Journal of Mathematics
A NEW BOUND FOR FINITE FIELD BESICOVITCH SETS
IN FOUR DIMENSIONS
T ERENCE T AO
V olume222No.2December2005PACIFIC JOURNAL OF MATHEMATICS
V ol.222,No.2,2005
A NEW BOUND FOR FINITE FIELD BESICOVITCH SETS
IN FOUR DIMENSIONS
T ERENCE T AO
Let F be afinitefield with characteristic greater than two.A Besicovitch set in F4is a set P⊆F4containing a line in every direction.The Kakeya conjecture asserts that P and F4have roughly the same size,in the sense that|P|/|F|4exceeds Cε|F|−εforε>0arbitrarily small,where Cεdoes not depend on P or F.Wolff showed that|P|exceeds a universal constant
times|F|3.Here we improve his exponent to3+1
16−εforε>0arbitrarily
small.On the other hand,we show that Wolff’s bound of|F|3is sharp if we
relax the assumption that the lines point in different directions.One new
feature in the argument is the use of some basic algebraic geometry.
1.Introduction
Let F be afinitefield with characteristic greater than2.For any n≥2,we define a Besicovitch set in F n to be a set P⊆F n containing a line in every direction(every equivalence class under parallelism).Thefinitefield Kakeya conjecture(see[Wolff 1998b],for example)asserts that|P|≥Cε|F|n−εfor anyε>0,where|P|denotes the cardinality of P and the quantities Cεare independent of|F|.This conjecture is thefinitefield analogue of the Euclidean Kakeya set conjecture,which is related to several other problems in harmonic analysis;see[Wolff1998b;Mockenhaupt and Tao2004]for further discussion on this.Basically,one can view thefinitefield Kakeya problem as a simplified model for the more interesting Euclidean Kakeya problem,where technical difficulties involving small separations,small angles,and multiple scales complicate the task(as discussed briefly in Section9).
Informally,the Kakeya conjecture asserts that lines pointing in different direc-tions in F n cannot have substantial overlap.This conjecture has been proved in two dimensions but remains open in higher dimensions.In[Wolff1998b](see also [Wolff1995;Mockenhaupt and Tao2004])it was shown that|P| |F|(n+2)/2,
MSC2000:42B25,05C35.
Keywords:Besicovitch sets,affine spaces overfinitefields,Kakeya conjecture,reguli.
This work was inspired by the April2002Instructional Conference on Combinatorial aspects of Mathematical Analysis at University of Edinburgh.The author is a Clay Prize Fellow and is sup-ported by the Packard Foundation.
337338TERENCE TAO
where A B means that A≥C−1B for some universal constant C.In fact,more
was proved:
Definition1.1.A family L of lines in F n is said to obey the Wolff axiom if for
every2≤k≤n−1,every k-dimensional affine subspace V⊂F n contains at
most O(|F|k−1)lines in L.(Here we view thefield F as being quite large,and the family L as depending on F.The implied constant in the O()notation may
depend on n and k but is uniform in F.Also Recall that an affine subspace is a
translate of a vector subspace of F n.)
Theorem1.2[Wolff1995;1998b].If L is a family of O(|F|n−1)lines obeying the Wolff axiom,and P⊆F n contains all the lines in L,then|P| |F|(n+2)/2.
In fact one only needs to use the Wolff axiom for k=2.From this theorem
and the observation that any family of lines that point in different directions auto-
matically obeys the Wolff axiom,we immediately see that Besicovitch sets have
cardinality |F|(n+2)/2.
In[Mockenhaupt and Tao2004](see also[Katz et al.2000])it was observed
that the statement of Theorem1.2is sharp in three dimensions,in the sense that
there existfinitefields F and collections of lines L in F3obeying the Wolff axiom
and a collection P of points containing all the lines in L,such that|P|∼|F|5/2
(where A∼B means that A B and B A).Indeed,if F contains a subfield G
of index2,with the accompanying involution z→¯z on F,one can take P to be
the Heisenberg group
P:={(z1,z2,z3)∈F3:Im(z3)=Im(z1¯z2)},
where Im(z):=(z−¯z)/2.(It is an interesting question whether an example similar
to this can be obtained if F does not contain a subfield of index2.)
Ourfirst observation is that Theorem1.2is also sharp in four dimensions: Proposition1.3.Let , :F4×F4→F be a nondegenerate symmetric quadratic form on F4.Let P be the“unit sphere”
(1–1)P:={x∈F4: x,x =1}
and let L be the set of all lines of the form{x+t v:t∈F},where x∈F4,v∈F4\\{0} are such that x,x =1, v,x =0,and v,v =0.Then L has cardinality |L|∼|F|3and obeys the Wolff axiom,while P has cardinality|P|∼|F|3and contains all the lines in L.
We prove this in Section3.A similar counterexample can be created inޒ4
as long as one chooses the form , to be indefinite.The proposition does not
contradict the Kakeya conjecture because the lines L do not all point in different
directions(despite obeying the Wolff axiom).Nevertheless,it seems of interestA NEW BOUND FOR BESICOVITCH SETS IN FOUR DIMENSIONS339 to extend this example(and the Heisenberg group)to higher dimensions,though perhaps the bound of|F|(n+2)/2in Theorem1.2need not be sharp for large n.
This example illustrates two things.Firstly,in order to progress toward the Kakeya conjecture in low dimensions one must make better use of the hypothesis that the lines in L point in different directions;merely assuming the Wolff axiom will not by itself suffice.(In high dimensions—say n≥9—there are other,more “arithmetic”arguments available to improve upon Theorem1.2.See[Bourgain 1999;Katz and Tao1999;2002b;Rogers2001;Mockenhaupt and Tao2004].) Secondly,the algebraic geometry of quadric surfaces may be relevant to the Kakeya problem.1In the three-dimensional case n=3,quadric surfaces are essen-tially the same thing as reguli,those ruled surfaces consisting of all the lines that intersect threefixed lines in general position.In particular,we have the“three-line lemma”,which asserts that given three mutually skew lines in F3,there are at most O(|F|)lines in different directions that intersect all three.
Reguli have already come up in the work of Schlag[1998],who used the three-line lemma to give a new proof of Bourgain’s estimate[1991]
(1–2)|P| |F|7/3
in three dimensions.While it is true that this bound has since been superseded by the estimate in Theorem1.2,we shall need to follow[Schlag1998]and make use of reguli and the three-line lemma in what follows.We are indebted to Nets Katz for pointing out the usefulness of reguli in the low-dimensional Kakeya problem. Indeed,our work here was inspired by similar work in three dimensions by Nets Katz(currently in preparation).
The main result of this paper is the following improved bound on the cardinality of Besicovitch sets in four dimensions.We use A B to denote the estimate A≤Cε|F|εB for anyε>0,where Cεis a quantity depending only onε. Theorem1.4.If P is a Besicovitch set in|F|4,then|P| |F|3+116.
