line (cid:96) passing through p and p . Thus Q vanishes on the five-dimensional (reducible)curve ˜ C x ∩ C ∗ ∪ (cid:96) ∪ (cid:96) ∪ (cid:96) . Since Q has degree at most 2 and ˜ C x has degree at44ost 2, we conclude that Q vanishes on ˜ C x . An identical argument shows that Q vanishes on C ∗ .Recall that for each T ∈ H ( T ) ∩ T (cid:48)(cid:48) , we have that Z ( Q ) vanishes on (cid:38) ( stκ/AL ) O (1) δ − distinct δ -separated points on T . Since Q is monic and has degree 2, we have | Q ( x ) | (cid:46) ( AL/stκ ) O (1) δ for all x ∈ T. By the definition of T (cid:48)(cid:48) , we have (cid:88) T ∈ H ( T ) ∩ T (cid:48)(cid:48) | H Y ( T ) | (cid:38) ( stκ/AL ) O (1) δ − . Thus there exists a set T (cid:48)(cid:48)(cid:48) with | T (cid:48)(cid:48)(cid:48) | (cid:38) ( stκ/AL ) O (1) δ − and (cid:12)(cid:12)(cid:12) Y ( T (cid:48) ) ∩ (cid:91) T ∈ H ( T ) Y ( T ) (cid:12)(cid:12)(cid:12) (cid:38) ( stκ/AL ) O (1) | T (cid:48) | for every T (cid:48) ∈ T (cid:48)(cid:48)(cid:48) . Since Q is monic and Z ( Q ) ∩ B (0 , (cid:54) = ∅ , we can assume that at least one non-constant term of Q has size ∼
1. We can also assume that at least one degree-two termof Q has magnitude (cid:38) ( κst/AL ) O (1) ; if this were not the case, then Z ( Q ) ∩ B (0 , ∼ ( κst/AL ) O (1) -neighborhood of a hyperplane H , and thus | N δ ( Z ) ∩ N ( κst/AL ) O (1) ( H ) | ≥ (cid:91) T ∈ H ( T ) Y ( T ) (cid:38) ( κst/AL ) O (1) δ, but this would contradict the estimate | N δ ( Z ) ∩ N ( κst/AL ) O (1) ( H ) | (cid:46) δ / ( κst/AL ) − O (1) coming from Lemma 4.3. Thus at least one degree-two term of Q must have magni-tude (cid:38) ( κst/AL ) O (1) . Next, the set { x ∈ B (0 ,
1) : |∇ Q ( x ) | ≤ ( κst/AL ) O (1) } is contained in the ( κst/AL ) O (1) –neighborhood of a hyperplane H (cid:48) . By the sameargument as above, we can choose a refinement T (iv) ⊂ T (cid:48)(cid:48)(cid:48) with | T (iv) | (cid:38) ( κst/AL ) O (1) δ − (70)and a shading Y (iv) ( T ) so that |∇ Q ( x ) | (cid:38) ( κsc/A ) O (1) for all x ∈ Y (iv) ( T ) and all T ∈ T (iv) . 45gain by pigeonholing, we can refine Y (iv) to get a shading Y (v) so that | T Y (v) ( x ) | (cid:38) ( κst/AL ) O (1) δ − for all x ∈ (cid:83) T ∈ T (iv) Y (v) ( T ). Now fix a point x ∈ (cid:83) T ∈ T (iv) Y (v) ( T ).We will show that T x ( Z ( Q )) ∩ Z ( Q ) is a ζ –non-degenerate cone, where ζ = ( AL/ ( stκ )) O (1) .Indeed, v ( T (iv) ( x )) is contained in the Aδ –neighborhood of the w –non-degeneratecone C x , and | T (iv) ( x ) | (cid:38) ( stκ/AL ) O (1) δ − . At most ( ζ/w ) / Aδ δ –separated vec-tors can be contained in the intersection of N A ( C x ) with the ζ –neighborhood of aplane. We conclude that the cone T x ( Z ( Q )) ∩ Z ( Q ) is ζ –non-degenerate for some ζ = ( sctκ/A ) O (1) .We conclude that if T ∈ T (iv) and x ∈ Y (v) ( T ), then v ( T ) makes