Improved bounds for the Kakeya maximal conjecture in higher dimensions
aa r X i v : . [ m a t h . C A ] A ug IMPROVED BOUNDS FOR THE KAKEYA MAXIMALCONJECTURE IN HIGHER DIMENSIONS
JONATHAN HICKMAN, KEITH M. ROGERS, AND RUIXIANG ZHANG
Abstract.
We adapt Guth’s polynomial partitioning argument for the Fourierrestriction problem to the context of the Kakeya problem. By writing out theinduction argument as a recursive algorithm, additional multiscale geomet-ric information is made available. To take advantage of this, we prove thatdirection-separated tubes satisfy a multiscale version of the polynomial Wolffaxioms. Altogether, this yields improved bounds for the Kakeya maximal con-jecture in R n with n = 5 or n > Introduction
For n > δ >
0, a δ -tube is a cylinder T ⊂ R n of unit heightand radius δ , with arbitrary position and arbitrary orientation dir( T ) ∈ S n − . Afamily T of δ -tubes is direction-separated if { dir( T ) : T ∈ T } forms a δ -separatedsubset of the unit sphere. Conjecture 1.1 (Kakeya maximal conjecture) . Let p > nn − . For all ε > , thereexists a constant C ε,n > such that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) L p ( R n ) C ε,n δ − ( n − − n/p ) − ε (cid:16) X T ∈ T | T | (cid:17) /p ( K p ) whenever < δ < and T is a direction-separated family of δ -tubes. By an application of H¨older’s inequality, one may readily verify that if ( K p )holds for p = nn − , then, for all ε >
0, there exists a constant c ε,n > (cid:12)(cid:12) [ T ∈ T T (cid:12)(cid:12) > c ε,n δ ε X T ∈ T | T | . This can be interpreted as the statement that any direction-separated family of δ -tubes is ‘essentially disjoint’. A more refined argument shows that if ( K p ) holdsfor a given p , then every Kakeya set in R n (that is, every compact set that containsa unit line segment in every direction) has Hausdorff dimension at least p ′ , the con-jugate exponent of p . Thus, Conjecture 1.1 would imply the Kakeya set conjecture ,that Kakeya sets in R n have Hausdorff dimension n ; see, for instance, [5, 42, 28].For n = 2, the set conjecture was proven by Davies [12] and the maximal con-jecture was proven by C´ordoba [11] in the seventies. Both conjectures remainchallenging and important open problems in higher dimensions; for partial results,see [13, 9, 10, 5, 41, 34, 39, 6, 26, 24, 31, 27, 4, 14, 16, 17, 15, 29] and referencestherein. Supported by the MINECO grants SEV-2015-0554 and MTM2017-85934-C3-1-P and the ERCgrant 834728. n = p > n = p > / / − ε Katz–Zahl [29, 30] 10 13 /
11 Theorem 1.24 1 . ... Katz–Zahl [30] 11 7 / /
13 Theorem 1.2 12 31 /
27 Theorem 1.26 4 / /
93 Theorem 1.27 34 /
27 Theorem 1.2 14 9 / /
17 Theorem 1.2 15 47 /
42 Theorem 1.2
Figure 1.
The state-of-the-art for the Kakeya maximal conjecturein low dimensions. New results are highlighted.In 1999, Bourgain [6] improved the state-of-the-art in higher dimensions usingsum-difference theory from additive combinatorics. This technique was refinedby Katz and Tao [26, 27, 28], proving that Conjecture 1.1 is true in the range p >
74 1 n − . The purpose of the present article is to extend this range using adifferent approach. Theorem 1.2.
Conjecture 1.1 is true in the range p > k n max n n ( n − n + ( k − k , n − k + 1 o . (1)When k = n , the first entry of the maximum of (1) takes the conjectured value;however, the second entry only reaches this value at the other extreme, when k = 2.A reasonable compromise can be found by taking k to be the closest integer to( √ − n + 1, at which point we find, for instance, that the Kakeya maximalconjecture holds in the range p > − √ n − , (2)which is an improvement over the Katz–Tao maximal bound [27]. See Figure 1 forthe state-of-the-art in low dimensions. Theorem 1.2 also implies improved boundsfor the Kakeya set conjecture in certain dimensions. Further discussion of thenumerology is contained in the final section of the article.The proof of Theorem 1.2 is based on the polynomial method , which was intro-duced in the context of the Kakeya problem by Dvir in his celebrated proof [14]of Wolff’s finite field Kakeya conjecture [42]. The polynomial method has beenadapted to analyse Kakeya sets in Euclidean space in, for instance, works of Guth[16, 17] and Guth and Zahl [22]. A key tool here is polynomial partitioning , in-troduced by Guth and Katz in their resolution of the two dimensional Erd˝os dis-tance conjecture [21]. Of most relevance to the present article is the recent workof Guth [18, 19] which adapted the partitioning technique to the context of theFourier restriction problem.In [18, 19, 22], polynomial partitioning was used to study collections of direction-separated tubes. This led to the consideration of configurations of tubes that are In all dimensions the range (1) is in fact strictly larger than (2): the latter is included to providea ready comparison with the maximal bounds from [27].
MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 3 partially contained in the neighbourhood of a real algebraic variety. Guth provedthe following cardinality estimate for direction-separated tubes in three dimensions[18, Lemma 4.9] and conjectured that it should hold in higher dimensions [19,Conjecture B.1]. This was confirmed by Zahl [44] in four dimensions and then ingeneral by Katz and the second author [25].
Theorem 1.3 ([18, 44, 25]) . For all n > k > , d > and ε > , there is aconstant C n,d,ε > such that n T ∈ T : | T ∩ B λ k ∩ N ρ Z k | > λ k | T | o C n,d,ε (cid:16) ρλ k (cid:17) n − k δ − ( n − − ε whenever < δ ρ λ k , T is a direction-separated family of δ -tubes and Z k ⊂ R n is a k -dimensional algebraic variety of degree d . Here N r E denotes the r -neighbourhood of E for any r > E ⊆ R n and B r is a choice of ball in R n of radius r . The relevant algebraic definitions are recalledin Section 3.1 below. In the language of [19], this theorem states that direction-separated tubes satisfy the polynomial Wolff axioms ; this terminology is recalledand discussed in further detail in the final section of the paper.After adapting Guth’s restriction argument [18, 19] to the context of the Kakeyamaximal problem, one finds that Theorem 1.3 can be used to obtain improvedbounds in certain intermediate dimensions: see the final section for more details.However, by rewriting Guth’s induction argument as a recursive algorithm, one isreadily able to take advantage of the the following strengthened version of Theo-rem 1.3. Theorem 1.4.
For all n > m > k > , d > and ε > , there is a constant C n,d,ε > such that m \ j = k n T ∈ T : | T ∩ B λ j ∩ N ρ Z j | > λ j | T | o C n,d,ε (cid:16) m − Y j = k ρλ j (cid:17)(cid:16) ρλ m (cid:17) n − m δ − ( n − − ε whenever < δ ρ λ k . . . λ m , T is a direction-separated family of δ -tubes, Z j ⊂ R n are j -dimensional algebraic varieties of degree d and the balls B λ j are nested: B λ k ⊆ . . . ⊆ B λ m ⊂ R n . Taking the varieties Z j to be nested j -planes reveals that the cardinality estimateof Theorem 1.4 is sharp up to the factor of C n,d,ε δ − ε . The proof will follow theargument of [25] once a relevant Wongkew-type volume bound (in the spirit of[43]) has been established. The mixture of trigonometric and algebraic argumentsinvolved in the proof of this volume bound constitutes the most novel part of thearticle. Remark 1.5.
In a late stage of the development of this project, the authors discov-ered that J. Zahl has proved the same maximal results as Theorem 1.2 using similarmethods. In particular, J. Zahl has independently established Theorem 1.4 and,moreover, was able to use this result to prove a strengthened version of Theorem 4.1involving k -linear (as opposed to k -broad) estimates.The remainder of the article is organised as follows: • In Section 2 some notational conventions are fixed.
JONATHAN HICKMAN, KEITH M. ROGERS, AND RUIXIANG ZHANG • In Section 3 the proof of Theorem 1.4 is presented after first establishingthe relevant Wongkew-type volume bound. • In Section 4 the proof of Theorem 1.2 is reduced to estimating the so-called k -broad norms for the Kakeya maximal function, paralleling work onoscillatory integrals from [7, 18, 19]. • In Section 5 basic properties of k -broad norms are reviewed. • In Section 6 the polynomial partitioning theorem from [19] is recalled andapplied to the k -broad norms. • In Section 7 the recursive algorithm is described, culminating in a structuralstatement of algebraic nature for the Kakeya maximal problem. • In Section 8 the structural statement is combined with Theorem 1.4 toconclude the proof of Theorem 1.2. • In Section 9 the applications to the Kakeya set conjecture and other relatedproblems are discussed. • Appended is a review of some facts from real algbraic geometry used inSection 3.
Acknowledgement.
The first author thanks both Larry Guth and Joe Karmazynfor helpful discussions during the development of this project.2.
Notational conventions
We call an n -dimensional ball B r of radius r an r -ball . The intersection of S n − with a ball is called a cap . The δ -neighbourhood of a set E will be denoted by N δ E .The arguments will involve the admissible parameters n , p and ε and the con-stants in the estimates will be allowed to depend on these quantities. Moreover, anyconstant is said to be admissible if it depends only on the admissible parameters.Given positive numbers A, B > L , the notation A . L B , B & L A or A = O L ( B ) signifies that A C L B where C L is a constant whichdepends only on the objects in the list and the admissible parameters. We write A ∼ L B when both A . L B and B . L A .The cardinality of a finite set A is denoted by A . A set A ′ is said to be a refinement of A if A ′ ⊆ A and A ′ & A . In many cases it will be convenientto pass to a refinement of a set A , by which we mean that the original set A isreplaced with some refinement.3. Multiscale polynomial Wolff axioms: Proof of Theorem 1.4
In this section we prove Theorem 1.4. A minor modification of the argumentused to prove Theorem 1.3 in [25] reduces matters to establishing a “Wongkew-typelemma”. The details of this reduction are described in Section 3.3 below. In thesimplest case where k = m (which corresponds to Theorem 1.3), after the reductionall that is needed is Wongkew’s original lemma [43], which is used to bound thevolume of the semialgebraic set Z k ∩ B λ k . In the general case the problem is toobtain bounds for the volume of other semialgebraic sets S m ( I m , ρ ) which do notfall directly under the scope of [43]. These sets arise from the multiscale hypothesesand are defined in Section 3.2.3.1. Algebraic definitions.
Before continuing, it is perhaps useful to clarify someof the terminology featured in the statement of Theorem 1.4 and also in the proof.
MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 5
Definition 3.1.
A set Z ⊆ R n will be referred to as a variety if it can be expressedas Z = Z ( P , . . . , P r ) for a collection of polynomials P i : R n → R for 1 i r where Z ( P , . . . , P r ) := { x ∈ R n : P ( x ) = · · · = P r ( x ) = 0 } . (3)For the case of interest (namely, where Z is a transverse complete intersection :see Definition 5.1 below), Z will always be a real smooth submanifold of R n . Herethe dimension dim Z is defined to be the dimension of Z as a real smooth manifold.The results of this section hold for more general varieties which potentially admitsingular points, with a suitably generalised definition of dimension, although wewill not discuss the details of this definition here (see, for instance, [1]). Definition 3.2.
Given a variety Z the degree of Z isdeg Z := inf r X j =1 deg P j , where the infimum is taken over all possible representations of Z of the form (3).The proof of Theorem 1.4 will involve the analysis of a more general class of sets. Definition 3.3.
A set S ⊂ R n is semialgebraic if there exists a finite collection ofpolynomials P i,j , Q i,j : R n → R for 1 i r , 1 j s such that S = r [ i =1 (cid:8) x ∈ R n : P i, ( x ) = · · · = P i,s ( x ) = 0 , Q i, ( x ) > , . . . , Q i,s ( x ) > (cid:9) . (4) Definition 3.4.
Given a semialgebraic set S ⊂ R n the complexity of S isinf (cid:16) X i,j deg P i,j + deg Q i,j (cid:17) where the infimum is taken over all possible representations of S of the form (4).A number of fundamental results in the theory of semialgebraic sets will be usedin the proof of Theorem 1.4, including the Tarski–Seidenberg projection theoremand Gromov’s algebraic lemma. For the reader’s convenience, the relevant state-ments are recorded in the appendix.3.2. A Wongkew-type lemma.
The main new ingredient in the proof of The-orem 1.4 will be a bound for the Lebesgue measure of certain semialgebraic sets S m ( I m , ρ ) given by unions of line segments. Before defining these sets some basicreductions are made and some useful notion is introduced.We choose our coordinates in such a way that the λ m -ball B λ m is centred at theorigin and a reasonably large proportion of our direction-separated δ -tubes havecore lines which can be parametrised by l a , d ( t ) := ( a ,
0) + t ( d , , t ∈ R , for some a , d ∈ [ − , n − . Then, for each j = k, . . . , m , we partition the orthog-onal projection of B λ j onto the t -axis into 4 nd disjoint intervals I j ⊂ [ − ,
1] oflength λ j / (2 nd ), where d bounds the degree of our varieties Z k , . . . , Z m . Note that here, in contrast with much of the algebraic geometry literature, the ideal generatedby the P i is not required to be irreducible. JONATHAN HICKMAN, KEITH M. ROGERS, AND RUIXIANG ZHANG N ρ Z j ∩ B λ j l a , d ( I j ) JI j S m ( J, ρ ) l a , d ( J ) Figure 2.