One can probably improve the to a by going through the argument in this paper more carefully,but we will not do so here in order to simplify the exposition.
The paper is organized as follows.After setting out our incidence geometry notation in Section2,we prove Proposition1.3in Section3.We then review some basic algebraic geometry in Section4,culminating in a“three-regulus lemma”in F4,which will be the analogue of the three-line lemma in F3.In Section5 we review some combinatorial preliminaries,before starting the proof of Theorem 1.4,which occupies the next three sections.Thefirst step is to use a standard
1There seems to be a parallel phenomenon in recent work on Szemerédi’s theorem on arithmetic progressions,in that while arithmetic progressions are rather“linear”quantities,they give rise rather naturally to other“quadratic”objects which then need to be studied.See[Gowers1998].340TERENCE TAO
“iterated popularity”argument(as in[Christ1998],for example),together with a rudimentary version of the“plate number”argument in[Wolff1998a],in order to refine the Besicovitch set to a uniform,nondegenerate collection of points and lines.After a sufficient number of refinements,we can construct a large number of reguli incident to many lines in the Besicovitch set,and eventually get about|F|3 lines incident to three distinct reguli(if|P|is too close to|F|3);this will contradict the three-regulus lemma mentioned earlier.
2.Incidence notation
We now set some notation for thefinitefield geometry of the affine space F4.A line in F4is a set of the form l={x+t v:t∈F}where x,v∈F4and v is nonzero. Two lines are parallel if they are translates of each other but not identical;a set of lines is said to point in different directions if no two lines in the set are parallel or identical.
A2-plane in F4is a set of the formπ={x+t1v1+t2v2:t1,t2∈F}where x,v1,v2∈F4and v1,v2are linearly independent.Two lines are coplanar if they lie in the same2-plane;observe that coplanar lines must either be identical,parallel, or intersect in a point.A pair of lines are skew if they are not coplanar.
A3-space in F4is a set of the formλ={x+t1v1+t2v2+t3v3:t1,t2,t3∈F} where x,v1,v2,v3∈F4and v1,v2,v3are linearly independent.Observe that any pair of skew lines lies in a unique3-space.Two3-spaces are parallel if they are disjoint,and one is the translate of the other.
We shall use the symbol p to refer to points,l to lines,πto2-planes,andλto 3-spaces.We use the symbol P to refer to sets of points,L to sets of lines, to sets of2-planes,and to sets of3-spaces.We use Gr(F4,1)to denote the space of all lines,Gr(F4,2)to denote the space of all2-planes,and Gr(F4,3)to denote the space of all3-spaces.(Note that these are the affine Grassmannians,in that the spaces do not need to contain the origin).
3.The counterexample
It is likely that Proposition1.3follows from the standard theory of Fano varieties of quadric surfaces,but we will just give an elementary argument.
Proof of Proposition1.3.Let P and L be as in the proposition.It is clear from the construction that the lines in L lie in P.Now we verify the cardinality bounds. We begin with a standard lemma on the number of ways of representing afield element as a quadratic form.A NEW BOUND FOR BESICOVITCH SETS IN FOUR DIMENSIONS341 Lemma3.1.Let , :F n×F n→F be a symmetric bilinear form on F n with rank at least1,and let Q(x):= x,x be the associated quadratic form.Then (3–1){(x1,...,x n)∈F n:Q(x1,...,x n)=x} |F|n−1
for all x∈F.
If we know that , has rank at least3,we can improve(3–1)to
(3–2){(x1,...,x n)∈F n:Q(x1,...,x n) =x}∼|F|n−1
for all x∈F,if|F|is sufficiently large.
Proof.By placing the quadratic form Q in normal form(recalling that char F=2) we may assume that
Q(x1,...,x n)=α1x21+···+αk x2k,
where k is the rank of Q andα1,...,αk are nonzero elements of F.We may assume that k=n since the general case n≥k follows by adding n−k dummy variables.In particularαj=0for j=1,...,n.
The bound(3–1)is now clear,since if wefix x1,...,x n−1and x then there are at most2choices for x n.Now let us assume n≥3,and prove(3–2).
We use Gauss sums.Wefix a nonprincipal character e of F,i.e.a multiplicative function e:F→S1that is not identically1.For instance,if F=ޚ/pޚfor some
prime p,one can take e(x):=exp(2πi x/p).
For any y∈F,let S(y)be the Gauss sum S(y):=
x∈F
e(yx2).As is well
known(see[Mockenhaupt and Tao2004],for example),S(0)is equal to|F|,while |S(y)|=|F|1/2for all nonzero values of y.
Fix x∈F.By expanding the Kronecker delta as a Fourier series,we see that the number of solutions to(3–1)can be written as
x1,...,x n∈F δ(α1x21+···+αn x2n−x)=1
|F|
y∈F
x1,...,x n∈F
e((α1x21+···+αn x2n−x)y)
=
1
|F|
y∈F
e(−xy)
n
i=1
S(αi y)
=|F|n−1+
1
|F|
y∈F\\{0}
e(−xy)
n
i=1
S(αi y)
=|F|n−1+
1
|F|
y∈F\\{0}
O(|F|n/2)
=|F|n−1+O(|F|n/2)
as desired,since n≥3.
From the lemma we see that|P|∼|F|3,as desired.Now we count the lines in L.
The lemma yields∼|F|3choices of null direction{v∈F4\\0: v,v =0}. For each such v,the space v⊥:={x∈F4: x,v =0}is3-dimensional(since Q is nondegenerate).Furthermore,since Q is nondegenerate on F4and v is a null vector,we see that Q must also be nondegenerate on v⊥.Restricting Q to v⊥(which is of course isomorphic to F3)we see from Lemma3.1that there are ∼|F|2choices for x.Thus there are∼|F|5possible pairs(x,v)that generate a line in L.However,each line in L is generated by∼|F|2such pairs(x,v),so we have|L|∼|F|3as desired.
It remains to verify the Wolff axiom.First pick a3-spaceλand consider the lines in L which go throughλ.
Pick an arbitrary point x0inλ,so thatλ−x0is a three-dimensional subspace of F4.By Lemma3.1,the number of null vectors{v∈λ−x0: v,v =0}is O(|F|2). Fix v as above.There are two cases.If v⊥≡(λ−x0),then there are O(|F|) choices of x∈λsuch that x,v =0and x,x =1.But if v⊥≡(λ−x0),then the number of choices for x could be as large as O(|F|2).But(λ−x0)⊥only has cardinality O(|F|),hence the number of v in the second category is at most O(|F|).Thus the number of pairs(x,v)which can generate a line inλis at most O(|F|3).But each line is generated by∼|F|2pairs(x,v).Thus the number of lines inλis at most O(|F|),which clearly implies the Wolff axiom for both k=2 and k=3.This completes the proof of Proposition1.3.