an angle (cid:46) ( AL/ ( stκ )) O (1) δ with the quadratic cone T x ( Z ( Q )) ∩ Z ( Q ) of Q at x . However, since Q is degree-two, if v is a vector contained in the quadratic cone of Q at x , then theline { x + vt : t ∈ R } is contained in Z ( Q ). Thus if T ∈ T (iv) with Y (v) ( T ) (cid:54) = ∅ , thenthere is a line (cid:96) contained in Z ( Q ) with (cid:96) ∩ B (0 , ⊂ N ( AL/ ( stκ )) O (1) δ ( T ).By (67) and (70), we have that | T (iv) | ≥ ( stκ/AL ) O (1) | T | (cid:38) (cid:0) swtκ/KL (cid:1) O (1) | T | (cid:38) (cid:0) swtκ/KL (cid:1) O (1) E δ (Σ) . Thus there is a set Σ (cid:48) ⊂ Σ (note that Σ (cid:48) need not be semi-algebraic) with E δ (Σ (cid:48) ) (cid:38) (cid:0) swtκ/KL (cid:1) O (1) E δ (Σ)so that for all (cid:96) (cid:48) ∈ Σ (cid:48) there is a line (cid:96) contained in Z ( Q ) withdist( (cid:96), (cid:96) (cid:48) ) (cid:46) (cid:0) swtκ/KL (cid:1) O (1) δ. The following result allows us to separate the lines in Σ δ ( Z ) into two sets—those thatcan be covered by a small number of one and two-dimensional rectangular prisms,and those that are amenable to Proposition 7.1. Proposition 8.1.
Let δ, s, u, c, κ be positive real numbers. Let P be a polynomial ofdegree at most D . Let Z ⊂ { x ∈ Z ( P ) ∩ B (0 , , ≤ |∇ P ( x ) | ≤ , (cid:107) II ( x ) (cid:107) ∞ ≥ κ } . (71)46 et Σ ⊂ Σ δ,c ( Z ) and Γ ⊂ Γ( N δ ( Z ) , Σ) . Suppose that Z, Σ , and Γ are semi-algebraicof complexity at most E . Let Φ ⊂ Z × S be associated to Γ , in the sense of Definition5.4. Suppose that | Γ( (cid:96) ) | ≥ c, for all (cid:96) ∈ Σ , (72) | ( v · ∇ ) j P ( z ) | ≤ Kδ, j = 1 , for each z ∈ Z and each v ∈ π S (Φ( z )) . (73) Then there is a number w (cid:38) D,E ( usκc ) O (1) and sets Σ (cid:48) , Σ (cid:48)(cid:48) , Σ (cid:48)(cid:48)(cid:48) , Z (cid:48)(cid:48)(cid:48) and Γ (cid:48)(cid:48)(cid:48) with Σ = Σ (cid:48) ∪ Σ (cid:48)(cid:48) ∪ Σ (cid:48)(cid:48)(cid:48) , Σ (cid:48)(cid:48)(cid:48) ⊂ Σ δ,c/ ( Z (cid:48)(cid:48)(cid:48) ) , and Γ (cid:48)(cid:48)(cid:48) ⊂ Γ( N δ ( Z (cid:48)(cid:48)(cid:48) ) , Σ (cid:48)(cid:48)(cid:48) ) , so that • The lines in Σ (cid:48) can be covered by O D,E ( c − O (1) s − ) rectangular prisms of dimen-sions × s × s × s . • The lines in Σ (cid:48)(cid:48) can be covered by O D,E (( sc ) − O (1) u − ) rectangular prisms ofdimensions × × u × u . • Σ (cid:48)(cid:48)(cid:48) , Z (cid:48)(cid:48)(cid:48) and Γ (cid:48)(cid:48)(cid:48) are semi-algebraic of complexity O D,E (1) . We have | N δ (Σ (cid:48)(cid:48)(cid:48) ) | (cid:46) D ( K/ ( usκc )) O (1) δ − . (74) Finally, let Φ (cid:48)(cid:48)(cid:48) be the set associated to Γ (cid:48)(cid:48)(cid:48) . Then Z (cid:48)(cid:48)(cid:48) ⊂ s (Φ (cid:48)(cid:48)(cid:48) ) ∩ (2 ,
2) -Broad w (Φ (cid:48)(cid:48)(cid:48) ) , (75) E δ ( Z (cid:48)(cid:48)(cid:48) ) (cid:38) ( scuκ ) O (1) δ − . (76) Proof.