The set S m ( J, ρ ) is formed by a union of line segments l a , d ( J ) which have the property that l a , d ( I j ) ⊆ N ρ Z j ∩ B λ j for k j m .Given any interval J ⊆ R , we define S m ( J, ρ ) := m \ j = k (cid:8) l a , d ( t ) : t ∈ J, ( a , d ) ∈ [ − , n − , l a , d ( I j ) ⊆ N ρ Z j ∩ B λ j (cid:9) ;see Figure 2 for a diagrammatic description of this set. The key problem will be toestimate the measure of these sets. Note that the measure of S m ( I m , ρ ) dependson the specific choice of I k , . . . , I m ˚; however˚, our bounds will be uniform over anychoice and so we suppress this dependence in the notation. An example of sucha bound follows from the m -dimensional version of Wongkew’s theorem [43] (seeTheorem A.1 in the appendix), which immediately implies that | S m ( I m , ρ ) | | N ρ Z m ∩ B λ m | . d λ mm ρ n − m . (5)This estimate only uses the m -dimensional information, and our first task is toimprove this bound using the additional lower dimensional information.In order to improve (5), we will consider both S ℓ ( I ℓ , ρ ) and S ℓ ( I ℓ +1 , ρ ), the latterof which need not be contained in either N ρ Z ℓ ∩ B λ ℓ or N ρ Z ℓ +1 ∩ B λ ℓ +1 . Roughlyspeaking, there are two steps to the argument: Step 1 : We bound | S ℓ +1 ( I ℓ +1 , ρ ) | in terms of | S ℓ ( I ℓ +1 , ρ ) | using trigonometry andWongkew’s theorem [43]. Step 2 : We bound | S ℓ ( I ℓ +1 , ρ ) | in terms of | S ℓ ( I ℓ , ρ ) | using an algebraic argumentthat borrows ideas from [25].Iterating these steps yields a bound for | S m ( I m , ρ ) | in terms of | S k ( I k , m − k ρ ) | ˚, atwhich point we can use the k -dimensional version of Wongkew’s theorem ratherthan the m -dimensional version. The resulting bound is presented in the followinglemma. Lemma 3.5.
For all n > m > k > , d > and ε > , | S m ( I m , ρ ) | . d ρ − ε (cid:16) m − Y j = k ρλ j (cid:17) λ mm ρ n − m MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 7 whenever < ρ/ λ k . . . λ m , the j -dimensional varieties Z j ⊂ R n havedegree d and B λ k ⊂ . . . ⊂ B λ m ⊂ R n . Taking the j -dimensional varieties Z j to be nested j -planes reveals that theestimate is sharp up to the factor of C n,d,ε ρ − ε . Proof (of Lemma 3.5).
The proof is somewhat involved and is broken into stages.
Initial reductions.
We may assume without loss of generality that ρ k − m +1 λ k . (6)Indeed, otherwise there exists a largest k ′ such that k + 1 k ′ m + 1 and ρ > k − m +1 λ j for all k j k ′ −
1. If k ′ = m + 1, then the result is trivial. If k ′ < m + 1, then we may drop the j th condition in S m ( I m , ρ ) for k j k ′ − k ′ as k , (6) now holds.It will also be useful to assume that the intervals I j have lengths given by somedyadic number: that is, λ j nd ∈ Z . (7)This is possible by slightly enlarging the set by appropriately rounding up the λ j ’s. Setting up the induction.
For all k ℓ m we will prove that | S ℓ ( I ℓ , ρ ) | . d ρ − ε (cid:16) ℓ − Y j = k ρλ j (cid:17) λ ℓℓ ρ n − ℓ (8)whenever ρ k − ℓ +1 λ k and the λ j satisfy (7). To do this, we induct on ℓ . Fortechnical reasons, it will be useful to slightly enlarge the sets by redefining S ℓ ( J, ρ ) := ℓ \ j = k (cid:8) l a , d ( t ) : t ∈ J, ( a , d ) ∈ Q n − ( ρ ) , l a , d ( I j ) ⊆ N ρ Z j ∩ B λ j + ρ (cid:9) where Q n − ( ρ ) := [ − − ρ, ρ ] n − . Clearly, any bound of the form (8) forthese enlarged sets implies the same bound holds for the original S ℓ ( I ℓ , ρ ).By the k -dimensional version of Wongkew’s theorem [43] (see Theorem A.1), | S k ( I k , ρ ) | . d λ kk ρ n − k whenever ρ λ k and this serves as the base case for the induction argument.Assuming (8) holds for some k ℓ m −
1, it suffices to prove that | S ℓ +1 ( I ℓ +1 , ρ ) | . d ρ − ε (cid:16) λ ℓ +1 λ ℓ (cid:17) ℓ +1 | S ℓ ( I ℓ , ρ ) | whenever ρ k − ℓ λ k and λ j / (2 nd ) ∈ Z . We may also assume the non-degeneracyhypothesis that | S ℓ +1 ( I ℓ +1 , ρ ) | > (cid:16) ℓ Y j = k ρλ j (cid:17) λ ℓ +1 ℓ +1 ρ n − ℓ − > λ ℓ +1 ρ n − , (9)as otherwise the induction step would have closed already. JONATHAN HICKMAN, KEITH M. ROGERS, AND RUIXIANG ZHANG
Dyadic decomposition.
Recall from our initial reductions that the I j are dyadicintervals. To prove the induction step we partition I ℓ +1 into the part close to I ℓ , { t ∈ I ℓ +1 : dist( t, I ℓ ) | I ℓ |} , (10)and dyadic parts further from I ℓ , (cid:8) t ∈ I ℓ +1 : 2 i | I ℓ | dist( t, I ℓ ) i +1 | I ℓ | (cid:9) , i > . (11)Let J denote the collection of all maximal dyadic subintervals of the sets in (10)or (11). We have S ℓ +1 ( I ℓ +1 , ρ ) = [ J ∈J S ℓ +1 ( J, ρ ) ⊂ [ J ∈J S ℓ ( J, ρ ) ∩ N ρ Z ℓ +1 , where the final inclusion follows directly from the definitions. Since the J ∈ J arecontained in I ℓ +1 and are pairwise disjoint, | S ℓ +1 ( I ℓ +1 , ρ ) | λ ℓ +1 ρ n − + X J ∈J| S ℓ ( J,ρ ) | > | J | ρ n − | S ℓ ( J, ρ ) ∩ N ρ Z ℓ +1 | . By (9), the first term on the right-hand side of the above display is at most halfthe term on the left-hand side. Thus, it suffices to estimate the right-hand sum.Given that the balls are nested, B λ k ⊂ . . . ⊂ B λ m ⊂ R n , we havemaxdist( I ℓ , J ) := sup (cid:8) | t − t ′ | : t ∈ I ℓ , t ′ ∈ J (cid:9) . λ ℓ +1 , so there are no more than 2 log( λ ℓ +1 / | I ℓ | ) . d log( ρ − ) intervals J ∈ J . Thus, itwill suffice to prove that | S ℓ ( J, ρ ) ∩ N ρ Z ℓ +1 | . d ρ − ε (cid:16) | J || I ℓ | (cid:17) ℓ +1 | S ℓ ( I ℓ , ρ ) | , (12)whenever J ∈ J satisfies | S ℓ ( J, ρ ) | > | J | ρ n − . Inductive step: the first bound.
We now turn to the precise version of Step 1 fromthe proof sketch at the beginning of the section.
Lemma 3.6. If J ∈ J satisfies dist( I ℓ , J ) > | J | , then | S ℓ ( J, ρ ) ∩ N ρ Z ℓ +1 | . d (cid:16) | J || I ℓ | (cid:17) ℓ +1 − n | S ℓ ( J, ρ ) | . Proof.
We first claim that it is possible to cover S ℓ ( J, ρ ) by a collection B of ballsof radius ρ | J | / | I ℓ | with cardinality B . d (cid:16) | I ℓ | ρ | J | (cid:17) n | S ℓ ( J, ρ ) | . (13)Temporarily assuming that this is so, one may argue as follows. For each of theballs B ∈ B one may apply Wongkew’s theorem [43] (see Theorem A.1) to deducethat | B ∩ N ρ Z ℓ +1 | . d (cid:16) ρ | J || I ℓ | (cid:17) ℓ +1 ρ n − ( ℓ +1) . MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 9 x n l a , d ( I ℓ ) l ˜a , ˜d ( I ℓ ) JI ℓ . ρ . ρ | J || I ℓ |∼ | J | y ∈ l ˜a , ˜d ( J ) x ∈ l a , d ( J ) z Figure 3.
The trigonometric argument.Thus, by (13), altogether we find that | S ℓ ( J, ρ ) ∩ N ρ Z ℓ +1 | X B ∈B | B ∩ N ρ Z ℓ +1 | . d (cid:16) ρ | J || I ℓ | (cid:17) ℓ +1 ρ n − ( ℓ +1) (cid:16) | I ℓ | ρ | J | (cid:17) n | S ℓ ( J, ρ ) | , as desired.It remains to verify the claim. Letting r ℓ := ρ | J | / (4 nd | I ℓ | ), by an elementarycovering argument it suffices to show that N r ℓ S ℓ ( J, ρ ) ∩ (cid:0) R n − × J (cid:1) ⊆ S ℓ ( J, ρ ) . (14)Fix a point y ∈ N r ℓ S ℓ ( J, ρ ) ∩ (cid:0) R n − × J (cid:1) so there exists some x ∈ S ℓ ( J, ρ ) with | x − y | < r ℓ . Furthermore, by the definition of S ℓ ( J, ρ ), there exists some ( a , d ) ∈ Q n − ( ρ ) and t ∈ J such that x = l a , d ( t ) and l a , d ( I j ) ⊆ N ρ Z j ∩ B λ j . Let z denote the midpoint of the line segment l a , d ( I ℓ ) and θ the angle ∠ xzy ; see Figure 3.The separation between J and I ℓ implies that | x − z | , | y − z | > | J | and therefore | tan θ | r ℓ | J | = 14 nd · ρ | I ℓ | . (15)The line passing through z and y can be parametrised by t l ˜a , ˜d ( t ) for some choiceof ( ˜a , ˜d ) ∈ Q n − (2 ρ ) and y = l ˜a , ˜d ( t ) for some t ∈ J . Moreover, the angle bound(15) implies that the segment l ˜a , ˜d ( I j ) is contained in a ρ -neighbourhood of l a , d ( I j )for k j ℓ . Thus, l ˜a , ˜d ( I j ) ⊆ N ρ Z j ∩ B λ j +2 ρ for k j ℓ and, consequently, y ∈ l ˜a , ˜d ( J ) ⊆ S ℓ ( J, ρ ) . This establishes (14) and concludes the proof. (cid:3)
Inductive step: the second bound.
We now turn to the precise version of Step 2 fromthe proof sketch at the beginning of the section. Loosely speaking, the followinglemma tells us that our line segments can never expand at an unexpectedly fastrate, even after leaving the constricted region.
Lemma 3.7. If J ∈ J satisfies | S ℓ ( J, ρ ) | > | J | ρ n − , then | S ℓ ( J, ρ ) | . d ρ − ε (cid:16) | J || I ℓ | (cid:17) n | S ℓ ( I ℓ , ρ ) | . To prove Lemma 3.7, we will apply the following elementary lemma which statesthat, although it is not possible to bound a polynomial at a point in terms of thevalue that it takes at another point (which could be a root), such a bound holds onaverage.
Lemma 3.8.
Let P : R → R be a polynomial of degree m , I ⊂ R be an intervaland t ∈ R . Then | P ( t ) | (cid:16) m max {| I | , dist( t, I ) }| I | (cid:17) m | I | ˆ I | P ( t ′ ) | dt ′ . The simple proof of this result is postponed until the end of the subsection.At this point it is also worth recalling that the ρ -neighbourhoods N ρ Z j of alge-braic varieties Z j = Z ( P , . . . , P n − j ) are semialgebraic sets. To see this we considerthe auxiliary set Y j = n ( x , y ) ∈ R n : P ( x ) , . . . , P n − j ( x ) = 0 , | y − x | < ρ o which is clearly semiaglebraic. Then the Tarski–Seidenberg theorem (see Theo-rem A.2) tells us that the orthogonal projection Π( Y j ) = N ρ Z j , where Π : ( x , y ) y , is also semialgebraic with compexity bounded in terms of n and d . Proof (of Lemma 3.7).
Consider slices of S ℓ ( J, ρ ) of the form S ℓ ( J, ρ ) t := S ℓ ( J, ρ ) ∩ (cid:0) R n − × { t } (cid:1) , t ∈ R , so that, by Fubini’s theorem, | S ℓ ( J, ρ ) | | J | ρ n − + ˆ { t ∈ J : | S ℓ ( J,ρ ) t | > ρ n − } | S ℓ ( J, ρ ) t | dt. By the hypothesis of the lemma, the first term on the right-hand side is at mosthalf the left-hand term. Therefore, is suffices to prove that | S ℓ ( J, ρ ) t ℓ | . d ρ − ε (cid:16) | J || I ℓ | (cid:17) n − | S ℓ ( I ℓ , ρ ) || I ℓ | (16)whenever t ℓ ∈ J and | S ℓ ( J, ρ ) t ℓ | > ρ n − . In order to prove (16), we write a ′ = a + t ℓ d and l ′ a ′ , d ( t ) := ( a ′ + ( t − t ℓ ) d , t ) sothat l ′ a ′ , d ( t ) = l a , d ( t ) and S ℓ ( J, ρ ) can be rewritten as ℓ \ j = k (cid:8) l ′ a ′ , d ( t ) : t ∈ J, ( a ′ − t ℓ d , d ) ∈ Q n − ( ρ ) , l ′ a ′ , d ( I j ) ⊆ N ρ Z j ∩ B λ j + ρ (cid:9) . Consider the associated sets of lines L ℓ ( ρ, t ℓ ) ≡ L ℓ ( ρ, t ℓ , I k , . . . , I m ) defined by L ℓ ( ρ, t ℓ ) := ℓ \ j = k (cid:8) ( a ′ , d ) : ( a ′ − t ℓ d , d ) ∈ Q n − ( ρ ) , l ′ a ′ , d ( I j ) ⊆ N ρ Z j ∩ B λ j + ρ (cid:9) . From the definitions,( a ′ , d ) ∈ L ℓ ( ρ, t ℓ ) if and only if l ′ a ′ , d ( J ) ⊆ S ℓ ( J, ρ ) (17)and, in particular, if either of these equivalent statements holds, then a ′ ∈ S ℓ ( J, ρ ) t ℓ .Recall from our earlier discussion that the sets N ρ Z j ∩ B λ j + ρ are semialgebraic.By quantifier elimination (that is, the Tarski–Seidenberg theorem), the sets L ℓ ( ρ, t ℓ )are also semialgebraic (see [25, Lemma 1.1] for an argument of this type). By anapplication of Lemma 2.2 of [25] (see also Corollary A.3 of the appendix), we can MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 11 t ℓ ∈ J l ′ x t ℓ ∈ J ( F ( x ) , t ℓ ) Figure 4.