4.Some basic algebraic geometry
Here we review some basic facts from algebraic geometry(for proofs see[Harris 1992],for example),and apply them to our Kakeya problem.The material we will need is not very advanced;basically,we need the concept of the dimension of an algebraic variety,and we need to know that this dimension behaves in the expected way with respect to intersections,projections,cardinality,etc.We shall also rely heavily on the basic fact that the dimension of an algebraic variety is always an integer(in contrast to,say,the“half-dimensional”field G mentioned in the introduction).
Let F denote the algebraic closure of F and n≥1.An algebraic variety in F n is defined to be the zero locus of a collection Q1,...,Q k of F-valued polynomials on the affine space F n.In this paper we shall always assume that our algebraic varieties have bounded degree,thus k=O(1)and all the polynomials Q1,...,Q k have degree O(1).
An algebraic variety V in F n has a well-defined integer-valued dimension0≤d≤n;there are several equivalent definitions of this dimension,for instance d isthe smallest nonnegative integer such that generic affine spaces in F n of codimen-sion greater than d are disjoint from V.(See[Harris1992]for more equivalent definitions of dimension).If V has dimension n then it must be all of F n,while if V has dimension0then it can only consist of at most O(1)points.Of course, the algebraic geometry notion of dimension is consistent with the linear algebra notion of dimension,thus for instance3-spaces have dimension3.
An algebraic variety is irreducible if it does not contain any proper sub-variety of the same dimension.Every algebraic variety of dimension k can be decomposed as a union of O(1)irreducible varieties of dimension at most k.
We define an algebraic variety in F n of dimension d to be a restriction to F n of an algebraic variety in F n of dimension d.Observe that if V is a variety in F n of dimension d then|V| |F|d(this can be shown,for instance,by taking generic intersections with affine spaces of codimension d).
Let L⊆Gr(F n,1)be a collection of lines which point in different directions.In the introduction we observed that this implies the Wolff axiom,that not too many lines in L can lie inside a k-space.In fact we can generalize this to k-dimensional varieties;see[Mockenhaupt and Tao2004,Proposition8.1]:
Lemma4.1(Generalized Wolff property).Let V⊆F n be an algebraic variety in F n of dimension k,and let L⊆Gr(F n,1)be a collection of lines in F n which point in different directions.Then
{l∈L:l⊆V}
|F|k−1.
Remark.The lines in Proposition1.3violate this property,but of course they do not point in different directions.(On the other hand,one can show that the lines arising from the Heisenberg example do obey this generalized Wolff property.It may be that in the three-dimensional theory,one needs to extend this lemma further, to cover not only varieties over F,but also over subfields of F such as G.) Proof.We may of course assume that|F| 1,since the claim is obvious for|F| bounded.
We can embed F n in the projective space P F n+1,which we think of as the union of F n with the hyperplane at infinity.By replacing the defining polynomials of V with their homogeneous counterparts,we can thus extend V to a k-dimensional variety V in P F n+1(see e.g.[Harris1992]).
We break up V into irreducible components,each of dimension at most k.We can assume that none of the irreducible components are contained inside the hyper-plane at infinity,since we could simply remove those components and still have an extension of V.In particular we see that the intersection of V with the hyperplane at infinity is at most k−1-dimensional.Let l be a line in L,which we can extend to be a projective line¯l in P F n+1by adding a single point at infinity(the direction of l).Observe that the restriction of V to¯l is an algebraic variety of dimension either0or1;in other words,either the projective line¯l lies inside V,or else¯l intersects V in at most O(1)points.Thus in order for l to be contained in V,the direction of l must lie inside V(assuming that|F|is sufficiently large).But by the previous paragraph the number of such directions is at most O(|F|k−1).Since the lines in L point in different directions, we are done.
As a consequence of this lemma we see that a Besicovitch set cannot have high intersection with an algebraic variety:
Corollary4.2.Let V⊆F n be an algebraic variety in F n of dimension at most n−1,and let L⊆Gr(F n,1)be a collection of lines in F n which point in different directions.Then
{(p,l)∈V×L:p∈l} |F|n−1.
The trivial upper bound for the left-hand side is|F||L| |F|n.Thus this lemma gains a power of|F|over the trivial bound.
Proof.As in the proof of Lemma4.1,we observe that every line l in F n is either contained in V,or else intersects V in at most O(1)points.The lines of the second type contribute at most O(|L|)=O(|F|n−1)incidences,while by Lemma 4.1the lines of thefirst type contribute at most O(|F||F|dim(V)−1)=O(|F|n−1) incidences,and we are done.
A further consequence is that the lines of a Besicovitch set cannot have large intersection with an algebraic variety:
Corollary4.3.Let L⊆Gr(F n,1)be a collection of lines in F n which point in different directions,and let P⊆F n be a set of points containing all the lines in L.Let W⊆Gr(F n,1)be an algebraic variety of lines of dimension at most n−1. Then
|L∩W| |F|n−2+|F|−1|P|.
Again,this lemma gains a power of|F|over the trivial bound of|F|n−1(as-suming P is not too huge).
Proof.Consider the set
X:={(p,l)∈F n×W:p∈l}.
This is an algebraic variety in F n×Gr(F n,1)of dimension at most n.Now consider the mapφ:X→F n given byφ(p,l):=p.Observe that for any p in the image ofφ, thefibersφ−1(p)are either0-dimensional(i.e.have cardinality O(1)),or at least 1-dimensional.This implies(see e.g.[Harris1992])that we have a decompositionφ(X):=P1∪P2,where thefibersφ−1(p)are0-dimensional for all p∈P1,and P2is contained in an algebraic variety of dimension at most n−1.
By the construction of P1we have
| (p,l)∈P
1×(L∩W):p∈l}
|
p∈P1:p∈l for some l∈L∩W}
|
p∈P1:p∈P}
≤|P|.
Also,by Corollary4.2we have
{(p,l)∈P2×(L∩W):p∈l}
|F|n−1.
Adding the two estimates,we see that
|F||L∩W|=
{(p,l)∈φ(X)×(L∩W):p∈l}
|F|n−1+|P|
and the claim follows.
To apply these results to our four-dimensional problem,we need some notation for reguli.
Definition4.4.A frame f is a quadruplet f=(l1,l2,l3,λ)whereλ∈Gr(F4,3) is a3-space in F4,and l1,l2,l3∈Gr(F4,1)are distinct,mutually skew lines in F4 which lie insideλ.If f is a frame,we writeλ(f)forλ.If f=(l1,l2,l3,λ)is a frame,we use L(f)to denote the set of lines l∈Gr(F4,1)which intersect l1,l2, and l3,and r(f)⊆λto denote the union of all the lines in L(f).