Let w (cid:38) D,E ( usκc ) O (1) be the constant from Proposition 6.2 associated to thevalues u, s, κ, c, D, E . Define Σ = Σ , Γ = Γ, and Φ = Φ . For each i = 1 , , (cid:48) i = { ( z, v ) ∈ Φ i : z (cid:54)∈ s (Φ i ) } , Φ (cid:48)(cid:48) i = { ( z, v ) ∈ Φ i : z ∈ (2 ,
2) -Narrow w (Φ i ) } , Φ (cid:48)(cid:48)(cid:48) i = Φ i \ (Φ (cid:48) i ∪ Φ (cid:48)(cid:48) i ) . Let Γ (cid:48) i , Γ (cid:48)(cid:48) i , and Γ (cid:48)(cid:48)(cid:48) i be the pre-images of Φ (cid:48) i , Φ (cid:48)(cid:48) i , and Φ (cid:48)(cid:48)(cid:48) i (respectively) under the mapΓ i → Φ i . Define Σ (cid:48) i = { (cid:96) ∈ Σ i : | Γ (cid:48) i ( (cid:96) ) | ≥ − i c } , Σ (cid:48)(cid:48) i = { (cid:96) ∈ Σ i : | Γ (cid:48)(cid:48) i ( (cid:96) ) | ≥ − i c } , Σ (cid:48)(cid:48)(cid:48) i = { (cid:96) ∈ Σ i : | Γ (cid:48)(cid:48)(cid:48) i ( (cid:96) ) | ≥ − i c } ,
47o Σ i ⊂ Σ (cid:48) i ∪ Σ (cid:48)(cid:48) i ∪ Σ (cid:48)(cid:48)(cid:48) i .By Proposition 6.1, the lines in Σ (cid:48) i can be covered by O D,E ( c − O (1) s − ) rectangularprisms of dimensions 2 × s × s × s ; these lines will be placed in Σ (cid:48) . By Proposition6.2, the lines in Σ (cid:48)(cid:48) i can be partitioned into two sets; the first set can be covered by O D,E ( c − O (1) s − ) rectangular prisms of dimensions 2 × s × s × s and the second setcan be covered by O D,E (( cs ) − O (1) u − ) rectangular prisms of dimensions 2 × × u × u .We will place these lines into Σ (cid:48) and Σ (cid:48)(cid:48) , respectively.Define Σ i +1 = Σ (cid:48)(cid:48)(cid:48) i and define Γ i +1 = Γ (cid:48)(cid:48)(cid:48) i . Define Z i +1 to be the image of Φ i under the map ( z, v ) (cid:55)→ z . Then we have • Z i +1 ⊂ Z i . • Σ i +1 ⊂ Σ δ, − i c ( Z i ). • Σ i +1 ⊂ Σ i . • Γ i +1 ⊂ Γ i . • Z i +1 ⊂ s (Φ i ) ∩ (2 ,
2) -Broad w (Φ i ) . (77) • | Γ i ( (cid:96) ) | ≥ − i c ≥ c/ . (78) • By (73), | ( v · ∇ ) j P ( z ) | ≤ Kδ, j = 1 , , for each z ∈ Z i and each v ∈ π S (Φ i ( z )) . (79)If Σ = ∅ then define Σ (cid:48)(cid:48)(cid:48) = ∅ , Z (cid:48)(cid:48)(cid:48) = ∅ and Γ (cid:48)(cid:48)(cid:48) = ∅ and we are done. If not,define Z (cid:48)(cid:48)(cid:48) = Z , Σ (cid:48)(cid:48)(cid:48) = Σ , and Γ (cid:48)(cid:48)(cid:48) = Γ . (75) follows from (77). Applying Lemma7.6 to the sets Z ⊂ Z ; Σ ⊂ Σ ; Γ ⊂ Γ (equations (77), (78) and (79) guaranteethat the hypotheses of Lemma 7.6 are met), we conclude that E δ ( Z (cid:48)(cid:48)(cid:48) ) = E δ ( Z ) (cid:38) ( scwκ ) O (1) δ − , which is (76).Finally, it remains to prove (74). But this follows from the observation that | N δ ( Z ) | (cid:46) D δ − , and that for each z ∈ Z , N δ (Φ (cid:48)(cid:48)(cid:48) ( N δ ( z )) (cid:46) D,E ( K/ ( usκc )) O (1) δ − .