Forming a semialgebraic section of the lines. Roughlyspeaking, the slice S ℓ ( J, ρ ) t ℓ (shown as a blue vertical line above)is parametrised by a polynomial mapping F : R n − → R n − . Wecan find another polynomial mapping G : R n − → R n − which“selects” a single line through each point ( F ( x ) , t ℓ ) ∈ S ℓ ( J, ρ ) t ℓ .Indeed, the line l ′ x := { ( F ( x ) + ( t − t ℓ ) G ( x ) , t ) : t ∈ R } has thisproperty.take a semialgebraic section of L ℓ ( ρ, t ℓ ) with complexity bounded by C ( n, d ), so thatthere is only one direction d for each possible position a ′ (this is in contrast with [25],where the section was taken to leave only one position for each direction). Callingthis section L ′ ℓ ( ρ, t ℓ ), we may use Gromov’s algebraic lemma (see Lemma A.4), as in[25, Section 3], to parametrise L ′ ℓ ( ρ, t ℓ ). In particular, taking s to be the first integerlarger than 2 n /ε , there exists some N ∈ N , depending only on the dimension n ,degree d and ε , and a collection of C s functions F i , G i : [0 , n − → R n − for1 i N such that:i) N [ i =1 ( F i , G i )([0 , n − ) = L ′ ℓ ( ρ, t ℓ ),ii) sup | α | s k ∂ α F i k ∞ , sup | α | s k ∂ α G i k ∞ , i = 1 , . . . , N .Again following [25, Section 3], we partition [0 , n − into cubes Q of smalldiameter cρ ε/n , with c to be chosen below. On each cube Q , we approximate the C s functions F i , G i : [0 , n − → R n − by polynomials F iQ , G iQ : R n − → R n − ofdegree s using Taylor’s theorem. Indeed, letting y Q denote the centre of Q , Taylor’stheorem yields polynomials that satisfy | F i ( y ) − F iQ ( y ) | , | G i ( y ) − G iQ ( y ) | s ! | y − y Q | s c s ρ n , y ∈ Q. (18) Using (17) and unpacking all the definitions, S ℓ ( J, ρ ) t ℓ ⊆ N [ i =1 [ Q F i ( Q ) . Furthermore, by (18), the boundary of F i ( Q ) belongs to the c s ρ n -neighbourhoodof the boundary of F iQ ( Q ) and, in particular, F i ( Q ) ⊆ N c s ρ n F iQ ( ∂Q ) ∪ F iQ ( Q ) . The set F iQ ( ∂Q ) is contained in a union of 2 n algebraic hypersurfaces so that, byWongkew’s theorem [43] (see Theorem A.1), | F i ( Q ) | C ( n, s ) c s ρ n + | F iQ ( Q ) | . By taking c sufficiently small, depending only on n , d and ε , | S ℓ ( J, ρ ) t ℓ | ρ n − + N X i =1 X Q | F iQ ( Q ) | (19)and, by the nondegeneracy hypothesis | S ℓ ( J, ρ ) t ℓ | > ρ n − , we have | S ℓ ( J, ρ ) t ℓ | N X i =1 X Q |S iQ ( J ) t ℓ | (20)where S iQ ( J ) := n ( F iQ ( y ) + ( t − t ℓ ) G iQ ( y ) , t ) ∈ R n − × J : y ∈ Q o . On the other hand, we also have that S iQ ( I ℓ ) ⊆ S ℓ ( I ℓ , ρ ). Indeed, fixing y ∈ Q ,it follows from the definition of the F i and G i , (17) and (18) that( F iQ ( y ) + ( t − t ℓ ) G iQ ( y ) , t ) ∈ N ρ Z j ∩ B λ j +2 ρ for all t ∈ I j and k j ℓ .In particular, if t ∈ I ℓ then ( F iQ ( y ) + ( t − t ℓ ) G iQ ( y ) , t ) ∈ S ℓ ( I ℓ , ρ ). Given that thereare fewer than C ( n, d, ε ) ρ − ε summands in (20), it therefore suffices to show |S iQ ( J ) t ℓ | . d (cid:16) | J || I ℓ | (cid:17) n − | S iQ ( I ℓ ) || I ℓ | (21)for any fixed choice of i and Q . Suppose F, G : R n − → R n − are polynomials ofdegree at most s such that det DF is not the zero polynomial, where DF denotesthe ( n − × ( n −
1) Jacobian matrix of F . It thus suffices to prove, more generally,that |S ( J ) t ℓ | (8( n − s ) n − (cid:16) max {| I | , maxdist( I, J ) }| I | (cid:17) n − |S ( I ) || I | (22)where I, J ⊆ R are arbitrary intervals, Q ⊂ [0 , n − is any measureable set and S ( I ) := n ( F ( y ) + ( t − t ℓ ) G ( y ) , t ) ∈ R n − × I : y ∈ Q o . Indeed, it follows from (19) that the polynomials det DF iQ are not zero and for thechoice of intervals I ℓ and J above we have max {| I ℓ | , maxdist( I ℓ , J ) } | J | . Hence(21) follows as a special case of (22).Now, by B´ezout’s theorem, F + ( t − t ℓ ) G is at most s n − -to-one on Q t = (cid:8) y ∈ Q : det( DF + ( t − t ℓ ) DG )( y ) = 0 (cid:9) . MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 13
Furthermore, since, by hypothesis, the polynomial ( y , t ) det( DF + tDG )( y ) isnon-zero, it follows by Fubini’s theorem that Q \ Q t is a Lebesgue null set for almostevery t ∈ R . Consequently,1 s n − ˆ I ˆ Q | det( DF + ( t − t ℓ ) DG )( y ) | d y dt ˆ I | ( F + ( t − t ℓ ) G )( Q t ) | dt |S ( I ) | . On the other hand, by an application of Lemma 3.8, we have that |S ( J ) t ℓ | = | F ( Q ) | ˆ Q | det DF ( y ) | d y (cid:0) n − (cid:1) n − max {| I | , maxdist( I, J ) } n − | I | n ˆ Q ˆ I | det( DF + ( t − t ℓ ) DG )( y ) | dtd y . Combining these displayed inequalities, via an application of Fubini’s theorem,yields (22) which completes the proof. (cid:3)
Closing the induction.
By the initial reductions, to close the inductive step (andthereby finish the proof of Lemma 3.5), it suffices to show (12). There are two casesto consider: • If J is a subinterval of (10), then | J | = | I ℓ | and maxdist( I ℓ , J ) | I ℓ | . Inthis case, (12) immediately follows from Lemma 3.7. • If J is a subinterval of one of the sets in (11), then dist( I ℓ , J ) = | J | andmaxdist( I ℓ , J ) | J | . In this case, (12) follows from a successive applica-tion of Lemma 3.6 and Lemma 3.7.This concludes the proof of Lemma 3.5. (cid:3) The elementary polynomial bound.
It remains to prove the elementary Lemma 3.8,which was used in the proof of Lemma 3.7.
Proof (of Lemma 3.8).
By translating so that I = [ − λ, λ ] for some λ >
0, factoris-ing the resulting polynomial, scaling t → t/λ and using the fact that the resultinginequality is symmetric over the origin, this reduces to proving | ( t − z ) · · · ( t − z m ) | (cid:0) m max {| t − | , } (cid:1) m ˆ − | ( t ′ − z ) · · · ( t ′ − z m ) | dt ′ whenever z , . . . , z k ∈ C . Supposing that | z | , . . . , | z k | > | z k +1 | , . . . , | z m | < ˆ − | ( t ′ − z ) · · · ( t ′ − z m ) | dt ′ > (cid:16) (cid:17) k | z | · · · | z k | ˆ − | ( t ′ − z k +1 ) · · · ( t ′ − z m ) | dt ′ > (cid:16) (cid:17) k | z | · · · | z k | (cid:16) m − k ) (cid:17) m − k , (23)where the second inequality follows because most values of t ′ ∈ [ − ,
1] must bereasonably far from the roots. Now the small roots, when j = k + 1 , . . . , m , satisfy | t − z j | | t − | + | − z j | {| t − | , } , and the large roots, when j = 1 , . . . , k , satisfy | z j || t − z j | > | z j || t − | + | − z j | > min n | z j | | t − | , | z j | | − z j | o >
12 max {| t − | , } . Together we find that | z | · · · | z k | > (cid:16)
14 max {| t − | , } (cid:17) m | ( t − z ) · · · ( t − z m ) | which can be plugged into (23) to complete the proof. (cid:3) Proof of Theorem 1.4.
Theorem 1.4 now follows by a minor adaptation ofthe argument from [25], applying Lemma 3.5 in one key step.
Proof (of Theorem 1.4).
Note first that when | T ∩ B λ j ∩ N ρ Z j | > λ j | T | , (24)there necessarily exists a line in the direction of T for which the one-dimensionalLebesgue measure of the line intersected with B λ j ∩ N ρ Z j is greater than or equalto λ j . By B´ezout’s theorem, this line can cross Z j at most d times, so that if T ∩ B λ j ∩ N ρ Z j satisfies (24), it must contain a line segment in the direction of T of length λ j / ( d + 1). Fattening this line segment, we obtain a truncated δ -tubecontained in B λ j ∩ N ρ Z j that projects onto an interval in the t -axis of length > λ j / ( nd ). This interval must contain one of the intervals I j of length λ j / (2 nd ) withwhich we partitioned the orthogonal projection of B λ j . Recalling that L m (2 ρ, , I k , . . . , I m ) := m \ j = k n ( a , d ) ∈ [ − , n − : l a , d ( I j ) ⊆ N ρ Z j ∩ B λ j o , we find that δ n − m \ j = k (cid:8) T ∈ T : | T ∩ B λ j ∩ N ρ Z j | > λ j | T | (cid:9) . X I k ,...,I m (cid:12)(cid:12) Π (cid:0) L m (2 ρ, , I k , . . . , I m ) (cid:1)(cid:12)(cid:12) , where Π : ( a , d ) d denotes the orthogonal projection onto the directions. This isbecause, for each of the δ -tubes of the original discrete set, there is a whole δ -ball’sworth of different directions contained in one of Π (cid:0) L m (2 ρ, , I k , . . . , I m ) (cid:1) , and theseballs finitely overlap due to the fact that T is direction-separated.Now by the Tarski–Seidenberg projection theorem, we can take another semi-algebraic section of L m (2 ρ, , I k , . . . , I m ), this time leaving only one position a foreach d as in [25, Lemma 1.2] (see Corollary A.3). Following the notation of [25],we call this section L ′ ( I k , . . . , I m ), and so we also have δ n − m \ j = k (cid:8) T ∈ T : | T ∩ B λ j ∩ N ρ Z j | > λ j | T | (cid:9) . X I k ,...,I m (cid:12)(cid:12) Π (cid:0) L ′ ( I k , . . . , I m ) (cid:1)(cid:12)(cid:12) . (25)Noting that there are no more than (4 nd ) m − k +1 summands in this sum, it re-mains to bound | Π( L ′ ( I k , . . . , I m )) | independently of the choice of I k , . . . , I m . Forthis we use Gromov’s algebraic lemma as in the previous section to parametrise L ′ ( I k , . . . , I m ) with C s functions F i and G i ; N [ i =1 ( F i , G i )([0 , n − ) = L ′ ( I k , . . . , I m ) . Then we partition [0 , n − into cubes Q again, this time of diameter cδ ε/n , andapproximate the functions F i and G i by polynomials F iQ and G iQ of degree s C ( n, ε ) using Taylor’s theorem. Assuming that | Π( L ′ ( I k , . . . , I m ) | > δ n − , as we MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 15 may, these polynomial approximations do not alter the total measure significantlyand we find that (cid:12)(cid:12) Π (cid:0) L ′ ( I k , . . . , I m ) (cid:1)(cid:12)(cid:12) N X i =1 X Q | G iQ ( Q ) | N X i =1 X Q ˆ Q | det DG iQ ( y ) | d y . For any fixed y ∈ R n − , provided det DG iQ ( y ) = 0, the polynomial t det( DF iQ + tDG iQ )( y ) can be expressed asdet DG iQ ( y ) · n − Y j =1 ( t − z j )for some family of complex roots z , . . . , z n − ∈ C . There exists a subset of I m ofmeasure at least λ m / | t − z j | > λ m n −
1) for j = 1 , . . . , n − | det( DF iQ + tDG iQ )( y ) | . λ n − m | det DG iQ ( y ) | and, con-sequently, (cid:12)(cid:12) Π (cid:0) L ′ ( I k , . . . , I m ) (cid:1)(cid:12)(cid:12) . N X i =1 X Q λ − nm ˆ I m ˆ Q | det( DF iQ + tDG iQ )( y ) | d y dt. (26)Now by an application of B´ezout’s theorem as in the previous section, the poly-nomials F iQ + tG iQ are at most s n − -to-one, so that each of the integrals on theright-hand side of (26) can be bounded by s n − ˆ I m | ( F iQ + tG iQ )( Q ) | dt s n − | S m ( I m , ρ ) | . Given that there are fewer than C ( n, d, ε ) δ − ε summands in (26), this yields (cid:12)(cid:12) Π (cid:0) L ′ ( I k , . . . , I m ) (cid:1)(cid:12)(cid:12) . d δ − ε λ − nm | S m ( I m , ρ ) | . Then the proof is completed by combining this with (25), bounding | S m ( I m , ρ ) | by an application of Lemma 3.5. (cid:3) Reduction to k -broad estimates Rather than attempt to prove ( K p ) directly, it is useful to work with a classof weaker inequalities known as k -broad estimates . This type of inequality was in-troduced by Guth [18, 19] in the context of oscillatory integral operators (and, inparticular, the Fourier restriction conjecture) and was inspired by the earlier mul-tilinear theory developed in [4] (see also [3] for a detailed discussion of multilinearKakeya inequalities or Proposition 5.7 below for a precise statement relating the k -broad and k -linear theory).In order to introduce the k -broad estimates, we decompose the unit sphere S n − into finitely-overlapping caps τ of diameter β , an admissible constant satisfying δ ≪ β ≪
1. We then perform a corresponding decomposition of T by writing thefamily as a disjoint union of subcollections T = [ τ T [ τ ] where each T [ τ ] satisfies dir( T ) ∈ τ for all T ∈ T [ τ ]. The ambient euclidean spaceis also decomposed into tiny balls B δ of radius δ . In particular, fix B δ a collectionof finitely-overlapping δ -balls which cover R n . For B δ ∈ B δ define µ T ( B δ ) := min V ,...,V A ∈ Gr( k − ,n ) max τ : ∠ ( τ,V a ) >β for 1 a A (cid:13)(cid:13)(cid:13) X T ∈ T [ τ ] χ T (cid:13)(cid:13)(cid:13) pL p ( B δ ) ! , where A ∈ N and Gr( k − , n ) is the Grassmannian manifold of all ( k − R n . Here ∠ ( τ, V a ) denotes the infimum of the (unsigned) angles ∠ ( v, v ′ ) over all pairs of non-zero vectors v ∈ τ and v ′ ∈ V a . For U ⊆ R n the k -broad norm over U is then defined to be (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( U ) := X B δ ∈B δ | B δ ∩ U || B δ | µ T ( B δ ) ! /p . The k -broad norms are not norms in any familiar sense, but they do satisfy weakanalogues of various properties of L p -norms. The basic properties of these objectsare described in Section 5 below.The main ingredient in the proof of Theorem 1.2 is the following estimate for k -broad norms. Theorem 4.1.