The set r(f)is called the regulus generated by the frame f.It is a quadric in λ,that is,the zero locus of a quadratic polynomial inλ,and hence an algebraic variety of dimension2.(The prototypical regulus is the set{(x,y,xy,0)∈F4: x,y∈F},where the lines l i are of the form{(x,y i,xy i,0):x∈F}for some distinct y1,y2,y3∈F.All reguli can be shown to be projectively equivalent to this example.)Since the lines in a frame are mutually skew,we see that this quadratic polynomial is irreducible(so the regulus is not a(double)plane,or the union of two planes),and that the lines L(f)have cardinality∼|F|and arefinitely overlapping. Corollary4.5(Three-regulus lemma).Let L⊆Gr(F4,1)be a collection of lines in F4which point in different directions,and let P⊆F4be a set of points containing all the lines in L.Let f1,f2,f3be three frames such that the3-spacesλ(f1),λ(f2),
λ(f3)are parallel and disjoint.Then
{l∈L:l∩r(f i)=∅for all i=1,2,3}
|F|2+|F|−1|P|.
Again,this bound improves by roughly|F|over the trivial bound of|F|3,if|P| is not much larger than|F|3.The hypothesis that the3-spacesλ(f i)are parallel can be substantially relaxed,but we will not need to do so here.Proof.Fix f1,f2,f3,and let W⊆Gr(F4,1)denote the set
(4–1)W:={l∈Gr(F4,1):l∩r(f i)=∅for all i=1,2,3}.
Since the r(f i)and l are algebraic varieties,it is clear(e.g.by using resultants; see e.g.[Harris1992])that the relationship l∩r(f i)=∅is equivalent to some finite set of explicit algebraic relations between the defining parameters of l and f i. Thus W is an algebraic variety in Gr(F4,1).In light of Corollary4.3,it will suffice to verify that W has dimension at most3.(We apologize to algebraic geometry sophisticates for the appalling crudeness of the following argument.)
Let p be a point in r(f1).Letφp be the stereographic projection fromλ(f2)to λ(f3),thusφp(x)=y if and only if p,x,y are collinear.W is isomorphic to
{(p,y)∈r(f1)×λ(f3):y∈r(f3)∩φp(r(f2))}
(basically because two points determine a line,and because the planesλ(f i)are disjoint).In other words,one can think of W as a bundle over r(f1)whosefiber at p is r(f3)∩φp(r(f2)).
Note thatφp:λ(f2)→λ(f3)is an invertible linear map,so thatφp(r(f2))is an irreducible quadric surface inλ(f3).The set r(f3)∩φp(r(f2))thus has dimension at most2,and in fact will have dimension at most1unlessφp(r(f2))≡r(f3) (by irreducibility).However,as p varies,the quadric surfacesφp(r(f2))move by translation.Since r(f2)is not a plane,we thus see that there can be at most a one-dimensional family of points p for whichφp(r(f2))≡r(f3).
To summarize,as p varies over the two-dimensional variety r(f1),thefiber r(f3)∩φp(r(f2))is at most one-dimensional,except possibly for a one-dimensional family of points p for which thefiber is two-dimensional.From this it is clear that W has dimension at most3,and we are done.(To compute the dimension properly one should work in the algebraically closedfield F here.But this causes no difficulty,since the above geometric considerations are valid for allfields of characteristic larger than two.)
Corollary4.5is the analogue of the three lines lemma used in[Schlag1998]. Our strategy will now be to start with a Besicovitch set and construct many frames f and many lines l∈L so that r(f)intersects L,in order to exploit the above Corollary.To do this we shall need some basic combinatorial tools,which we now pause to review.
5.Some basic combinatorics
We shall frequently use the following elementary observation:If B is a finite set and µ:B →ޒ+is a function such that b ∈B
µ(b )≥X ,
then
b ∈B :µ(b )≥X /2|B |µ(b )≥X /2.
We refer to this as a popularity argument ,since we are restricting B to the values b which are popular in the sense that µis large.The argument will be used iteratively many times.
We shall frequently use a version of the Cauchy–Schwarz and Hölder inequali-ties:
Lemma 5.1.Let A,B be finite sets,and let ∼be a relation connecting pairs (a ,b )∈A ×B such that {(a ,b )∈A ×B :a ∼b } X
for some X |B |.Then
{(a ,a ,b )∈A ×A ×B :a =a ;a ,a ∼b } X 2|B |
and
{(a ,a ,a ,b )∈A ×A ×B :a ,a ,a distinct ;a ,a ,a ∼b } X 3|B |2
Proof.Define for each b ∈B ,define µ(b ):=|{a ∈A :a ∼b }|.By hypothesis,we have b ∈B
µ(b ) X .
In particular,by the popularity argument we have
b ∈B :µ(b ) X /|B |
µ(b ) X .
By hypothesis,we have X /|B | 1.From this and the foregoing,we obtain
b ∈B :µ(b ) X /|B |
µ(b )(µ(b )−1) X (X /|B |)
and
b ∈B :µ(b ) X /|B |µ(b )(µ(b )−1)(µ(b )−2) X (X /|B |)(X /|B |).
The claims follow.
A typical application of Lemma5.1is the following standard incidence bound: Corollary5.2.For an arbitrarily collection P⊆F n of points and L⊆Gr(F n,1) of lines,we have
(5–1)
{(p,l)∈P×L:p∈l}
|P|1/2|L|+|P|
Proof.We may of course assume that the left-hand side of(5–1)is |P|,since the claim is trivial otherwise.From Lemma5.1we have
{(p,l,l )∈P×L×L:p∈l∩l ;l=l }
|P|−1
{(p,l)∈P×L:p∈l}
2
.
On the other hand,|l∩l |has cardinality O(1)if l=l ,thus
{(p,l,l )∈P×L×L:p∈l∩l ;l=l }
|L|2.
Combining the two estimates we obtain the result.
The preceding estimate will be most useful when|L|is small—in particular if |L|=O(|F|).When|L|is large,we have an alternate estimate:
Proposition5.3[Mockenhaupt and Tao2004].Let the notation be as in Corollary 5.2.If we further assume that the lines in L point in different directions,then
(5–2)
{(p,l)∈P×L:p∈l}
|P|1/2|L|3/4|F|1/4+|P|+|L|.
Proof.A proof is given after[Mockenhaupt and Tao2004,Proposition8.6];the argument there is essentially due to Nets Katz,but the original result of this type
dates back to Wolff[1995;1998b].
For the convenience of the reader we now sketch an informal“probabilistic”
derivation of(5–2).Let I denote the set in(5–2).We may assume that|I|
|P|,|L|since the claim is trivial otherwise.