48e are now ready to prove Theorem 1.2. First, we will state a slightly moretechnical version that will be useful for applications.
Theorem 1.2, technical version.
Let P ∈ R [ x , x , x , x ] be a polynomial ofdegree D , and let Z = Z ( P ) ∩ B (0 , . Let δ, κ, u, s ∈ (0 , be numbers satisfying < δ < u < s < and δ < κ < (if these conditions are not satisfied the theoremis still true, but it has no content).Define Σ = { (cid:96) ∈ L : | (cid:96) ∩ N δ ( Z ) | ≥ } , and let Σ (cid:48) ⊂ Σ be a semi-algebraic set of complexity at most E . Then we can write Σ (cid:48) = Σ ∪ Σ ∪ Σ ∪ Σ , where • There is a collection of O D,E (cid:0) | log δ | O (1) s − (cid:1) rectangular prisms of dimensions × s × s × s so that every line from Σ is covered by one of these prisms. • There is a collection of O D,E (cid:0) ( | log δ | /s ) O (1) u − (cid:1) rectangular prisms of dimen-sions × × u × u so that every line from Σ is covered by one of these prisms. • There is a collection of O D,E (cid:0) | log δ | O (1) (cid:1) rectangular prisms of dimensions × × × κ so that every line in Σ is covered by one of these prisms. • There is a set Σ (cid:48) ⊂ Σ with E δ (Σ (cid:48) ) (cid:38) D,E (cid:0) usκ/ | log δ | (cid:1) O (1) E δ (Σ ) and a quadratic hypersurface Q so that for every line (cid:96) (cid:48) ∈ Σ (cid:48) , there is a line (cid:96) contained in Z ( Q ) with dist( (cid:96), (cid:96) (cid:48) ) (cid:46) D (cid:0) | log δ | / ( usκ ) (cid:1) O (1) δ. Proof.
Apply Proposition 3.1 to Σ and P . Let P , . . . , P b , Σ , . . . , Σ b , and Γ , . . . , Γ b ,b = O D,E ( | log δ | ) be the output from the proposition. By Item 5 from Proposition3.1, there exists a number c (cid:38) D | log δ | − so that | Γ j ( (cid:96) ) | ≥ c for each j = 1 , . . . , b andeach (cid:96) ∈ Σ j .For each index j , define Z j = { x ∈ Z ( P j ) ∩ B (0 ,
1) : 1 ≤ |∇ P j ( x ) | ≤ } . By Item 3 from Proposition 3.1, we have that for all (cid:96) ∈ Σ j and all x ∈ Γ j ( (cid:96) ), x ∈ N δ ( Z j ). Let Φ j ⊂ Z j × S be associated to Γ j , in the sense of Definition 5.4. Wehave that for each z ∈ Z j and each v ∈ Π S (Φ j ( z )), | ( v ( (cid:96) ) · ∇ ) i P j ( z ) | (cid:46) D | log δ | δ, i = 1 , . (80)49efine Z (cid:48) j = { z ∈ Z j : (cid:107) II ( z ) (cid:107) ∞ ≤ κ } ,Z (cid:48)(cid:48) j = { z ∈ Z j : (cid:107) II ( z ) (cid:107) ∞ > κ } . Define Φ (cid:48) j = { ( z, v ) ∈ Φ j : z ∈ Z (cid:48) j } , Φ (cid:48)(cid:48) j = { ( z, v ) ∈ Φ j : z ∈ Z (cid:48)(cid:48) j } . Let Γ (cid:48) j and Γ (cid:48)(cid:48) j be the pre-images of Φ (cid:48) j and Φ (cid:48)(cid:48) j , respectively, under the map