Let p > n ( n − n +( k − k . For all ε > , there is an A ∼ suchthat (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) . δ − ( n − − n/p ) − ε (cid:16) X T ∈ T | T | (cid:17) /p (BL pk ) whenever < δ < and T is a direction-separated family of δ -tubes. The proof of Theorem 4.1, which is based on the polynomial partitioning methodand closely follows the arguments of [18, 19, 23], will be presented in Sections 5–8.The key feature which distinguishes the k -broad norm from its L p counterpartis that the former vanishes whenever the tubes of T cluster around a ( k − pk ) is substantially weaker than ( K p ).Nevertheless, a mechanism introduced by Bourgain and Guth [7] allows one to passfrom k -broad to linear estimates, albeit under a rather stringent condition on theexponent. Proposition 4.2 (Bourgain–Guth [7], Guth [19]) . Let p > n − k +2 n − k +1 , ε > and A ∼ . Suppose that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) . δ − ( n − − n/p ) − ε (cid:16) X T ∈ T | T | (cid:17) /p (BL pk ) whenever < δ < and T is a direction-separated family of δ -tubes. Then (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) L p ( R n ) . δ − ( n − − n/p ) − ε (cid:16) X T ∈ T | T | (cid:17) /p (K p ) whenever < δ < and T is a direction-separated family of δ -tubes. Thus, combining Theorem 4.1 and Proposition 4.2 yields Theorem 1.2. In con-trast with the range of Lebesgue exponents in Theorem 4.1, the range in whichProposition 4.2 applies shrinks as k increases. The optimal compromise betweenthe constraints in Theorem 4.1 and Proposition 4.2 is given by (1). MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 17
We end this section with a proof of Proposition 4.2, which is a minor modificationof the argument in [7] (see also [19]).
Proof (of Proposition 4.2).
The proof is by an induction-on-scale argument.For the base case, fix δ ∼ T be a family of direction-separated δ -tubes.If B is a cover of R n by finitely-overlapping balls of radius 1, then (cid:13)(cid:13)(cid:13)X T ∈ T χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) X B ∈B (cid:13)(cid:13)(cid:13) X T ∈ T T ∩ B = ∅ χ T (cid:13)(cid:13)(cid:13) pL p ( B ) . X B ∈B { T ∈ T : T ⊂ B } p . The direction separation condition implies that T . p )follows from H¨older’s inequality and the fact that any tube T ∈ T can belong to atmost O (1) of the balls 3 B .Now let C be a fixed constant, chosen sufficiently large so as to satisfy therequirements of the forthcoming argument, and fix some small δ > Induction hypothesis:
Suppose the inequality (cid:13)(cid:13)(cid:13)X e T ∈ e T χ e T (cid:13)(cid:13)(cid:13) L p ( R n ) C ˜ δ − ( n − − n/p ) − ε (cid:16) X e T ∈ e T | e T | (cid:17) /p holds whenever ˜ δ ∈ [2 δ,
1) and e T is a direction-separated family of ˜ δ -tubes.Let T be a direction-separated family of δ -tubes. Fix a δ -ball B δ ∈ B δ andsubspaces V , . . . , V A ∈ Gr( n, k −
1) which obtain the minimum in the definition of µ T ( B δ ); thus µ T ( B δ ) = max τ : ∠ ( τ,V a ) >β for 1 a A (cid:13)(cid:13)(cid:13) X T ∈ T [ τ ] χ T (cid:13)(cid:13)(cid:13) pL p ( B δ ) . Since A ∼ { τ : ∠ ( τ, V a ) β } ∼ β − ( k − , by the triangle inequality followedby H¨older’s inequality, ˆ B δ (cid:12)(cid:12) X T ∈ T χ T (cid:12)(cid:12) p . ˆ B δ (cid:12)(cid:12) X τ : ∠ ( τ,V a ) >β for 1 a A X T ∈ T [ τ ] χ T (cid:12)(cid:12) p + A X a =1 ˆ B δ (cid:12)(cid:12) X τ : ∠ ( τ,V a ) β X T ∈ T [ τ ] χ T (cid:12)(cid:12) p . β − ( n − p µ T ( B δ ) + β − ( k − p − X τ ˆ B δ (cid:12)(cid:12) X T ∈ T [ τ ] χ T (cid:12)(cid:12) p . Summing the estimate over all the balls B δ ∈ B δ , we find that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) . β − ( n − p (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( R n ) + β − ( k − p − X τ (cid:13)(cid:13)(cid:13) X T ∈ T [ τ ] χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) . The first term on the right-hand side of the above display is estimated usingthe hypothesised broad estimate. For the second term, we apply a linear rescaling L : R n → R n so that (cid:13)(cid:13)(cid:13) X T ∈ T [ τ ] χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) = β n − (cid:13)(cid:13)(cid:13) X T ∈ T [ τ ] χ L ( T ) (cid:13)(cid:13)(cid:13) pL p ( R n ) (27)where { L ( T ) : T ∈ T [ τ ] } is essentially a collection of ˜ δ -tubes with ˜ δ := β − δ . To bemore precise, let ω ∈ S n − denote the centre of the cap τ and choose L so that itfixes the 1-dimensional space spanned by ω and acts as a dilation by a factor of β −
18 JONATHAN HICKMAN, KEITH M. ROGERS, AND RUIXIANG ZHANG on the orthogonal complement ω ⊥ . Writing x ∈ R n as x = ( x ′ , x n ) with x ′ ∈ ω ⊥ ,for any T ∈ T [ τ ] with v := dir( T ) there exists some u ∈ R n such that T ⊆ (cid:8) x ∈ R n : | x ′ − u ′ − tv ′ | . δ for some | t | | x n − u n | / (cid:9) , Applying L one obtains L ( T ) ⊆ (cid:8) y ∈ R n : | y ′ − β − u ′ − tβ − v ′ | . β − δ for some | t | | y n − u n | / (cid:9) and the right-hand side can be covered by a bounded number of ˜ δ -tubes. Further-more, the family of ˜ δ -tubes L ( T ) is also direction-separated.Combining (27) with the induction hypothesis we find that (cid:13)(cid:13)(cid:13) X T ∈ T [ τ ] χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) . β n − C p ( β − δ ) − ( n − p + n − pε ( β − δ ) n − T [ τ ] . Recalling that P τ T [ τ ] = T , by plugging the preceding estimate into our L p ( R n )-norm bound, (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) C (cid:16) C b ( β ) + C p β e ( p,n,k )+ pε (cid:17) δ − ( n − p + n − pε (cid:16) X T ∈ T | T | (cid:17) ;here C b ( β ) depends, amongst other things, on the implied constant in (BL pk ) whilst C is a constant depending only on n and p (and, in particular, is independent ofthe choice of β ) and e ( p, n, k ) := ( n − k + 1) p − ( n − k + 2) . By assumption, p > n − k +2 n − k +1 and therefore e ( p, n, k ) >
0. Consequently, β may bechosen sufficiently small, depending only on the admissible parameters n , p and ε ,so that Cβ e ( p,n,k )+ pε . Moreover, if C is chosen sufficiently large from the outset, it follows that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) pL p ( R n ) C p δ − ( n − p + n − pε (cid:16) X T ∈ T | T | (cid:17) , which closes the induction and completes the proof. (cid:3) Basic properties of the k -broad norms Vanishing property.
The proof of Theorem 4.1 will involve analysing collections oftubes which enjoy certain tangency properties with respect to algebraic varieties.
Definition 5.1.
Given any collection of polynomials P , . . . , P n − m : R n → R , recallthat the common zero set Z ( P , . . . , P n − m ) := { x ∈ R n : P ( x ) = · · · = P n − m ( x ) = 0 } is referred to as a variety. It will often be convenient to work with varieties whichsatisfy the additional property that n − m ^ j =1 ∇ P j ( z ) = 0 for all z ∈ Z = Z ( P , . . . , P n − m ). (28)In this case the zero set forms a smooth m -dimensional submanifold of R n with a(classical) tangent space T z Z at every point z ∈ Z . A variety Z which satisfies (28)is said to be an m -dimensional transverse complete intersection . MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 19
Definition 5.2.
Let 0 < δ < r < x ∈ R n and Z ⊆ R n be a transverse completeintersection. A δ -tube T ⊂ R n is tangent to Z in B ( x , r ) ifi) T ∩ B ( x , r ) ∩ N δ Z = ∅ ;ii) If x ∈ T and z ∈ Z ∩ B ( x , r ) satisfy | z − x | δ , then ∠ (dir( T ) , T z Z ) c tang δr . Here 0 < c tang is an admissible constant which is chosen small enough to ensurethat, whenever i) and ii) hold, T ∩ B ( x , r ) ⊆ N δ Z . (29)The fact that such a choice is possible follows from a simple calculus exercise (see,for instance, [20, Proposition 9.2] for details of an argument of this type).The raison d’ˆetre for the k -broad norms is the following lemma, which roughlystates that the broad norms vanish if the tubes in T cluster around a low dimensionalvariety. Lemma 5.3 (Vanishing property) . Given ε ◦ > and < β < there exists some < c < such that the following holds. Let < δ < c , r > δ − ε ◦ , x ∈ R n and Z ⊆ R n be a transverse complete intersection of dimension at most k − . Then (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( B ( x ,r )) = 0 whenever T is a family of δ -tubes which are tangent to Z in B ( x , r ) . Proof.
Fix B δ ∈ B δ with B δ ∩ B ( x , r ) = ∅ . Recalling the definition of the k -broadnorm, it suffices to show that there exists some V ∈ Gr( k − , n ) such thatmax τ : ∠ ( τ,V ) >β ˆ B δ (cid:12)(cid:12) X T ∈ T [ τ ] χ T (cid:12)(cid:12) p = 0 . This would follow if V has the property thatif T ∈ T satisfies T ∩ B δ = ∅ , then ∠ (dir( T ) , V ) β . (30)Without loss of generality, one may assume there exists some T ∈ T such that T ∩ B δ = ∅ (otherwise (30) vacuously holds for any choice of ( k − T ∩ B δ ⊆ T ∩ B ( x , r ) ⊆ N δ Z and therefore there exists some z ∈ Z such that | z − y | < δ for some y ∈ T ∩ B δ .Let V be a ( k − T z Z . Given any T ∈ T , if x ∈ T ∩ B δ then | x − z | < δ and property ii) of the tangency hypothesis implies ∠ (dir( T ) , V ) . δr . Since r > δ − ε ◦ , it follows that ∠ (dir( T ) , V ) β provided δ is sufficiently smalldepending only on ε ◦ and β , which completes the proof. (cid:3) Here the parameter β appears implicitly in the definition of the k -broad norm. Triangle and logarithmic convexity inequalities.
The k -broad norms satisfy weakvariants of certain key properties of L p -norms. Lemma 5.4 (Finite subadditivity) . Let U , U ⊆ R n , p < ∞ and A ∈ N .Then (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( U ∪ U ) (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( U ) + (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( U ) whenever T is a family of δ -tubes. Lemma 5.5 (Triangle inequality) . Let U ⊆ R n , p < ∞ and A ∈ N . Then (cid:13)(cid:13)(cid:13) X T ∈ T ∪ T χ T (cid:13)(cid:13)(cid:13) BL pk, A ( U ) . (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( U ) + (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( U ) whenever T and T are families of δ -tubes. Lemma 5.6 (Logarithmic convexity) . Let U ⊆ R n , p, p , p < ∞ and A ∈ N .Suppose that θ ∈ [0 , satisfies p = 1 − θp + θp . Then (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk, A ( U ) . (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) − θ BL p k,A ( U ) (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) θ BL p k,A ( U ) whenever T is a family of δ -tubes. These estimates are entirely elementary. The proofs are identical to those usedto analyse broad norms in the context of the Fourier restriction problem [19]. It isremarked that the parameter A appears in the definition of the k -broad norm toallow for these weak triangle and logarithmic convexity inequalities. k -broad versus k -linear estimates. Although not required for the proof of Theo-rem 1.2, it is perhaps instructive to note the relationship between the k -broadnorms and the multilinear expressions appearing in the work of Bennett–Carbery–Tao [4]. Proposition 5.7.
Let T be a collection of δ -tubes in R n . Then (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) . X ( τ ,...,τ k ) ∼ β k − − trans . (cid:13)(cid:13)(cid:13) k Y j =1 (cid:16) X T j ∈ T [ τ j ] χ N δ T j (cid:17) /k (cid:13)(cid:13)(cid:13) pL p ( R n ) ! /p where the sum is over all k -tuples ( τ , . . . , τ k ) of caps of diameter β which are ∼ β k − -transversal in the sense that | V kj =1 ω j | & β k − for all ω j ∈ τ j . Thus, any k -linear inequality of the type featured in [4, 16, 7] is stronger thanthe corresponding k -broad estimate (given that β is admissible).The proof of Proposition 5.7 is a simple exercise and is omitted (see [20] forsimilar results in the (more complicated) context of oscillatory integral operators). MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 21 Polynomial partitioning
In this section the algebraic and topological ingredients for the proof of The-orem 4.1 are reviewed. In particular, the key polynomial partitioning theorem isrecalled, which is adapted from [18, 19] (see also [40]) and previously appearedexplicitly in [23].Given a polynomial P : R n → R consider the collection cell( P ) of connectedcomponents of R n \ Z ( P ). Each O ′ ∈ cell( P ) is referred to as a cell cut out bythe variety Z ( P ) and the cells are thought of as partitioning the ambient euclideanspace into a finite collection of disjoint regions.In order to account for the choice of scale δ > δ -tubes, it will be useful to consider the family of δ -shrunken cells defined byØ := (cid:8) O ′ \ N δ Z ( P ) : O ′ ∈ cell( P ) (cid:9) . (31)An important consequence of this definition is the following simple observation:A δ -tube T can enter at most deg P +1 of the shrunken cells O ∈ Ø.Indeed, this is a simple and direct consequence of the fundamental theorem ofalgebra (or B´ezout’s theorem) applied to the core line of T . Theorem 6.1 (Guth [19]) . Fix < δ < r , x ∈ R n and suppose F ∈ L ( R n ) isnon-negative and supported on B ( x , r ) ∩ N δ Z where Z is an m -dimensional trans-verse complete intersection with deg Z d . At least one of the following cases holds: Cellular case.