Observe that a randomly chosen point p∈P and a randomly chosen line l∈L
have a probability|I|/|P||L|of being incident(so that p∈l).Thus,given two
random lines l1,l2∈L and a random point p∈P,we expect2the chance that p is incident to both l1and l2is(|I|/|P||L|)2.Since there are|P|possible values for
p,the chance that two random lines l1,l2∈L intersect at all is thus heuristically |P|(|I|/|P||L|)2.
As a consequence,the probability that three random lines l1,l2,l3∈L form a triangle is heuristically(|P|(|I|/|P||L|)2)3.(There is the chance that this triangle 2This of course assumes independence of various random events,which is usually not the case. To make the argument rigorous one must use such tools as Lemma5.1,which can be viewed as a statement that certain events are positively correlated.See[Mockenhaupt and Tao2004,Proposition 8.6]for details.is degenerate,but the hypothesis|I| |P|,|L|can be used to show that the prob-
ability of this occurring is low).On the other hand,given two intersecting lines
l1,l2∈L,there are at most O(|F|)lines l3∈L which can intersect them both,since we may apply the Wolff axiom to the2-plane spanned by l1and l2.Combining
these estimates we obtain
|P| |I|
|P||L|
2 3
|P|
|I|
|P||L|
2|F|
|L|
and(5–2)follows.3 Remark.One only requires the Wolff axiom on L to obtain(5–2).In particular one can easily obtain Theorem1.2as a consequence of(5–2).It is likely that one can generalize Theorem1.4to obtain a further improvement to(5–2),but we do not pursue this question here.
6.A heuristic proof of Theorem1.4
We now give a heuristic explanation for why we can improve upon Theorem1.2 in four dimensions,in the spirit of the probabilistic arguments in Proposition5.3. In later sections we shall make this heuristic argument rigorous.
Suppose for contradiction we have a family L⊆Gr(F4,1)of lines in different directions of cardinality|L|∼|F|3which are contained in a set P⊆F4,also of cardinality|P|∼|F|3.Arguing as in Proposition5.3we see that any two lines in L have a(heuristic)probability∼1/|F|of intersecting.
Also,a random line l∈L and a random3-spaceλ∈Gr(F4,3)have a probability 1/|F|2of being incident(so that l⊆λ).Thus we expect a3-spaceλto contain |L|/|F|2∼|F|lines in L.
Now consider the set of all quintuples(l1,l2,l1,l2,l3)∈L5of lines such that l i intersects l j for all i=1,2,j=1,2,3.From the above heuristics we see that there should be about|L|5(1/|F|)6∼|F|9such quintuples.On the other hand,for generic quintuples(l1,l2,l1,l2,l3)of the above form,the lines l1,l2,l3must lie in a3-spaceλ,and l1,l2must lie in the regulus generated by the frame(l1,l2,l3,λ). (For this heuristic argument we ignore the possibility that the quintuple could de-generate).
To count the number of possible reguli,observe that there are|L|2∼|F|6choices for l1,l2,which determinesλ.From our previous heuristic we see thatλcan contain at most O(|F|)choices for l3,thus there are at most O(|F|7)reguli.
3It is an instructive exercise to obtain similar heuristic probabilistic derivations of such estimates as(5–1)(using the fact that two random lines intersect in at most one point)or(1–2)(using the fact that a random regulus contains at most O(|F|)lines).See also Section6below.Dividing|F|9by|F|7,we thus see that a generic regulus r(f)of the above type must contain at least|F|2pairs(l1,l2)of lines in L.But r(f)only has O(|F|)lines to begin with.Thus a generic regulus r(f)must have extremely large intersection with P,so that|r(f)∩P|∼|r(f)|∼|F|2.
Since a random p∈P and l∈L have a probability1/|F|2of being incident, this means that a random line l∈L and a random regulus r(f)have a probability ∼1of intersecting.In particular,if we select three parallel reguli r(f1),r(f2), r(f3),a large fraction of lines in L must be incident to all three reguli.But this contradicts Corollary4.5,since|L|∼|F|3and|P|∼|F|3.
7.Preliminary refinements
We now begin the rigorous proof of Theorem1.4,which will broadly follow the heuristic outline of the previous section.
Let P0⊆F4be a Besicovitch set.We may assume that
(7–1)|P0| |F|3+116.
since the claim is trivial otherwise.We may also assume that|F| 1for similar reasons.
Since P0is a Besicovitch set,there exists a set L0⊆Gr(F4,1)of lines in different directions such that|L0|∼|F|3and P0contains every line in L0.In particular the incidence set
I0:={(p,l)∈P0×L0:p∈l}
has cardinality|I0|=|F||L0|∼|F|4.
Given any line l in L0and a randomly selected3-spaceλin Gr(F4,3),the probability that l lies inλis∼1/|F|2.Since|L0|∼|F|3,one thus expects every 3-spaceλcontains about|F|lines in L0on the average.A similar heuristic leads us to expect every2-planeπ∈Gr(F4,2)to contain at most O(1)lines on the average.
Although these statements need not be true for all3-spacesλ,certain variants do hold if we refine L0and P0slightly,as stated in the next result.We write A≈B to mean A B and A B.
Proposition7.1.There exists a quantity
(7–2)1 α N116,
a subset4P1of P0and a subset L1of L0such that the following properties hold.
4In the course of this argument we shall need to refine the set P
0to a slightly smaller set P1, and then further to P2,and similarly refine L0to L1and then L2,while also refining some auxiliary sets H0to H1,andᏲ0toᏲ1toᏲ2toᏲ3.These refinements are largely technical and as afirst
Many incidences:We have the incidence bound
(7–3)
{(p ,l )∈P 1×L 1:p ∈l } |F |4.Cardinality and multiplicity bounds:We have the cardinality bound
(7–4)
|P 1| α|F |3and the multiplicity bound
(7–5)
{l ∈L 1:p ∈l } ≈α−1|F |for all p ∈P 1.
No 3-space degeneracy:For any 3-space λ∈Gr (F 4,3),we have (7–6) {l ∈L 1:l ⊂λ} α2|F |.
No 2-plane degeneracy:For any 2-plane π∈Gr (F 4,2),we have (7–7)
{l ∈L 1:l ⊂π} α4.The quantity αmeasures the improvement over Wolff’s bound |P 0| |F |3.As one can see from (7–2),it is rather close to 1.
Proof.We follow standard “iterated refinement”arguments (see [Wolff 1998a ;Łaba and Tao 2001b ;Christ 1998;Tao and Wright 2003];our argument here is particularly close to that in [Łaba and Tao 2001b ]).The purpose of the iteration is mainly to obtain the property (7–7).
Define the multiplicity function µ0on P 0by
µ0(p ):=|{l ∈L 0:p ∈l }|.