fromΦ j → Γ j . Define Σ (cid:48) j = { (cid:96) ∈ Σ j : | Γ (cid:48) j ( (cid:96) ) | ≥ c/ } , Σ (cid:48)(cid:48) j = { (cid:96) ∈ Σ j : | Γ (cid:48)(cid:48) j ( (cid:96) ) | ≥ c/ } . Apply Proposition 4.1 to P j , Σ (cid:48) j and Γ (cid:48) j . Define Σ (1) j = Σ (cid:48) j . We conclude thatthe lines in Σ (1) j can be covered by (cid:46) D,E ( | log δ | /c ) O (1) (cid:46) D,E | log δ | O (1) rectangularprisms of dimensions 2 × × × κ .Apply Proposition 8.1 to P j , Z (cid:48)(cid:48) j , Σ (cid:48)(cid:48) j , and Γ (cid:48)(cid:48) j , with the parameters δ, s, u, c/ , and κ . We obtain a number w (cid:38) D,E ( usκc ) O (1) (cid:38) D,E ( usκ/ | log δ | ) O (1) ; sets of lines Σ (2) j ,Σ (3) j , and Σ (4) j ; and sets Z (4) j and Γ (4) j so that Σ (cid:48)(cid:48) j = Σ (2) j ∪ Σ (3) j ∪ Σ (4) j , and • The lines in Σ (2) j can be covered by O D,E ( c − O (1) s − ) = O D,E ( | log δ | O (1) s − )rectangular prisms of dimensions 2 × s × s × s . • The lines in Σ (3) j can be covered by O D,E (( sc ) − O (1) u − ) = O D,E (( | log δ | /s ) O (1) u − )rectangular prisms of dimensions 2 × × u × u . • Σ (4) j , Z (4) j and Γ (4) j are semi-algebraic of complexity O D,E (1). If Φ (4) j is the setassociated to Γ (4) j , then Z (4) j ⊂ s (Φ (4) j ) ∩ (2 ,
2) -Broad w (Φ (4) j ) , (81) E δ ( Z (4) j ) (cid:38) D,E ( scuκ ) O (1) δ − (cid:38) D,E ( suκ/ | log δ | ) O (1) δ − . (82)The sets Z (4) j , Σ (4) j , and Γ (4) j satisfy the hypotheses of Proposition 7.1. Thus thereexists a set (Σ (4) j ) (cid:48) ⊂ Σ (4) j and a quadratic polynomial Q j so that E δ (cid:0) (Σ (4) j ) (cid:48) (cid:1) (cid:38) D,E (cid:0) suκ/ | log δ | (cid:1) O (1) E δ (cid:0) Σ (4) j (cid:1) , (83)50nd for every (cid:96) (cid:48) ∈ (Σ (4) j ) (cid:48) , there is a line (cid:96) ⊂ Z ( Q j ) withdist( (cid:96), (cid:96) (cid:48) ) (cid:46) (cid:0) suκ/ | log δ | (cid:1) − O (1) δ. For i = 1 , , ,
4, define Σ i = b (cid:91) j =1 Σ ( i ) j . Then Σ = Σ ∪ Σ ∪ Σ ∪ Σ , and the sets Σ , Σ , Σ , and Σ satisfy the conclusions ofTheorem 1.2 (the set Σ (cid:48) is the set of the form (Σ (4) j ) (cid:48) that maximizes E δ (cid:0) (Σ (4) j ) (cid:48) (cid:1) ). Corollary 1.2 will be proved by combining induction on scale and re-scaling argu-ments. First, we will state a variant of Corollary 1.2 that is more amenable toinduction.
Proposition 9.1.
For each
D, ε > , there exists a constant C D,ε so that the follow-ing holds for all < δ ≤ .Let P ∈ R [ x , . . . , x ] be a polynomial of degree at most D and let Z = Z ( P ) ∩ B (0 , . Then E δ (cid:0) v (Σ δ ( Z )) (cid:1)(cid:1) ≤ C D,ε δ − − ε . Lemma 9.1.