There exists a polynomial P : R n → R of degree O ( d ) with the follow-ing properties:i) P ) ∼ d m and each O ∈ cell( P ) has diameter at most r/ .ii) One may pass to a refinement of cell( P ) such that if Ø is defined as in (31) ,then ˆ O F ∼ d − m ˆ R n F for all O ∈ Ø . Algebraic case.
There exists an ( m − -dimensional transverse complete intersec-tion Y of degree at most O ( d ) such that ˆ B ( x ,r ) ∩ N δ Z F . log d ˆ B ( x ,r ) ∩ N δ Y F. This theorem is based on an earlier discrete partitioning result which played acentral role in the resolution of the Erd˝os distance conjecture [21]. The proof isessentially topological, involving the polynomial ham sandwich theorem of Stone–Tukey [36], which is itself a consequence of the Borsuk–Ulam theorem (see, forinstance, [32]), combined with a pigeonholing argument.The theorem is applied to k -broad norms by taking F = X B δ ∈B δ µ T ( B δ ) 1 | B δ | χ B δ . • If the cellular case holds, then it follows that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B ( x ,r ) ∩ N δ Z ) . d m (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( O ) for all O ∈ Øwhere Ø is the collection of cells produced by Theorem 6.1. • If the algebraic case holds, then it follows that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B ( x ,r ) ∩ N δ Z ) . log d (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B ( x ,r ) ∩ N δ Y ) where Y is the variety produced by Theorem 6.1.7. Finding polynomial structure
In this section, the recursive argument used to study the Fourier restrictionproblem in [23] (which, in turn, is adapted from [19]) is reformulated so as to applyto the Kakeya problem. As in [23], the argument will be presented as two separatealgorithms: • [alg 1] effects a dimensional reduction, essentially passing from an m -dimensional to an ( m − • [alg 2] consists of repeated application of the first algorithm to reduce toa minimal dimensional case.The final outcome is a method of decomposing any given k -broad norm intopieces which are either easily controlled or enjoy special algebraic structure. Thisdecomposition applies to arbitrary families of δ -tubes. In the following section, wewill specialise to the case where the tube family is direction-separated and use thisadditional information to prove Theorem 4.1. The first algorithm.
Throughout this section let p > < ε ◦ ≪ ε ≪ Input . [alg 1] will take as its input: • A choice of small scale 0 < δ ≪ r ∈ [ δ − ε ◦ , δ ε ◦ ]. • A transverse complete intersection Z of dimension m ∈ { , . . . , n } . • A family T of δ -tubes which are tangent to Z on a ball B r of radius r . • A large integer A ∈ N . Output . [alg 1] will output a finite sequence of sets ( E j ) Jj =0 , which are constructedvia a recursive process. Each E j is referred to as an ensemble and contains all therelevant information coming from the j th step of the algorithm. In particular, theensemble E j consists of: • A word h j of length j in the alphabet { a , c } , referred to as a history .The a is an abbreviation of “algebraic” and c “cellular”. The words h j are recursively defined by successively adjoining a single letter. Each h j records how the cells O j ∈ Ø j were constructed via repeated application ofthe polynomial partitioning theorem. • A large scale r j ∈ [ δ − ε ◦ , δ ε ◦ ]. The r j will in fact be completely determinedby the initial scales and the history h j . In particular, let σ k : [0 , → [0 , σ k ( r ) := r if the k th letter of h j is c r ε ◦ if the k th letter of h j is a for each 1 k j . With these definitions, r j := σ j ◦ · · · ◦ σ ( r ) . MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 23
Note that each σ k is a decreasing function and r j δ ε ◦ (1+ ε ◦ ) a ( j ) and r j − c ( j ) δ ε ◦ (32)where a ( j ) and c ( j ) denote the number of occurrences of a and c in thehistory h j , respectively. • A family of subsets Ø j of R n which will be referred to as cells . Each cell O j ∈ Ø j is contained in B r and will have diameter at most 2 r j . • An assignment of a subfamily T [ O j ] of δ -tubes to each of the cells O j . • A large integer d ∈ N which depends only on deg Z and the admissibleparameters n , p and ε .Moreover, the components of the ensemble are defined so as to ensure that, forcertain coefficients C j ( d ) := d c ( j ) ε ◦ d a ( j )( n + ε ◦ ) and A j := 2 − a ( j ) A ∈ N , the following properties hold:Property I. The function P T ∈ T χ T on B r can be compared with functions definedover the T [ O j ]: (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B r ) C j ( d ) X O j ∈ Ø j (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . (I) j Property II. The tube families T [ O j ] satisfy X O j ∈ Ø j T [ O j ] C j ( d ) d c ( j ) T . (II) j Property III. Furthermore, each individual T [ O j ] satisfies T [ O j ] C j ( d ) d − c ( j )( m − T . (III) j The initial step.
The initial ensemble E is defined by taking: • h := ∅ to be the empty word; • r to be the large scale; • Ø the collection consisting of the single ball O := B r ; • T [ O ] := T .All the desired properties then vacuously hold.At this point it is also convenient to fix some large d ∈ N , to be determined later,which depends only on deg Z and the admissible parameters n , p and ε .With these definitions, it is trivial to verify that Properties I, II and III hold. The recursive step.
Assume the ensembles E , . . . , E j have been constructed forsome j ∈ N and that they all satisfy the desired properties. Stopping conditions . The algorithm has two stopping conditions which are la-belled [tiny] and [tang] . Stop:[tiny]
The algorithm terminates if r j δ − ε ◦ . Stop:[tang]
Let C tang and C alg be fixed constants, chosen large enough to satisfy theforthcoming requirements of the proof. The algorithm terminates if theinequalities X O j ∈ Ø j (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) C tang log d X S ∈S (cid:13)(cid:13)(cid:13) X T ∈ T [ S ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj/ ( B [ S ]) and X S ∈S T [ S ] C tang δ − nε ◦ X O j ∈ Ø j T [ O j ];max S ∈S T [ S ] C tang max O j ∈ Ø j T [ O j ]hold for some choice of: • S a collection of transverse complete intersections in R n all of equal dimen-sion m − C alg d ; • An assignment of a subfamily T [ S ] of T and a max { r ε ◦ j , δ − ε ◦ } -ball B [ S ]to each S ∈ S with the property that each T ∈ T [ S ] is tangent to S in B [ S ]in the sense of Definition 5.2.The stopping condition [tang] can be roughly interpreted as forcing the algo-rithm to terminate if one can pass to a lower dimensional situation. Indeed, by theinclusion property (29), the broad norm over B [ S ] could instead be taken over a4 δ -neighbourhood of S .If either of the above conditions hold, then the stopping time is defined to be J := j . Recalling (32), the stopping condition [tiny] implies that the algorithmmust terminate after finitely many steps and, moreover, a ( J ) . ε − ◦ log( ε − ◦ ) and c ( J ) . log δ − . Note that there can be relatively few algebraic steps a ( j ) but there can manycellular steps c ( j ). The first of the above estimates can also be used to show that C j ( d ) . d,ε ◦ d c ( j ) ε ◦ always holds. Furthermore, by choosing A > ε − ◦ , say, onemay ensure that the A j defined above are indeed integers. Recursive step . Suppose that neither stopping condition [tiny] nor [tang] ismet. One proceeds to construct the ensemble E j +1 as follows.Given O j ∈ Ø j , apply the polynomial partitioning theorem with degree d to (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ∩ N δ Z ) = (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . For each O j ∈ Ø j either the cellular or the algebraic case holds, as defined inTheorem 6.1. Let Ø j, cell denote the subcollection of Ø j consisting of all cells forwhich the cellular case holds and Ø j, alg := Ø j \ Ø j, cell . Thus, by (I) j , one maybound k P T ∈ T χ T k p BL pk,A ( B r ) by C j ( d ) h X O j ∈ Ø j, cell (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) + X O j ∈ Ø j, alg (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) i ; MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 25 the analysis is splits into two cases depending on which term in the above sumdominates. ◮ Cellular-dominant case.
Suppose that the inequality X O j ∈ Ø j, alg (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) X O j ∈ Ø j, cell (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) holds so that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B r ) C j ( d ) X O j ∈ Ø j, cell (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . (33)Definition of E j +1 . Define h j +1 by adjoining the letter c to the word h j . Thus, itfollows from the definitions that r j +1 = r j , c ( j + 1) = c ( j ) + 1 and a ( j + 1) = a ( j ) . (34)The next generation of cells Ø j +1 arise from the cellular decomposition guar-anteed by Theorem 6.1. Fix O j ∈ Ø j, cell so that there exists some polynomial P : R n → R of degree O ( d ) with the following properties:i) P ) ∼ d m and each O ∈ cell( P ) has diameter at most 2 r j +1 .ii) One may pass to a refinement of cell( P ) such that ifØ j +1 ( O j ) := (cid:8) O \ N δ Z ( P ) : O ∈ cell( P ) } denotes the corresponding collection of δ -shrunken cells, then (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . d m (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j +1 ) for all O j +1 ∈ Ø j +1 ( O j ).Given O j +1 ∈ Ø j +1 ( O j ), define T [ O j +1 ] := (cid:8) T ∈ T [ O j ] : T ∩ O j +1 = ∅ (cid:9) . Recall that, by the fundamental theorem of algebra (or B´ezout’s theorem), any δ -tube T can enter at most O ( d ) cells O j +1 ∈ Ø j +1 ( O j ) and, consequently, X O j +1 ∈ Ø j +1 ( O j ) T [ O j +1 ] . d · T [ O j ] . (35)By the pigeonhole principle, one may pass to a refinement of Ø j +1 ( O j ) such that T [ O j +1 ] . d − ( m − T [ O j ] for all O j +1 ∈ Ø j +1 ( O j ). (36)Finally, define Ø j +1 := [ O j ∈ Ø j, cell Ø j +1 ( O j ) . This completes the construction of E j +1 and it remains to check that the newensemble satisfies the desired properties. In view of this, it is useful to note that C j ( d ) = d − ε ◦ C j +1 ( d ) and A j = A j +1 , (37)which follows immediately from (34) and the definition of the C j ( d ) and A j . Property I. Fix O j ∈ Ø j, cell and observe that j +1 ( O j ) ∼ d m and (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . d m (cid:13)(cid:13)(cid:13) X T ∈ T [ O j +1 ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j +1 ) for all O j +1 ∈ Ø j +1 ( O j ). Averaging, (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . X O j +1 ∈ Ø j +1 ( O j ) (cid:13)(cid:13)(cid:13) X T ∈ T [ O j +1 ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j +1 ) and, recalling (33) and (37), one deduces that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B r ) Cd − ε ◦ C j +1 ( d ) X O j +1 ∈ Ø j +1 (cid:13)(cid:13)(cid:13) X T ∈ T [ O j +1 ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj +1 ( O j +1 ) . Provided d is chosen large enough so as to ensure that the additional d − ε ◦ factorabsorbs the unwanted constant C , one deduces (I) j +1 . This should be comparedwith the approach of Solymosi and Tao to polynomial partitioning [35].Property II. By the construction, X O j +1 ∈ Ø j +1 T [ O j +1 ] = X O j ∈ Ø j X O j +1 ∈ Ø j +1 ( O j ) T [ O j +1 ] . d X O j ∈ Ø j T [ O j ] , where the inequality follows from a term-wise application of (35). Thus, (II) j , (34)and (37) imply that X O j +1 ∈ Ø j +1 T [ O j +1 ] . d − ε ◦ C j +1 ( d ) d c ( j +1) T . Provided d is chosen sufficiently large, one deduces (II) j +1 .Property III. Fix O j +1 ∈ Ø j +1 ( O j ) and recall from (36) that T [ O j +1 ] . d − ( m − T [ O j ] . Thus, (III) j , (34) and (37) imply that T [ O j +1 ] . d − ε ◦ C j +1 ( d ) d − c ( j +1)( m − T [ O j ] . Provided d is chosen sufficiently large as before, one deduces (III) j +1 . ◮ Algebraic-dominant case.