Then we have
p ∈P 0µ0(p )=|I 0|.
If we divide µ0(p )into dyadic “pigeonholes”and apply the dyadic pigeonhole principle (observing that log |F |≈1),we conclude that there exists a multiplicity α−1|F |such that p ∈P 0:µ0(p )∼α−1|F |
µ0(p )≈|I 0|≈|F |4.
Fix this α,and define
P 0:={p ∈P 0:µ0(p )∼α−1|F |}
approximation one can view these sets as being essentially the same (although the sets Ᏺ2,Ᏺ3are significantly smaller than Ᏺ0,Ᏺ1).
I 0:={(p,l)∈P 0×L0:p∈l}⊆I0.
Then by construction,|I
| |I0|∼|F|4,and
|P
|≈|I0|/(α−1|F|)≈α|F|3.
By(7–1)we thus haveα N116.To get the other half of(7–2),we observe from Proposition5.3that
|I 0| |P
|1/2|L0|3/4|F|1/4+|P
|+|L0|;
applying the above estimates,we thus obtainα 1.Thus(7–2)holds.
Set N:=log log|F|;the point of this choice of N is that both|F|C/N and C N are≈1for anyfixed choice of constant C.We shall inductively construct sets (7–8)P 0=:P(0)⊃P(1)⊃···⊃P(N)
and
(7–9)L0=:L(0)⊃L(1)⊃···⊃L(N)
as follows.5
As indicated above,we set P(0):=P
0and L(0):=L0.Now suppose inductively
that P(k)and L(k)have already been constructed for some0≤k Clearly we have l∈L(k) |l∩P(k)|=|I(k)|. Thus if we set L(k+1):= l∈L(k):|l∩P(k)|≥ |I(k)| 2|L(k)| then by the popularity argument we have l∈L(k+1) |l∩P(k)|≥|I(k)|/2. We rewrite this as p∈P(k) {l∈L(k+1):p∈l} ≥|I(k)|/2. 5The use of such a large number of refinements is of course overkill(one could probably get away with N=5,in fact),but reducing the number of refinements used does not alter the exponent1 16 "c": P(k+1):= p∈P(k): {l∈L(k+1):p∈l} ≥ |I(k)| 4|P(k)| , we get,again by the popularity argument, p∈P(k+1) {l∈L(k+1):p∈l} ≥|I(k)|/4 or in other words |I(k+1)|≥|I(k)|/4. We repeat this construction for k=0,1,...,N−1,creating a nested sequence of sets of points(7–8)and sets of lines(7–9).By construction and the fact that 4−N≈1,we clearly have |I(k)|≈|I(0)|=|I | |F|4 for all k.Furthermore, |P(k)|≤|P | α|F|3and|L(k)|≤|L0| |F|3. Thus,setting P1:=P(N)and L1:=L(N−1),we see that(7–3),(7–4),and(7–5) hold.(To get the upper bound in(7–5),simply bound the left-hand side byµ0(p).) It remains only to verify the nondegeneracy conditions(7–6),(7–7). Wefirst verify(7–6).Letλbe a3-space.Sinceλis clearly an algebraic variety of dimension3,we can invoke Corollary4.2and conclude that {(p,l)∈λ×L(N−2):p∈l} |F|3. From the construction of P(N−1)we thus have |P(λ)| |F|2α where P(λ):=λ∩P(N−1). Let L(λ)denote those lines in L1which lie inλ.By the construction of L1we have {(p,l)∈P(λ)×L(λ):p∈l} |F||L(λ)|. On the other hand,from Proposition5.3we have {(p,l)∈P(λ)×L(λ):p∈l} |P(λ)|1/2|L(λ)|3/4|F|1/4+|P(λ)|+|L(λ)|. Combining all three estimates and using(7–2)we obtain |L(λ)| α2|F| which is(7–6). In fact,this argument gives(7–6)if L1is replaced by L(k)for any1≤k≤N−1. We now prove (7–7).Following [Wolff 1998a ;Łaba and Tao 2001b ],we define the plate number p k for 0≤k ≤N −1to be the quantity p k :=sup π∈Gr (F 4,2) {l ∈L k :l ⊂π} .We observe the bounds (7–10)1≤p k |F |; the former bound comes since L k is nonempty,while the latter bound comes since a 2-plane can contain at most O (|F |)lines in different directions. Clearly the plate numbers are nonincreasing in k .From this,(7–10),the pigeon-hole principle and the fact that |F |1/N ≈1,we can find 2≤k ≤N −1such that (7–11)p k −1≈p k . Fix this k .We can find a 2-plane π∈Gr (F 4,2)such that the set L k (π):={l ∈L k :l ⊂π} has cardinality p k . Fix π,and let P k (π)denote the set P k (π):=P k ∩π. By the construction of L k ,every line in L k contains |F |points in P k ,thus every line in L k (π)contains |F |points in P k (π).In particular we see that {(p ,l )∈P k (π)×L k (π):p ∈l } |F |p k . Applying Corollary 5.2we conclude that |F |p k |P k (π)|1/2p k +|P k (π)|; from this and (7–10)we thus have (7–12)|P k (π)| |F |p k . Let P k (π)denote the space of all points p in P k (π)such that at least half of all the lines in {l ∈L k −1:p ∈l }are contained in L k −1(π).We have two cases.Case 1(parallel case ):|P k (π)|≥12|P k (π)|.In this case we have {(p ,l )∈P k (π)×L k −1(π):p ∈l } ≥ {(p ,l )∈P k (π)×L k −1(π):p ∈l } ≥12 {(p ,l )∈P k (π)×L k −1:p ∈l } |P k (π)|α−1|F | |P k (π)|α−1|F |, |L k−1(π)|≤p k−1. Applying Corollary5.2we thus see that |P k(π)|α−1|F| |P k(π)|1/2p k−1+|P k(π)|, which by(7–2)implies that |P k(π)||F|2 α2p2k−1. But combining this with(7–11),(7–12)we obtain p k α−2|F|3. But this contradicts(7–10)by(7–2).Hence this case cannot occur. Case2(transverse case):|P k (π)|≤1 2 |P k(π)|.In this case we have(by a compu- tation similar to Case1) {(p,l)∈P k(π)×L k:p∈l;l⊂π} α−1|P k(π)||F|. Thus,if L∗ k−1denotes the lines l∈L k−1which are incident to a point in P k(π)but are not contained inπ,then we have (7–13)|L∗k−1| α−1|P k(π)||F| α−1|F|2p k by(7–12). We now use Wolff’s hairbrush argument[Wolff1995],[Wolff1998b],as mod-ified to deal with plates in[Wolff1998a],[Łaba and Tao2001b].We can foliate L∗ k−1 as the disjoint union of L∗k−1(λ):={l∈L∗k−1:l∈λ} whereλranges over the3-spaces containingπ.For each suchλ,observe from the analogue of(7–6)for L(k−1)that (7–14)|L∗k−1(λ)| α2|F|. Also,if we define P∗k−1(λ):={p∈P k−1:p∈λ\\π} then by the construction of L k−1,we have {(p,l)∈P∗ k−1(λ)×L∗k−1(λ):p∈l} |L∗k−1(λ)||F|. Applying Corollary5.2we obtain |L∗ k−1 (λ)||F| |P∗k−1(λ)|1/2|L∗k−1(λ)|+|P∗k−1(λ)|,which by(7–14),(7–2)implies that |P∗ k−1 (λ)| α−2|L∗k−1(λ)||F|. Summing inλ,we obtain |P k−1| α−2|L∗ k−1 ||F| α−3|F|3p k. Since|P k−1| α|F|3by construction,we obtain p k α4,and the claim follows. 