Fix D and ε > and let δ > . Suppose that Proposition 9.1 holdsfor all values of δ (cid:48) with δ < δ (cid:48) ≤ , and let C D,ε be the associated constant. Let P be a polynomial of degree at most D , and let Z = Z ( P ) ∩ B (0 , . Let R be arectangular prism (of arbitrary orientation) that has ≤ d ≤ “long” directions and − d “short” directions; suppose that R has length in the long directions and length t in the short directions (i.e. inside B (0 , , R is comparable to the t –neighborhoodof a d -dimensional affine hyperplane). Then E δ (cid:0) v (Σ δ ( Z ∩ R )) (cid:1) ≤ C · C D,ε t − d + ε δ − − ε , (84) where C is an absolute constant.Proof. Let H be a d -dimensional hyperplane with N t ( H ) ∩ B (0 ,
1) comparable to R ∩ B (0 , H contains the origin.51irst, we can assume t ≤ / H makes an angle ≤ / e , since otherwise Σ δ ( Z ∩ N t ( H ))is empty. Apply a rotation so that H is the d -dimensional hyperplane given by x d +1 = 0 , . . . , x = 0. After applying this rotation, it is still true that every line inΣ δ ( Z ∩ N t ( H )) makes an angle ≤ / e direction. Note that the map { v ∈ S ⊂ R : ∠ ( v, e ) ≤ / } → R , ( v , v , v , v ) (cid:55)→ ( v , v , v ) , is bi-Lipschitz with constant ∼
1. In particular, if (cid:96), (cid:96) (cid:48) ∈ Σ δ ( Z ∩ N t ( H )) with ∠ ( (cid:96), (cid:96) (cid:48) ) ≥ δ , and if v = v ( (cid:96) ) , v (cid:48) = v ( (cid:96) (cid:48) ), then max( | v − v (cid:48) | , | v − v (cid:48) | , | v − v (cid:48) | ) (cid:38) δ .Note as well that if (cid:96) ∈ Σ δ ( Z ∩ N t ( H )) and if v = v ( (cid:96) ), then | v d +1 | , . . . , | v | (cid:46) t .Let f : R → R be the linear map that dilates x d +1 , . . . , x by a factor of 1 /t andleaves x , . . . , x d unchanged. Observe that if (cid:96) ∈ Σ δ ( Z ∩ N t ( H )), and if v = v ( (cid:96) ) , ˜ v = v ( f ( (cid:96) )), then v i ∼ ˜ v i /t for i = d + 1 , . . . ,
4, and v i ∼ ˜ v i for i = 2 , . . . , d . Thus if (cid:96), (cid:96) (cid:48) ∈ Σ δ ( Z ∩ N t ( H )) with ∠ ( (cid:96), (cid:96) (cid:48) ) ≥ δ , and if ˜ v = v ( f ( (cid:96) )) and ˜ v (cid:48) = v ( f ( (cid:96) (cid:48) )), thenmax( | ˜ v − ˜ v (cid:48) | , . . . , | ˜ v d − ˜ v (cid:48) d | , t | ˜ v d +1 − ˜ v (cid:48) d +1 | , . . . , t | ˜ v − ˜ v (cid:48) | ) (cid:38) δ. (85)Let L ⊂ Σ δ ( Z ∩ N t ( H )) be a set of lines pointing in δ -separated directions. Wewill “thin out” the set of lines in L by a factor of t − in each of the d − e , . . . , e d . More precisely, let L ⊂ L be a set of lines with |L | (cid:38) t d − |L | , sothat if (cid:96), (cid:96) (cid:48) ∈ L are distinct, and if v = v ( (cid:96) ) , v (cid:48) = v ( (cid:96) (cid:48) ), then at least one of t | v − v (cid:48) | , . . . , t | v d − v (cid:48) d | , or at least one of | v d +1 − v (cid:48) d +1 | , . . . , | v − v (cid:48) | ) is (cid:38) δ. By (85), we have that if (cid:96), (cid:96) (cid:48) ∈ L , then ∠ (cid:0) v ( f ( (cid:96) )) , v ( f ( (cid:96) (cid:48) )) (cid:1) ≥ δ/t . For each (cid:96) ∈ L , we have that f ( (cid:96) ) ∈ Σ δ/t (cid:0) B (0 , ∩ f ( Z ∩ N t ( H )) (cid:1) . Applying Proposition 9.1 with δ (cid:48) = δ/t and the same values of D and ε as above, weconclude that |L | ≤ C D,ε ( δ/t ) − − ε , and thus |L | (cid:46) C D,ε t − d + ε δ − − ε . Since L was an arbitrary set of lines in Σ δ ( Z ∩ N t ( H )) pointing in δ -separateddirections, we obtain (84). 52 .2 Proof of Proposition 9.1 Proof.