Suppose the hypothesis of the cellular-dominantcase fails so that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B r ) C j ( d ) X O j ∈ Ø j, alg (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . (38)Each cell in Ø j, alg satisfies the condition of the algebraic case of Theorem 6.1; thisinformation is used to construct the ( j + 1)-generation ensemble. MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 27
Definition of E j +1 . Define h j +1 by adjoining the letter a to the word h j . Thus, itfollows from the definitions that r j +1 = r ε ◦ j , c ( j + 1) = c ( j ) and a ( j + 1) = a ( j ) + 1 . (39)The next generation of cells is constructed from the varieties which arise from thealgebraic case in Theorem 6.1. Fix O j ∈ Ø j, alg so that there exists a transversecomplete intersection Y j of dimension m − Y j C alg d such that (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . log d (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ∩ N δ Y j ) . Let B ( O j ) be a cover of O j ∩ N δ Y j consisting of finitely-overlapping balls of radiusmax { r j +1 , δ − ε ◦ } . For each B ∈ B ( O j ) let T B denote the family of T ∈ T [ O j ] forwhich T ∩ B ∩ N δ Y j = ∅ . This set is partitioned into the subsets T B, tang := (cid:8) T ∈ T B : T is tangent to Y j on B (cid:9) , T B, trans := T B \ T B, tang ;here the notion of tangency is that given in Definition 5.2.By hypothesis, [tang] fails and, consequently, one may deduce that X O j ∈ Ø j, alg (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . log d X O j ∈ Ø j, alg B ∈B ( O j ) (cid:13)(cid:13)(cid:13) X T ∈ T B, trans χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj +1 ( B j ) (40)where, for notational convenience, B j := B ∩ N δ Y j . Indeed, provided C tang > X O j ∈ Ø j, alg X B ∈B ( O j ) T B, tang C tang δ − nε ◦ X O j ∈ Ø j T [ O j ];max O j ∈ Ø j, alg max B ∈B ( O j ) T B, tang max O j ∈ Ø j T [ O j ] . (41)Consequently, the failure of the stopping condition [tang] forceslog d X O j ∈ Ø j X B ∈B ( O j ) (cid:13)(cid:13)(cid:13) X T ∈ T B, tang χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj +1 ( B ) < C tang X O j ∈ Ø j (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) (since the estimates in (41) show all other conditions for [tang] are met for S , T [ S ]and B [ S ] appropriately defined). On the other hand, by the triangle inequality forbroad norms (Lemma 5.5), using the fact that A j +1 = A j /
2, the left-hand sideof (40) is dominated bylog d X O j ∈ Ø j, alg X B ∈B ( O j ) h(cid:13)(cid:13) X T ∈ T B, tang χ T (cid:13)(cid:13) p BL pk,Aj +1 ( B j ) + (cid:13)(cid:13) X T ∈ T B, trans χ T (cid:13)(cid:13) p BL pk,Aj +1 ( B j ) i . For a suitable choice of constant C tang , combining the information in the two pre-vious displays yields (40).For O j ∈ Ø j, alg defineØ j +1 ( O j ) := (cid:8) B ∩ N δ Y j : B ∈ B ( O j ) (cid:9) and let T [ O j +1 ] := T B, trans for O j +1 = B ∩ N δ Y j ∈ Ø j +1 ( O j ). The collection ofcells Ø j +1 is then given by Ø j +1 := [ O j ∈ Ø j, alg Ø j +1 ( O j ) . It remains to verify that the ensemble E j +1 satisfies the desired properties. In viewof this, it is useful to note that C j ( d ) = d − ( n + ε ◦ ) C j +1 ( d ) , (42)which follows directly from the definition of C j ( d ) and (39).Property I. By combining (40) together with the various definitions one obtains X O j ∈ Ø j, alg (cid:13)(cid:13)(cid:13) X T ∈ T [ O j ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj ( O j ) . log d X O j +1 ∈ Ø j +1 (cid:13)(cid:13)(cid:13) X T ∈ T [ O j +1 ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj +1 ( O j +1 ) . Recalling (38) and (42), if c ( d ) := Cd − ( n + ε ◦ ) log d for an appropriate choice ofadmissible constant C , then (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) p BL pk,A ( B r ) c ( d ) C j +1 ( d ) X O j +1 ∈ Ø j +1 (cid:13)(cid:13)(cid:13) X T ∈ T [ O j +1 ] χ T (cid:13)(cid:13)(cid:13) p BL pk,Aj +1 ( O j +1 ) . Provided d is sufficiently large, c ( d ) j +1 .Property II. Fix O j ∈ Ø j, alg and note that X O j +1 ∈ Ø j +1 ( O j ) T [ O j +1 ] = X B ∈B ( O j ) T B, trans (43)by the definition of T [ O j +1 ]. To estimate the latter sum one may invoke the fol-lowing algebraic-geometric result of Guth, which appears in Lemma 5.7 of [19]. Lemma 7.1 ([19]) . Suppose T is an infinite cylinder in R n of radius δ and centralaxis ℓ and Y is a transverse complete intersection. For α > let Y >α := (cid:8) y ∈ Y : ∠ ( T y Y , ℓ ) > α (cid:9) . The set Y >α ∩ T is contained in a union of O (cid:0) (deg Y ) n (cid:1) balls of radius δα − . Since T ∩ B ∩ N δ Y = ∅ by the definition of T B , a tube T ∈ T B belongs to T B, trans if and only if the angle condition ii) from Definition 5.2 fails to be satisfied. Thus,given any T ∈ S B ∈B T B, trans , it follows from the definitions that ∠ (dir( T ) , T y Y ) & δr j +1 for some y ∈ Y ∩ B with | y − x | . δ for some x ∈ T . This implies that N Cδ T ∩ B ∩ Y >α j +1 = ∅ where α j +1 ∼ δ/r j +1 . Consequently, by Lemma 7.1, any T ∈ S B ∈B ( O j ) T B, trans liesin at most O ( d n ) of the sets T B, trans and so X B ∈B ( O j ) T B, trans . d n T [ O j ] . Combining this inequality with (43) and summing over all O j ∈ Ø j, alg , X O j +1 ∈ Ø j +1 T [ O j +1 ] . d n X O j ∈ Ø j T [ O j ] . MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 29
Applying (II) j , (39) and (42), one concludes that X O j +1 ∈ Ø j +1 T [ O j +1 ] . d − ε ◦ C j +1 ( d ) d c ( j +1) T . Provided d is chosen to be sufficiently large to absorb the implicit constant, onededuces (II) j +1 .Property III. Fix O j ∈ Ø j, alg and O j +1 ∈ Ø j +1 ( O j ). By definition, T [ O j +1 ] ⊆ T [ O j ] and so T [ O j +1 ] T [ O j ] C j +1 ( d ) d − c ( j +1)( m − T , by (III) j and (39). The second algorithm.
The algorithm [alg 1] is now applied repeatedly inorder to arrive at a final decomposition of the k -broad norm. This process formspart of a second algorithm, referred to as [alg 2] .Throughout this section let p ℓ , with k ℓ n , denote some choice of Lebesgueexponents satisfying p k > p k +1 > . . . > p n =: p > . The numbers 0 Θ ℓ p ℓ byΘ ℓ := (cid:16) − p ℓ (cid:17) − (cid:16) − p (cid:17) so that Θ n = 1. Also fix 0 < ε ◦ ≪ ε ≪ [alg 2] , which can roughly be described as follows: • The recursive stage : P T ∈ T χ T is repeatedly decomposed into pieces withfavourable tangency properties with respect to varieties of progressivelylower dimension. • The final stage : P T ∈ T χ T is further decomposed into very small scalepieces.To begin, the recursive stage of [alg 2] is described. Input . [alg 2] will take as its input: • A choice of small scale 0 < δ ≪ • A large integer A ∈ N . • A family of δ -tubes T which are non-degenerate in the sense that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) = 0 . (44)Note that the process applies to essentially arbitrary families of δ -tubes (in partic-ular, the direction-separated hypothesis does not appear at this stage). Output . The ( n + 1 − ℓ )th step of the recursion will produce: • An ( n + 1 − ℓ )-tuple of: – scales ~δ ℓ = ( δ n , . . . , δ ℓ ) satisfying δ ε ◦ = δ n > · · · > δ ℓ > δ − ε ◦ ; – large and (in general) non-admissible parameters ~D ℓ = ( D n , . . . , D ℓ ); – integers ~A = ( A n , . . . , A ℓ ) satisfying A = A n > A n − > · · · > A ℓ .Each of these ( n + 1 − ℓ )-tuples is formed by adjoining a component to thecorresponding ( n − ℓ )-tuple from the previous stage. • A family ~ S ℓ of ( n + 1 − ℓ )-tuples of transverse complete intersections ~S ℓ =( S n , . . . , S ℓ ) satisfying dim S i = i and deg S i = O (1) for ℓ i n . • An assignment of a δ ℓ -ball B [ ~S ℓ ] and a subfamily T [ ~S ℓ ] of δ -tubes to each ~S ℓ ∈ ~ S ℓ with the property that the tubes T ∈ T [ ~S ℓ ] are tangent to S ℓ in B [ ~S ℓ ] (here S ℓ is the final component of ~S ℓ ).This data is chosen so that the following properties hold: Notation.
Throughout this section a large number of harmless δ − ε ◦ -factors appearin the inequalities. For notational convenience, given A, B > A / B or B ' A denote A . δ − cε ◦ B for some c > n and p .Property 1. The inequality (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) / C ( ~D ℓ ; ~δ ℓ )[ δ n T ] − Θ ℓ (cid:16) X ~S ℓ ∈ ~ S ℓ (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk,Aℓ ( B [ ~S ℓ ]) (cid:17) Θ ℓpℓ (45)holds for C ( ~D ℓ ; ~δ ℓ ) := n − Y i = ℓ (cid:16) δ i δ (cid:17) Θ i +1 − Θ i D (1+ ε ◦ )(Θ i +1 − Θ ℓ )+ ε ◦ i . Property 2. For ℓ n −
1, the inequality X ~S ℓ ∈ ~ S ℓ T [ ~S ℓ ] / D ε ◦ ℓ X ~S ℓ +1 ∈ ~ S ℓ +1 T [ ~S ℓ +1 ]holds.Property 3. For ℓ n −
1, the inequalitymax ~S ℓ ∈ ~ S ℓ T [ ~S ℓ ] / D − ℓ + ε ◦ ℓ max ~S ℓ +1 ∈ ~ S ℓ +1 T [ ~S ℓ +1 ]holds.By the inclusion property (29), the broad norms over B [ ~S ℓ ] on the right-handside of (45) could be replaced by broad norms over 4 δ -neighbourhoods of S ℓ . First step . Vacuously, the tubes belonging to T are tangent to the n -dimensionalvariety R n . Let B ◦ denote a collection of finitely-overlapping balls of radius δ ε ◦ which cover S T ∈ T T and define • δ n := δ ε ◦ ; D n := 1 and A n := A ; • S n is the collection consisting of repeated copies of the 1-tuple ( R n ), withone copy for each ball in B ◦ ; • For each ~S n ∈ S n assign a ball B [ ~S n ] ∈ B ◦ and let T [ ~S n ] := (cid:8) T ∈ T : T ∩ B [ ~S n ] = ∅ (cid:9) . By a straightforward orthogonality argument (identical to that used to establishthe base case in the proof of Proposition 4.2), Property 1 can be shown to holdwith C ( ~D n ; ~δ n ) = 1 and Θ n = 1. MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 31 ( n + 2 − ℓ )th step . Let ℓ > n +1 − ℓ steps. Since each family T [ ~S ℓ ] consists of δ -tubes which are tangentto S ℓ on B [ ~S ℓ ], one may apply [alg 1] to bound the k -broad norm (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) BL pℓk,Aℓ ( B [ ~S ℓ ]) . One of two things can happen: either [alg 1] terminates due to the stoppingcondition [tiny] or it terminates due to the stopping condition [tang] . The cur-rent recursive process terminates if the contributions from terms of the former typedominate:
Stopping condition . The recursive stage of [alg 2] has a single stopping condi-tion, which is denoted by [tiny-dom] . Stop:[tiny-dom]
Suppose that the inequality X ~S ℓ ∈ ~ S ℓ (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk,Aℓ ( B [ ~S ℓ ]) X ~S ℓ ∈ ~ S ℓ, tiny (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk,Aℓ ( B [ ~S ℓ ]) (46)holds, where the right-hand summation is restricted to those S ℓ ∈ ~ S ℓ forwhich [alg 1] terminates owing to the stopping condition [tiny] . Then [alg 2] terminates.Assume that the condition [tiny-dom] is not met. Necessarily, X ~S ℓ ∈ ~ S ℓ (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk,Aℓ ( B [ ~S ℓ ]) X ~S ℓ ∈ ~ S ℓ, tang (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk,Aℓ ( B [ ~S ℓ ]) , (47)where the right-hand summation is restricted to those S ℓ ∈ ~ S ℓ for which [alg 1] does not terminate owing to [tiny] and therefore terminates owing to [tang] .Consequently, for each ~S ℓ ∈ ~ S ℓ, tang the inequalities (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk,Aℓ ( B [ ~S ℓ ]) / D ε ◦ ℓ − X S ℓ − ∈S ℓ − [ ~S ℓ ] (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) p ℓ BL pℓk, Aℓ − ( B [ ~S ℓ − ]) , (48)and X S ℓ − ∈S ℓ − [ ~S ℓ ] T [ S ℓ − ] / D ε ◦ ℓ − T [ S ℓ ]; (49)max S ℓ − ∈S ℓ − [ ~S ℓ ] T [ S ℓ − ] / D − ( ℓ − ε ◦ ℓ − T [ S ℓ ] (50)hold for some choice of: • Scale δ ℓ − satisfying δ ℓ > δ ℓ − > δ − ε ◦ ; non-admissible number D ℓ − andlarge integer A ℓ − satisfying A ℓ − ∼ A ℓ ; • Family S ℓ − [ ~S ℓ ] of ( ℓ − O (1); • Assignment of a subfamily T [ ~S ℓ − ] = T [ ~S ℓ ][ S ℓ − ] of δ -tubes for every S ℓ − ∈S ℓ − [ ~S ℓ ] such that each T ∈ T [ ~S ℓ − ] is tangent to S ℓ − on B [ ~S ℓ − ]. Each inequality (48), (49) and (50) is obtained by combining the definition of thestopping condition [tang] with Properties I, II and III from [alg 1] , respectively.Indeed, we take r := δ ℓ , δ ℓ − := max { r ε ◦ J , δ − ε ◦ } , and D ℓ − := d c ( J ) , using the notation from [alg 1] .The δ ℓ − , D ℓ − and A ℓ − can depend on the choice of ~S ℓ , but this dependencecan be essentially removed by pigeonholing. In particular, c ( J ) depends on ~S ℓ ,but satisfies c ( J ) = O (log δ − ). Thus, since there are only logarithmically manypossible different values, one may find a subset of the S ℓ, tang over which the D ℓ − allhave a common value and, moreover, the inequality (46) still holds except that theconstant 1 / δ − ε ◦ . A brief inspection of [alg 1] showsthat both δ ℓ − and A ℓ − are determined by D ℓ − and so the desired uniformity isimmediately inherited by these parameters.Letting ~ S ℓ − denote the structured set ~ S ℓ − := (cid:8) ( ~S ℓ , S ℓ − ) : ~S ℓ ∈ ~ S ℓ, tang and S ℓ − ∈ S ℓ − [ ~S ℓ ] (cid:9) , where ~ S ℓ, tang is understood to be the refined collection described in the previousparagraph, it remains to verify that the desired properties hold for the newly con-structed data. Property 2 follows immediately from (49) and Property 3 from (50),so it remains only to verify Property 1.By combining the inequality (45) from the previous stage of the algorithmwith (47) and (48), one deduces that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) / D ε ◦ ℓ − C ( ~D ℓ ; ~δ ℓ )[ δ n T ] − Θ ℓ (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) Θ ℓ ℓ pℓ BL pℓk, Aℓ − ( ~ S ℓ − ) where, for any 1 q < ∞ and M ∈ N , we write (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) ℓ q BL qk,M ( ~ S ℓ − ) := X ~S ℓ − ∈ ~ S ℓ − (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) q BL qk,M ( B [ ~S ℓ − ]) ! /q . Taking q = p ℓ and M = 2 A ℓ − , the logarithmic convexity inequality (Lemma 5.6)dominates the preceding expression by (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) − Θ ℓ − / Θ ℓ ℓ BL k,Aℓ − ( ~ S ℓ − ) (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) Θ ℓ − / Θ ℓ ℓ pℓ − BL pℓ − k,Aℓ − ( ~ S ℓ − ) . Observe that, trivially, one has (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) ℓ BL k,Aℓ − ( ~ S ℓ − ) . (cid:16) δ ℓ − δ (cid:17) δ n X ~S ℓ − ∈ ~ S ℓ − T [ ~S ℓ − ] . and, by Property 2 for the tube families { T [ ~S i ] : ~S i ∈ ~ S i } for ℓ − i n −