8.Construction of reguli We now continue the proof of Theorem1.4.We begin by refining P1and L1a little further.By(7–3)we have l∈L1 |l∩P1|≈|F|4. Thus if we set L2:= l∈L1:|l∩P1|≈|F| then by the popularity argument we get l∈L2 |l∩P1|≈|F|4 or equivalently p∈P1 {l∈L2:p∈l} ≈|F|4. Thus if we set P2:= p∈P1:|{l∈L2:p∈l}|≈α−1|F| then by(7–4),(7–5),and the popularity argument we have (8–1) p∈P2 {l∈L2:p∈l} ≈|F|4. In particular, (8–2)|P2|≈α|F|3. The next task is to generate a large number of frames,and a large number of lines in L incident to the reguli generated by these frames.As a frame is a fairly complicated combinatorial object(consisting of three lines and a3-space),we will first begin by counting some simpler objects which eventually will be combined together to form frames. By (8–1)we have {(p ,l )∈P 2×L 2:p ∈l } ≈|F |4. Since |L 2| |F |3,we thus see from Lemma 5.1that {(p 1,p 2,l )∈P 2×P 2×L 2:p 1,p 2∈l ;p 1=p 2} |F |8/|L 2|≈|F |5. By the definition of P 2,we see that for each (p 1,p 2,l )as above there are α−1|F |lines l 1∈L 2that contain p 1but are distinct from l ,and similarly there are α−1|F |lines l 2∈L 2containing p 2but distinct from l .Thus |H 0| α−2|F |7, where H 0:= (p 1,p 2,l ,l 1,l 2)∈P 2×P 2×L 2×L 2×L 2: p 1,p 2∈l ;p 1=p 2;p 1∈l 1;p 2∈l 2;l =l 1,l 2 is the space of “H-shaped”objects. Let H 1⊆H 0be the set of elements (p 1,p 2,l ,l 1,l 2)in H 0such that l 1and l 2are skew.We claim that |H 0\\H 1| α−1|F |6. Indeed,to choose an element (p 1,p 2,l ,l 1,l 2)in H 0\\H 1(which is a degenerate H,i.e.a triangle),we first choose p 1∈P 2(of which there are α|F |3choices),and then choose the distinct lines l ,l 1incident to p 1(of which there are (α−1|F |)2choices).Since l 2must lie in the 2-plane generated by l and l 1,and the lines of L 1point in different directions,there are only O (|F |)choices for l 2.Since p 2is uniquely determined as p 2=l ∩l 2,the claim follows. From the bounds above and (7–2)we see that (8–3)|H 1| α−2|F |7. By construction,if h =(p 1,p 2,l ,l 1,l 2)∈H 1,then l 1and l 2are skew.Thus l 1and l 2lie in a unique 3-space λ(h ),which then must also contain p 1,p 2,l . Let S 0⊂L 2×L 2denote the pairs (l 1,l 2)of skew lines in L 2.For each pair (l 1,l 2)∈S 0,we define the connecting set C (l 1,l 2)⊂L 2to be the set of all lines l ∈L 2which are distinct from l 1,l 2,but intersect both l 1,l 2in points p 1∈P 2and p 2∈P 2respectively.Observe the identity (l 1,l 2)∈S 0 |C (l 1,l 2)|=|H 1|. Since |S 0|≤|L 2|2 |F |6,we thus see from (8–3)that if we define S 1:= (l 1,l 2)∈S 0:|C (l 1,l 2)| α−2|F | , (8–4) (l1,l2)∈S1 |C(l1,l2)| α−2|F|7. If(l1,l2)∈S1,we define the set C(3)(l1,l2)⊆C(l1,l2)3to be the space of all triplets(l1,l2,l3)∈C(l1,l2)3such that the six points l i∩l j for i=1,2,3,j=1,2 are all disjoint. We now use the nondegeneracy property(7–7)to obtain a lower bound for the size of C(3)(l1,l2). Lemma8.1(Many triple connections between skew lines).For any(l1,l2)∈S1, we have|C(3)(l1,l2)| α−4|F|2|C(l1,l2)|. Proof.Fix l1,l2.We choose l1∈C(l1,l2)arbitrarily;of course,there are|C(l1,l2)| choices for l1. Fix l1.From(7–7)we have {l2∈C(l1,l2):l2∩l1=l1∩l1} α4 (since such lines lie in the2-plane spanned by l1∩l1and l2.Similarly if the roles of l1and l2are interchanged.Since |C(l1,l2)| α−2|F|, we thus see from(7–2)that there are α−2|F|choices for l2such that l2∩l j=l1∩l j for j=1,2. Fix l2.Arguing as above we see that there are α−2|F|choices for l3such that l3∩l j=l i∩l j for i=1,2and j=1,2.The claim follows. From this lemma and(8–4)we see that (l1,l2)∈S1 |C(3)(l1,l2)| α−6|F|9. Observe that if(l1,l2)∈S1and(l1,l2,l3)∈C(3)(l1,l2),the various incidence assumptions in the definition of S1and C(3)(l1,l2)force f:=(l1,l2,l3,λ)to be a frame,whereλis the unique3-space spanned by l1and l2.Observe that l1,l2 both lie in L2∩L(f).Thus,ifᏲ0denotes the space of all frames generated in this manner,then (8–5) f∈Ᏺ0 |L2∩L(f)|2 α−6|F|9. Let f∈Ᏺ0.Since the lines in L(f)are contained in a regulus,they havefinite overlap.Since each line in L2contains≈|F|points in P1by construction,we thussee that6 |P1∩r(f)| |F||L2∩L(f)| so by(8–5)we have f∈Ᏺ0 |P1∩r(f)|2 α−6|F|11. By(7–5),each point in P1is incident to≈α−1|F|lines in L1.Thus we have f∈Ᏺ0 {l∈L1:l∩r(f)∩P1=∅} 2 α−8|F|13. We observe the cardinality bound (8–6)|Ᏺ0| α2|F|7. Indeed,to choose a frame(l1,l2,l3,λ)inᏲ0,we observe that there are O(|L1|2)=O(|F|6) choices for the skew pair(l1,l2).This determinesλ,and then by(7–6)we thus see that there are O(α2|F|)choices for l3,and(8–6)follows.In particular,if we define (8–7)Ᏺ1:={f∈Ᏺ0: {l∈L1:l∩r(f)∩P1=∅} α−5|F|3} then by the popularity argument (8–8) f∈Ᏺ1 {l∈L1:l∩r(f)∩P1=∅} 2 α−8|F|13. Since the summand on the left-hand side can be crudely bounded by|L1|2= O(|F|6),we thus have the crude bound7 (8–9)|Ᏺ1| α−8|F|7 (compare with(8–6)). For any frame f∈Ᏺ1,there are only O(|F|3)possible orientations forλ(f).By (8–9)and the pigeonhole principle,there therefore exists a3-spaceλ0∈Gr(F4,3) such that (8–10)|Ᏺ2| α−8|F|4 6One could also obtain this bound using Corollary5.2and the crude bound |L2∩L(f)|≤|L(f)| |F|. 