For each fixed value of D and ε , we will prove Proposition 9.1 by induction on δ . Fix D and ε , and suppose that Proposition 9.1 has been proved for all δ < δ (cid:48) ≤ C D,ε be the corresponding constant. We will show that if C D,ε is sufficiently large(depending only on D and ε ), then Proposition 9.1 holds for δ .Let s, κ, and u be parameters that will be determined below; for the impatientreader, s, κ and u will be of size roughly | log δ | − O ε (1). Let P be a polynomial ofdegree at most D . Apply Lemma 2.6 to the map v : Σ δ ( Z ) → S to select a semi-algebraic set Σ ⊂ Σ δ ( Z ) whose lines point in different directions. We have thatΣ ⊂ S is a semi-algebraic set of complexity O D (1), and E δ (cid:0) v (Σ) (cid:1) = E δ (cid:0) v (Σ δ ( Z )) (cid:1) . Apply Theorem 1.2 to Z ( P ) and Σ, and let Σ , Σ , Σ , Σ be the resulting sets oflines. The lines in Σ can be covered by O D ( | log δ | O (1) s − ) rectangular prisms ofdimensions 2 × s × s × s . Applying Lemma 9.1 to each of these prisms, we concludethat E δ (cid:0) v (Σ ) (cid:1) (cid:46) D (cid:16) | log δ | O (1) s − (cid:17) C D,ε s ε δ − − ε ≤ (cid:16) C D | log δ | O (1) s ε (cid:17) C D,ε δ − − ε . Thus there exist constants c > C , depending only on D and ε , so that if wedefine s = c | log δ | − C then E δ (cid:0) v (Σ ) (cid:1) ≤ C D,ε δ − − ε . (86) The lines in Σ can be covered by O D (cid:0) ( | log δ | /s ) O (1) u − (cid:1) rectangular prismsof dimensions 2 × × u × u . Applying Lemma 9.1 to each of these prisms, we concludethat E δ (cid:0) v (Σ ) (cid:1) (cid:46) D (cid:16) ( | log δ | /s ) O (1) u − (cid:17) C D,ε u ε δ − − ε ≤ (cid:16) C D | log δ | O (1) s − O (1) u ε (cid:17) C D,ε δ − − ε . Thus there exist constants c > C , depending only on D and ε (also on c and C , but this depends only on D and ε ), so that if we define u = c | log δ | − C then E δ (cid:0) v (Σ ) (cid:1) ≤ C D,ε δ − − ε . (87)53 . The lines in Σ can be covered by O D (cid:0) | log δ | O (1) (cid:1) rectangular prisms of di-mensions 2 × × × κ . Applying Lemma 9.1 to each of these prisms, we concludethat E δ (cid:0) v (Σ ) (cid:1) (cid:46) D (cid:16) | log δ | O (1) (cid:17) C D,ε κ ε δ − − ε ≤ (cid:16) C D | log δ | O (1) κ ε (cid:17) C D,ε δ − − ε . Thus there exist constants c > C , depending only on D and ε , so that if wedefine κ = c | log δ | − C then E δ (cid:0) v (Σ ) (cid:1) ≤ C D,ε δ − − ε . (88) Finally, there exists a set Σ (cid:48) ⊂ Σ and a quadratic polynomial Q so that E δ (Σ (cid:48) ) (cid:38) D ( sκu/ | log δ | ) O (1) E δ (Σ ), and for every line (cid:96) (cid:48) ∈ Σ (cid:48) , there is a line (cid:96) con-tained in Z ( Q ) with dist( (cid:96), (cid:96) (cid:48) ) (cid:46) D (cid:0) | log δ | / ( sκu ) (cid:1) O (1) δ. Note that for every quadraticpolynomial Q , we have E δ (cid:16)(cid:8) (cid:96) ∈ L : there exists (cid:96) (cid:48) ⊂ Z ( Q ) with dist( (cid:96), (cid:96) (cid:48) ) (cid:46) D ( sκu ) − O (1) δ (cid:9)(cid:17) (cid:46) D ( sκu ) − O (1) δ − , and thus E δ (cid:0) v (Σ (cid:48) ) (cid:1) (cid:46) D (cid:0) | log δ | /sκu (cid:1) − O (1) δ − , so E δ (cid:0) v (Σ ) (cid:1) (cid:46) D | log δ | O (1) ( sκu ) − O (1) δ − . Thus there exist constants c > C , depending only on D and ε (also on c , c , c , C , C , C ) , but these in turn only depend on D and ε ) so that E δ (cid:0) v (Σ ) (cid:1) ≤ c | log δ | − C δ − . If C D,ε is sufficiently large (depending only on D and ε ), then c | log δ | − C ≤ C D,ε δ − − ε ,and thus E δ (cid:0) v (Σ ) (cid:1) ≤ C D,ε δ − − ε . (89)Combining (86), (87), (88) and (89), we conclude that E δ (cid:0) v (Σ δ ( Z )) (cid:1) ≤ C D,ε δ − − ε . This completes the induction and concludes the proof.54 R δ -tube is the δ -neighborhood of a unit line segment contained in B (0 , Definition . Let T be a set of δ -tubes. For each T ∈ T , let Y ( T ) ⊂ T . We say that the tubes in T satisfy the two-ends condition with exponent ρ and error α if for all T ∈ T and for all balls B ( x, r ) of radius r , we have | Y ( T ) ∩ B ( x, r ) | ≤ αr ρ | Y ( T ) | . (90) Definition . Let T be a set of δ -tubes. Foreach T ∈ T , let Y ( T ) ⊂ T . We say that T is β –robustly transverse if for all x ∈ R and all vectors v , we have |{ T ∈ T : x ∈ Y ( T ) , ∠ ( T, v ) < β }| ≤ |{ T ∈ T : x ∈ Y ( T ) }| . (91) Definition . Let T be a set of δ -tubes. We say that T satisfies the linear Wolffaxioms if for every rectangular prism R of dimensions 1 × t × t × t , at most100 t t t δ − tubes from T can be contained in R .With these two definitions, we can now state Proposition 6.2 from [11]: Proposition 10.1 ( [11], Proposition 6.2) . For each (cid:15) > and ρ > there existconstants c , C , and D so that the following holds. Let T be a set of δ –tubes in R .Suppose that T satisfies the linear Wolff axioms. Suppose furthermore that for everyinteger ≤ E ≤ D, for every polynomial P ∈ R [ x , x , x , x ] of degree E , for everyball B ( x, r ) of radius r , and for every w > , we have |{ T ∈ T : T ∩ B ( x, r ) ⊂ N δ ( Z ) }| ≤ K E,w r − δ − − w . (92) For each T ∈ T , let Y ( T ) ⊂ T with λ ≤ | Y ( T ) | / | T | ≤ λ . Suppose that ( T , Y ) is s –robustly transverse and that each tube T ∈ T satisfies the two-ends condition withexponent ρ and error α . Then (cid:12)(cid:12)(cid:12) (cid:91) T ∈ T Y ( T ) (cid:12)(cid:12)(cid:12) ≥ c s α − C λ / K − δ − / (cid:15) (cid:0) δ | T | (cid:1) , (93) where K = max ≤ E ≤ D K E . T is a set of δ -tubes pointing in δ -separated directions, thenfor every polynomial P ∈ R [ x , x , x , x ] of degree E , for every ball B ( x, r ) of radius r , and for every w >
0, we have |{ T ∈ T : T ∩ B ( x, r ) ⊂ N δ ( Z ) }| ≤ C E,w r − ( δ/r ) − − w ≤ C E,w r − δ − − w . Since every set of δ -tubes pointing in δ -separated directions satisfies the linear Wolffaxioms, we obtain the following variant of Proposition 10.1. Proposition 10.2.
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