1, itfollows that (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) ℓ BL k,Aℓ − ( ~ S ℓ − ) . (cid:16) δ ℓ − δ (cid:17)(cid:16) n − Y i = ℓ − D ε ◦ i (cid:17) δ n T . MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 33
One may readily verify that D ε ◦ ℓ − C ( ~D ℓ ; ~δ ℓ ) · (cid:16) δ ℓ − δ n − Y i = ℓ − D ε ◦ i (cid:17) Θ ℓ − Θ ℓ − = C ( ~D ℓ − ; ~δ ℓ − )and so, combining the above estimates, (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) / C ( ~D ℓ − ; ~δ ℓ − )[ δ n T ] − Θ ℓ − (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S ℓ − ] χ T (cid:13)(cid:13)(cid:13) Θ ℓ − ℓ pℓ − BL pℓ − k,Aℓ − ( ~ S ℓ − ) , which is Property 1. The final stage . If the algorithm has not stopped by the k th step, then it neces-sarily terminates at the k th step. Indeed, otherwise (45) would hold for ℓ = k − T [ ~S k − ] of δ k − -tubes which are tangent to some transverse completeintersection of dimension k −
1. By the vanishing property of the k -broad norms asdescribed in Lemma 5.3, one would then have (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S k − ] χ T (cid:13)(cid:13)(cid:13) BL pk − k,Ak − ( B [ ~S k − ]) = 0 , which, by (45), would contradict the non-degeneracy hypothesis (44).Suppose the recursive process terminates at step m , so that m > k . For each ~S m ∈ ~ S m, tiny let Ø[ ~S m ] denote the final collection of cells output by [alg 1] (thatis, the collection denoted by Ø J in the notation of the previous subsection) whenapplied to estimate the broad norm k P T ∈ T [ ~S m ] χ T k BL pmk,Am ( B [ ~S m ]) . By PropertiesI, II and III of [alg 1] one has (cid:13)(cid:13)(cid:13) X T ∈ T [ ~S m ] χ T (cid:13)(cid:13)(cid:13) p m BL pmk,Am ( B [ ~S m ]) . X O ∈ Ø[ ~S m ] (cid:13)(cid:13)(cid:13) X T ∈ T [ O ] χ T (cid:13)(cid:13)(cid:13) p m BL pmk,Am − ( O ) , for some A m − ∼ A m where the families T [ O ] satisfy X O ∈ Ø[ ~S m ] T [ O ] . D ε ◦ m − T [ ~S m ] (51)and max O ∈ Ø[ ~S m ] T [ O ] . D − ( m − ε ◦ m − T [ ~S m ] (52)for D m − a large and (in general) non-admissible parameter. Once again, by pi-geonholing, one may pass to a subcollection of S m, tiny and thereby assume that the D m − (and also the A m − ) all share a common value.If Ø denotes the union of the Ø[ ~S m ] over all ~S m belonging to subcollection of S m, tiny described above, then [alg 2] outputs the following inequality. First key estimate. (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) / C ( ~D m ; ~δ m )[ δ n T ] − Θ m X O ∈ Ø (cid:13)(cid:13)(cid:13) X T ∈ T [ O ] χ T (cid:13)(cid:13)(cid:13) p m BL k,Am − ( O ) ! Θ mpm . Proof of Theorem 4.1
Henceforth, fix T to be a direction-separated family of δ -tubes in R n . Withoutloss of generality, we may assume that T satisfies the non-degeneracy hypothe-sis (44). The algorithms described in the previous section can be applied to thistube family, leading to the final decomposition of the broad norm described in thefirst key estimate. One therefore wishes to show, using the direction-separated hy-pothesis, that the quantity on the right-hand side of the first key estimate can beeffectively bounded, provided that the exponents p k , . . . , p n are suitably chosen.Since each O ∈ Ø is contained in a ball of radius at most δ − ε ◦ , trivially onemay bound (cid:13)(cid:13)(cid:13) X T ∈ T [ O ] χ T (cid:13)(cid:13)(cid:13) p m BL pmk,Am − ( O ) / δ n (cid:0) T [ O ] (cid:1) p m . Recalling that Θ m (1 − p m ) = 1 − p , this yields X O ∈ Ø (cid:13)(cid:13)(cid:13) X T ∈ T [ O ] χ T (cid:13)(cid:13)(cid:13) p m BL pmk,Am − ( O ) ! Θ mpm / (cid:0) max O ∈ Ø T [ O ] (cid:1) − p (cid:16) δ n X O ∈ Ø T [ O ] (cid:17) Θ mpm . Now (51) and repeated application of Property 2 from [alg 2] imply X O ∈ Ø T [ O ] / (cid:16) n − Y i = m − D ε ◦ i (cid:17) T . Combining this with the first key estimate and the definition of C ( ~D m ; ~δ m ), oneconcludes that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) / C ( ~D ; ~δ ) (cid:0) max O ∈ Ø T [ O ] (cid:1) − p (cid:16) δ X T ∈ T | T | (cid:17) p (53)where, taking δ m − := δ , the constant takes the form C ( ~D ; ~δ ) := n − Y i = m − (cid:16) δ i δ (cid:17) Θ i +1 − Θ i D Θ i +1 − (1 − p )+ O ( ε ◦ ) i . In order to bound the maximum appearing on the right-hand side of (53), by (52)and repeated application of Property 3 of [alg 2] , it follows thatmax O ∈ Ø T [ O ] / (cid:16) ℓ − Y i = m − D − i + ε ◦ i (cid:17) max ~S ℓ ∈ ~ S ℓ T [ ~S ℓ ]whenever m ℓ n . Recall, for each tube family T [ ~S ℓ ] produced by [alg 2] and each ℓ i n − δ i -ball B δ i := B [ ~S i ] such that every δ -tube T ∈ T [ ~S ℓ ] is tangent to S i in B δ i ; in particular, T ∩ B δ i ∩ N δ S i = ∅ and T ∩ B δ i ⊆ N δ S i for ℓ i n − . Here S i is a transverse complete intersection of dimension i and deg S i dependsonly on the admissible parameters n , p and ε . Thus, Theorem 1.4 implies that T [ ~S ℓ ] n − \ i = ℓ n T ∈ T : | T ∩ B δ i ∩ N δ S i | > δ i | T | o / δ − ( n − n − Y i = ℓ (cid:16) δ i δ (cid:17) − , MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 35 where the first inequality follows from elementary geometric considerations. Com-bining these observations,max O ∈ Ø T [ O ] / (cid:16) ℓ − Y i = m − D − i + ε ◦ i (cid:17) δ − ( n − n − Y i = ℓ (cid:16) δ i δ (cid:17) − . for all m ℓ n . Finally, these n − m + 1 different estimates can be combinedinto a single inequality by taking a weighted geometric mean, yielding: Second key estimate.
Let γ m , . . . , γ n satisfy P nj = m γ j = 1 . Then max O ∈ Ø T [ O ] / (cid:16) n − Y i = m − (cid:16) δ i δ (cid:17) − P ij = m γ j D − i (1 − P ij = m γ j )+ O ( ε ◦ ) i (cid:17) δ − ( n − . When i = m − δ i /δ ) − P ij = m γ j factor is understood to be equal to 1.Substituting the second key estimate into (53), one obtains (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) BL pk,A ( R n ) / (cid:16) n − Y i = m − (cid:16) δ i δ (cid:17) X i D Y i + O ( ε ◦ ) i (cid:17) δ − ( n − − np ) (cid:16) X T ∈ T | T | (cid:17) p where X i := Θ i +1 − Θ i − (cid:16) i X j = m γ j (cid:17)(cid:16) − p (cid:17) ; Y i := Θ i +1 − (cid:16) i (cid:0) − i X j = m γ j (cid:1)(cid:17)(cid:16) − p (cid:17) . One now chooses the various exponents so that X i , Y i = 0 for all m i n − Y m − = 0. This ensures that the ( δ i /δ ) X i and D Y i i factors in the above expressionare admissible but does not allow one to control the D O ( ε ◦ ) i factors, which may stillbe non-admissible. To deal with the D O ( ε ◦ ) i one may perturb the p exponent whichresults under the conditions X i , Y i = 0, so that Y i becomes negative, and thenchoose ε ◦ sufficiently small depending on the choice of perturbation. This yields anopen range of k -broad estimates, which can then be trivially extended to a closedrange via interpolation through logarithmic convexity (the interpolation argumentrelies on the fact that one is permitted an δ − ε -loss in the constants in the k -broadinequalities).The condition X i = 0 is equivalent to (cid:16) − p i +1 (cid:17) − − (cid:16) − p i (cid:17) − = i X j = m γ j (54)whilst the condition Y i − = 0 is equivalent to (cid:16) − p i (cid:17) − = i − ( i − i − X j = m γ j . (55) Choose p m := mm − so that (55) holds in the i = m case. The remaining p i are thendefined in terms of the γ j by the equation (cid:16) − p i (cid:17) − = m + i − X j = m ( i − j ) γ j (56)so that each of the n − m constraints in (54) is met.It remains to solve for the n − m + 1 variables γ m , . . . , γ n . By comparing theright-hand sides of (55) and (56), it follows that i − X j = m (2 i − j − γ j = i − m for m + 1 i n . (57)To solve this linear system, let κ i denote the left-hand side of (57) and observe that κ i +1 + κ i − − κ i = ( i + 1) γ i − ( i − γ i − for m + 1 i n − κ m := 0. On the other hand, by considering the right-hand side of (57), itis clear that κ i +1 + κ i − − κ i = 0. Combining these observations gives a recursiverelation γ m := 1 m + 1 , γ i = (cid:16) i − i + 1 (cid:17) γ i − for m + 1 i n and from this one deduces that γ j = 1 m + 1 j − Y i = m i − i + 2 = ( m − m ( j − j ( j + 1) for m j n − p n ,corresponding to the exponent featured in Theorem 4.1. It follows from (55) that (cid:16) − p n (cid:17) − = n − ( n − m − m n − X j = m j − j ( j + 1)= n − ( n − n − ( m − m n . This is smallest when m = k , which directly yields the desired range of p , as statedin Theorem 4.1, completing the proof.9. Remarks on the numerology and related results
In this section we discuss the relationship between the main result of this paperand the existing literature on the Kakeya set conjecture. The first step is to obtain amore explicit range of exponents for Theorem 1.2, which is treated in the followingsubsection. Later, we also discuss applications of the method of this article tocertain variants of the Kakeya maximal problem.9.1.
Numerology.
Recall that the range of exponents in Theorem 1.2 is given by p > k n max n nn ( n −
1) + k ( k − , n − k + 1 o . (58)As claimed in the introduction, this guarantees that Conjecture 1.1 holds in therange p > − √ n − . (59) MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 37
In fact, in many dimensions a somewhat better bound is obtained. To see this,allowing k to be non-integer for a moment, one finds that the minimum value in(58) is attained when k = k where k = k ( n ) is chosen so that2 nn ( n −
1) + k ( k −
1) = 1 n − k + 1 . Solving the quadratic, one deduces that k = ( √ − n + 12 + √ n (cid:16)(cid:0) n + 18 n (cid:1) / − (cid:17) ( √ − n + 12 + 1 √ √ n , where the upper bound follows by Bernoulli’s inequality. Let ˜ k denote the expres-sion appearing on the last line of the above display. Since the sequence ( √ − n is equidistributed modulo 1, for any ε > n for which the interval [˜ k , ˜ k + ε ] contains an integer. For any such value of n itfollows that Conjecture 1.1 is true in the range p > − √ n + − √ − √ n + ε. On the other hand, considering the worst case scenario, when k is not close to aninteger, we can at least find an integer in [ k , k + 1], with k = k ( n ) chosen sothat 2 nn ( n −
1) + k ( k −
1) = 1 n − ( k + 1) + 1 . Calculating k and bounding from above using Bernoulli’s inequality as before, wefind that, in any dimension, Conjecture 1.1 is true in the range p > − √ n − − √ n . This range is always larger than the one stated in (59).9.2.
Implications for the Kakeya set conjecture.
As mentioned in the intro-duction, a maximal estimate of the form ( K p ) implies that the Hausdorff dimensionof any Kakeya set must be greater than or equal to p ′ , where 1 /p + 1 /p ′ = 1. It isinstructive to compare the Hausdorff dimension bounds obtained from Theorem 1.2with the current best known high dimensional results on the Kakeya set conjecturedue to Katz and Tao [27]. In particular, in [27] it was shown that Kakeya sets in R n have Hausdorff dimension greater than or equal to (2 −√ n − Consideringthe best case scenario from the previous section, we are able to obtain the followingimprovement.
Corollary 9.1.
For every ε > there exists an infinite sequence of dimensions n such that every Kakeya set K ⊆ R n satisfies dim H K > (2 − √ n + 32 − √ − ε. This is an improved range over what can be obtained directly from the maximal estimate in [27];an even larger bound for the Minkowski dimension is obtained in [27] for dimensions n > n = dim H > n = dim H > / ε Katz–Zahl [29] 10 15 − √ . ... Katz–Zahl [30] 11 17 − √ / / − √ − √ / − √ − √ Figure 5.
The state-of-the-art for the Kakeya set conjecture inlow dimensions. New results are highlighted.Provided ε > ε > n for which our results do not provide a betterbound than dim H K > (2 − √ n + 12 + ε. Provided ε > − √ n + O (1) numerology as the (completely different) sum-difference approach employed by Katz and Tao [27].9.3. Further variants of the Kakeya problem.
It is an interesting problem todetermine what can be said when the direction-separation hypothesis in Conjec-ture 1.1 is weakened; indeed, results of this kind have greatly influenced the currentunderstanding of the Kakeya conjecture (see, for instance, [38]). One classical the-orem in this direction is due to Wolff [41] and considers families of tubes whichsatisfy the following hypothesis.
Definition 9.2.
Let N > T be a family of δ -tubes in R n . We say that T satisfies the ( N ) -linear Wolff axiom if (cid:8) T ∈ T : T ⊆ E (cid:9) N δ − ( n − | E | whenever E ⊆ R n is a rectangular box of arbitrary dimensions.In [41], Wolff showed that the maximal inequality (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) L p ( R n ) . n,ε N − /p δ − ( n − − np ) − ε (cid:16) X T ∈ T | T | (cid:17) /p (60)holds for the restricted range p > n +2 n whenever T satisfies the ( N )-linear Wolffaxiom. Furthermore, it is not difficult to see that any direction-separated T satisfies Strictly speaking, Wolff’s theorem [41] holds under a slightly less restrictive condition referredto simply as the Wolff axiom . See [22] for a comparison of these conditions.
MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 39 the ( N )-linear Wolff axiom for some N ∼ T in dimensions n > N )-linear Wolff axiom with N ∼
1, but for which (60) fails to hold forthe whole range p > nn − ; see [37]. In particular, when n = 4 one may constructsuch T for which (60) is only valid in Wolff’s range p > /
2. Examples of thiskind are not direction-separated and therefore do not provide counterexamples toConjecture 1.1.To go beyond p > / N )-linear Wolff axiom. Definition 9.3.
We say that T satisfies the ( D, N ) -polynomial Wolff axiom if (cid:8) T ∈ T : | T ∩ E | > λ | T | (cid:9) N δ − ( n − λ − n | E | whenever λ > δ and E ⊆ R n is a semialgebraic set of complexity at most D .In this language, Theorem 1.3 states that for all D ∈ N and all ε > C ( D, ε, n ) such that any direction-separated family T satisfies the ( D, N )-polynomial Wolff axiom with N = C ( D, ε, n ) δ − ε . Thus, the following conjecture ofGuth and Zahl [22, Conjecture 1.1] is stronger than the Kakeya maximal conjecture. Conjecture 9.4 (Guth–Zahl [22]) . Let p > nn − . Then, for all ε > , there is acomplexity D = D ε,n ∈ N and a constant C ε,n > such that (cid:13)(cid:13)(cid:13) X T ∈ T χ T (cid:13)(cid:13)(cid:13) L p ( R n ) C ε,n N − /p δ − ( n − − n/p ) − ε (cid:16) X T ∈ T | T | (cid:17) /p whenever < δ < , N > and T satisfies the ( D, N ) -polynomial Wolff axiom. It is easy to adapt C´ordoba’s L -argument [11] to prove Conjecture 9.4 for n = 2.Guth and Zahl [22] showed that in four dimensions, under the polynomial Wolffaxioms, the p > / p > / Later, Katz and Zahl[29] obtained a slight improvement over the Wolff bound p > / p > n +2 n provides theprevious best known result under the polynomial Wolff axioms alone. By carryingout the analysis of this paper, but only using the polynomial Wolff axiom ratherthan the nested estimates from Theorem 1.3, one obtains the following range ofestimates. Theorem 9.5.
Conjecture 9.4 is true in the range p > k n max n (cid:16) nn − (cid:17) n − k , n − n − k + 1 o n − . (61)The above range of exponents is larger than Wolff’s when n = 5 or n >
7. Tosee this, note that for any 0 < r < k n satisfying k ∈ [ r ( n −
1) + 1 , r ( n −
1) + 2). Writing the endpoint in (61) as 1 + α n n − , it followsthat α n < inf
57 but contained an arithmetic error, ashighlighted in [29]. n = p > n = p > / Theorem 9.53 5 / − ε Katz–Zahl [29] 10 1 + 10 / Theorem 9.54 121 /
81 Guth–Zahl [22] 11 7 / / Theorem 9.5 12 1 + 12 / Theorem 9.56 4/3 Wolff [41] 13 8 / / Theorem 9.5 14 1 + 14 / Theorem 9.58 1 + 8 / Theorem 9.5 15 1 + 15 / Theorem 9.5
Figure 6.
The current state-of-the-art for Conjecture 9.4 in lowdimensions.Here the omega constant Ω ∈ (1 / ,
1) is the solution to e Ω = Ω − . In particular,Theorem 9.5 implies that Conjecture 9.4 is true in the range p > Ω − n − , yieldingan improvement over Wolff’s bound when n >
9. Calculating the precise valueof p n for lower n , we find that Theorem 9.5 also improves the state-of-the-art forConjecture 9.4 in dimensions n = 5 , ,
8; see Figure 6 for some explicit values for(61).
Appendix A. Tools from real algebraic geometry
For the reader’s convenience, here we recall the definitions and results from realalgebraic geometry that play a role in our arguments in Section 3.
Wongkew’s theorem.
We make considerable use of the following theorem of Wongkew[43] (see also [18, 45]), which bounds the volume of neighbourhoods of algebraic va-rieties.
Theorem A.1 (Wongkew [43]) . Suppose Z is an m -dimensional variety in R n with deg Z d . For any < ρ λ and λ -ball B λ the neighbourhood N ρ ( Z ∩ B λ ) can becovered by O d (( λ/ρ ) m ) balls of radius ρ .The Tarski–Seidenberg projection theorem. A fundamental result in the theory ofsemialgebraic sets is the Tarski–Seidenberg projection theorem, which is also re-ferred to as “quantifier elimination”. A useful reference for this material is [2].
Theorem A.2 (Tarski–Seidenberg) . Let Π be the orthogonal projection of R n intoits first n − coordinates. Then for every E > , there is a constant C ( n, E ) > so that, for every semialgebraic S ⊂ R n of complexity at most E , the projection Π( S ) has complexity at most C ( n, E ) . We repeatedly use Theorem A.2 to form semialgebraic sections of semialgebraicsets.
Corollary A.3.
Let S ⊂ R n be a compact semialgebraic set of complexity at most E . Let Π be the orthogonal projection into the final n coordinates ( a , d ) d . Thenthere is a constant C ( n, E ) > , depending only on n and E , and a semialgebraicset Z , of complexity at most C ( n, E ) , so that Z ⊂ S, Π( Z ) = Π( S ) , MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 41 and so that for each d , there is at most one a with ( a , d ) ∈ Z. This is Lemma 2.2 from [25]. It is a direct consequence of Theorem A.2, asdiscussed in [25].
Gromov’s algebraic lemma.
The final key tool is the existence of useful parameter-isations of semialgebraic sets, as guaranteed by the following lemma.
Lemma A.4 (Gromov) . For all integers
E, n, r > , there exists M ( E, n, r ) < ∞ with the following properties. For any compact semialgebraic set A ⊂ [0 , n ofdimension m and complexity at most E , there exists an integer N M ( E, n, r ) and C r maps φ , . . . , φ N : [0 , m −→ [0 , n such that N [ j =1 φ j ([0 , m ) = A and k φ j k C r := max | α | r k ∂ α φ j k ∞ . This result was originally stated by Gromov. Detailed proofs were later givenby Pila and Wilkie [33] and Burguet [8].
References [1] J. Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry,
Ergebnisse der Mathematikund ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] , 36, Springer-Verlag, Berlin, 1998.[2] S. Basu, R. Pollack and M.-F. Roy, Algorithms in real algebraic geometry,
Algorithms andComputation in Mathematics , 10, Springer-Verlag, Berlin, 2003.[3] J. Bennett, Aspects of multilinear harmonic analysis related to transversality, in
HarmonicAnalysis and PDE , 1–28, Contemp. Math. , Amer. Math. Soc., Providence, RI.[4] J. Bennett, A. Carbery and T. Tao,
On the multilinear restriction and Kakeya conjectures ,Acta Math. (2006), no. 2, 261–302.[5] J. Bourgain,
Besicovitch-type maximal operators and applications to Fourier analysis , Geom.Funct. Anal. (1991), no. 2, 147–187.[6] , On the dimension of Kakeya sets and related maximal inequalities , Geom. Funct.Anal. (1999), no. 2, 256–282.[7] J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinearestimates , Geom. Funct. Anal. (2011), no. 6, 1239–1295.[8] D. Burguet, A proof of Yomdin–Gromov’s algebraic lemma , Israel J. Math. (2008),291–316.[9] M. Christ,
Estimates for the k -plane transform , Indiana Univ. Math. J. (1984), no. 6,891–910.[10] M. Christ, J. Duoandikoetxea and J. L. Rubio de Francia, Maximal operators associated tothe Radon transform and the Calder´on-Zygmund method of rotations , Duke Math. J. (1986), no. 1, 189–209.[11] A. C´ordoba, The Kakeya maximal function and the spherical summation multipliers , Amer.J. Math. (1977), no. 1, 1–22.[12] R. O. Davies, Some remarks on the Kakeya problem , Proc. Cambridge Philos. Soc. (1971),417–421.[13] S. W. Drury, L p estimates for the X-ray transform , Illinois J. Math. (1983), no. 1, 125–129.[14] Z. Dvir, On the size of Kakeya sets in finite fields , J. Amer. Math. Soc. (2009), no. 4,1093–1097.[15] B. Green and I. Z. Ruzsa, On the arithmetic Kakeya conjecture of Katz and Tao , Preprint:arXiv:1712.02108.[16] L. Guth,
The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture , ActaMath. (2010), no. 2, 263–286.[17] ,
Degree reduction and graininess for Kakeya-type sets in R , Rev. Mat. Iberoam. (2016), no. 2, 447–494.[18] , A restriction estimate using polynomial partitioning , J. Amer. Math. Soc. (2016),no. 2, 371–413. [19] , Restriction estimates using polynomial partitioning II , Acta Math. (2018), 81–142.[20] L. Guth, J. Hickman and M. Iliopoulou,
Sharp estimates for oscillatory integral operatorsvia polynomial partitioning , Preprint: arXiv:1710.10349.[21] L. Guth and N. H. Katz,
On the Erd¨os distinct distances problem in the plane , Ann. of Math.(2) (2015), no. 1, 155–190.[22] L. Guth and J. Zahl,
Polynomial Wolff axioms and Kakeya-type estimates in R , Proc. Lond.Math. Soc. (3) (2018), no. 1, 192–220.[23] J. Hickman and K. M. Rogers, Improved Fourier restriction estimates in higher dimensions ,Preprint: arXiv:1807.10940.[24] N. H. Katz, I. Laba and T. Tao,
An improved bound on the Minkowski dimension of Besi-covitch sets in R , Ann. of Math. (2) (2000), no. 2, 383–446.[25] N. H. Katz and K. M. Rogers, On the polynomial Wolff axioms , Geom. Funct. Anal. (2018), 1706–1716.[26] N. H. Katz and T. Tao, Bounds on arithmetic projections, and applications to the Kakeyaconjecture , Math. Res. Lett. (1999), no. 5-6, 625–630.[27] , New bounds for Kakeya problems , J. Anal. Math. (2002), 231–263, Dedicated tothe memory of Thomas H. Wolff.[28] , Recent progress on the Kakeya conjecture , in Harmonic Analysis and Partial Differ-ential Equations (El Escorial, 2000). Publ. Mat. 2002, 161–179.[29] N. H. Katz and J. Zahl,
An improved bound on the Hausdorff dimension of Besicovitch setsin R , J. Amer. Math. Soc. (2019), no. 1, 195–259.[30] , A Kakeya maximal function estimate in four dimensions using planebrushes ,arXiv:1902.00989.[31] I. Laba and T. Tao,
An improved bound for the Minkowski dimension of Besicovitch sets inmedium dimension , Geom. Funct. Anal. (2001), 773–806.[32] J. Matouˇsek, Using the Borsuk-Ulam theorem , Universitext, Lectures on topological methodsin combinatorics and geometry, Written in cooperation with Anders Bj¨orner and G¨unter M.Ziegler, Springer-Verlag, Berlin, 2003, xii+196.[33] J. Pila and A. Wilkie,
The rational points of a definable set , Duke Math. J. (2006),591–616.[34] W. Schlag,
A geometric inequality with applications to the Kakeya problem in three dimen-sions , Geom. Funct. Anal. (1998), no. 3, 606–625.[35] J. Solymosi and T. Tao, An incidence theorem in higher dimensions , Discrete Comput. Geom. (2012), no. 2, 255–280.[36] A. H. Stone and J. W. Tukey, Generalized “sandwich” theorems , Duke Math. J. (1942),356–359.[37] T. Tao, A new bound for finite field Besicovitch sets in four dimensions , Pacific J. Math. (2005), no. 2, 337–363.[38] T. Tao,
Stickiness, graininess, planiness, and a sum-product approach to the Kakeya problem ,blog post: https://terrytao.wordpress.com/2014/05/07/ .[39] T. Tao, A. Vargas and L. Vega,
A bilinear approach to the restriction and Kakeya conjectures ,J. Amer. Math. Soc. (1998), no. 4, 967–1000.[40] H. Wang, A restriction estimate in R using brooms , Preprint: arXiv:1802.04312.[41] T. Wolff, An improved bound for Kakeya type maximal functions , Rev. Mat. Iberoamericana (1995), no. 3, 651–674.[42] , Recent work connected with the Kakeya problem , Prospects in mathematics (Prince-ton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 129–162.[43] R. Wongkew,
Volumes of tubular neighbourhoods of real algebraic varieties , Pacific J. Math. (1993), no. 1, 177–184.[44] J. Zahl,
A discretized Severi-type theorem with applications to harmonic analysis , Geom.Funct. Anal. (2018), no. 4, 1131–1181.[45] R. Zhang, Polynomials with dense zero sets and discrete models of the Kakeya conjectureand the Furstenberg set problem , Selecta Math. (2017), no. 1, 275–292. MPROVED BOUNDS FOR THE KAKEYA MAXIMAL CONJECTURE 43
School of Mathematics, James Clerk Maxwell Building, The King’s Buildings, PeterGuthrie Tait Road Edinburgh, EH9 3FD, UK
E-mail address : [email protected] Instituto de Ciencias Matem´aticas CSIC-UAM-UC3M-UCM, Madrid 28049, Spain
E-mail address : [email protected] Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madi-son, WI-53706, USA
E-mail address ::