7The bounds on|Ᏺ 1|,and later on|Ᏺ2|,|Ᏺ3|,might not be best possible,however an improve-ment on this part of the argument does not directly improve the gain1 16 . where Ᏺ2:={f∈Ᏺ1:λ(f)is a translate ofλ0}. Fix thisλ0.LetᏲ3be a maximal subset ofᏲ2such that the reguli{r(f):f∈Ᏺ3} are all distinct.Since each r(f)contains at most O(|F|)lines,each regulus can arise from at most O(|F|3)frames.We thus see from(8–10)that (8–11)|Ᏺ3| α−8|F|. From(8–7)we have f∈Ᏺ3 {l∈L1:l∩r(f)∩P1=∅} α−5|F|3|Ᏺ3|. From(7–2),(8–11)the right-hand side is |F|3 |L1|.Thus we can use Lemma 5.1,and obtain f1,f2,f3∈Ᏺ3 f1,f2,f3distinct {l∈L1:l∩r(f i)∩P1=∅for i=1,2,3} α−15|F|3|Ᏺ3|3. From the pigeonhole principle,we may thusfind distinct frames f1,f2,f3inᏲ3 such that (8–12)|L∗| α−15|F|3, where L∗⊆L1is the collection of lines L∗:={l∈L1:l∩r(f i)∩P1=∅for i=1,2,3}. Now we consider the problem of obtaining upper bounds on|L∗|.The crude up-per bound of|L1|∼|F|3is clearly not enough to obtain a contradiction.However, thanks to the three-regulus lemma we can improve this bound by about|F|: Proposition8.2.We have (8–13)|L∗| |F|2+116. Proof.If the3-spacesλ(f1),λ(f2),λ(f3)are disjoint,this follows directly from Corollary4.5and(7–1). By symmetry it remains to consider the case whenλ(f1)andλ(f2)(for instance) are equal.Then the lines in L∗must either be parallel toλ(f1),or else intersect P1∩r(f1)∩r(f2).There are at most|F|2lines in thefirst category(in fact there are far fewer,thanks to(7–6)).In the second category,we observe that r(f1)∩r(f2)is at most one-dimensional(since r(f1),r(f2)are irreducible and distinct)and hence has cardinality O(|F|).On the other hand,by(7–5)every point in r(f1)∩r(f2)∩P1 is incident to≈α−1|F|lines in L1.Thus we certainly have |F|2+116incidences in this case as well.Combining(8–13)with(8–12)we obtain α |F|116 and hence by(8–2) |P0| |P2|≈α|F|3 |F|3+116 as desired.This concludes the proof of Theorem1.4. 9.Remarks It seems likely that this Theorem can be generalized in several ways.The expo- nent1 16is probably not sharp,and also the result should have extension to other dimensions,perhaps through more sophisticated use of algebraic geometry.In di-mensions5and higher there are other,more“arithmetic”arguments that give slight improvements to|F|(n+2)/2for Besicovitch sets;see[Bourgain1999;Katz and Tao 1999;2002a;2002b;Mockenhaupt and Tao2004;Rogers2001].Nevertheless,if one can make an improvement of the order of1 16in,say,five dimensions by these “geometric”techniques,this will be quite competitive with the results in,say,[Katz and Tao2002b].In the Euclidean setting one can improve the bound(n+2)/2in all dimensions n≥3by a small number(10−10)for the upper Minkowski dimension problem for Besicovitch sets[Katz et al.2000;Łaba and Tao2001a;2001b],but this argument seems special to the upper Minkowski problem and does not directly impact thefinitefield question. Also,the argument can probably be extended to obtain an estimate on the Kakeya maximal function forfinitefields;see[Mockenhaupt and Tao2004].In principle,thefinitefield results should also extend to the Euclidean settingޒn, but there are unpleasant technical difficulties in the process,due,for instance,to the presence of near-degenerate reguli inޒn.Also in thefinitefield case one is aided considerably by the fact that dimensions must be integer;for instance,the intersection of two lines is either empty,0-dimensional(a point),or1-dimensional (a line).In the(δ-discretized)Euclidean case there is a continuum of cases:two distinct1×δtubes can intersect in a set of length∼δ/θ,whereδ<θ<1is the angle between the two tubes.This introduces a new dyadic parameterθinto the analysis(measuring the degeneracy of the angle),and often the cases of smallθand largeθneed to be treated separately.See[Wolff1995],for example.Here we have more complicated algebraic objects,such as the variety(4–1),and to capture the possible degeneracies of this object seems to require a large number of additional dyadic parameters.It is possible that various rescaling arguments,such as the two-ends and bilinear reductions mentioned above,may be used to reduce the number of such parameters,but the extension of this argument to the Euclidean case still appears to be quite nontrivial.Difficulties of these kinds cause considerable complication in such papers as [Schlag1998],although some could perhaps be alleviated using the“two-ends”reduction in[Wolff1995]and the“bilinear reduction”in[Tao et al.1998]. Acknowledgments The author thanks Tony Carbery,Nets Katz,Wilhelm Schlag,and Jim Wright for helpful discussions,and is also indebted to David Gieseker and Allen Knutson for their explanation of some of the basics of algebraic geometry.The author is particularly indebted to Nets Katz for emphasizing the importance of reguli to this problem.Finally,the author thanks the anonymous referee for a careful reading and the detection of several misprints. References [Bourgain1991]J.Bourgain,“Besicovitch type maximal operators and applications to Fourier anal-ysis”,Geom.Funct.Anal.1:2(1991),147–187.MR92g:42010Zbl0756.42014 [Bourgain1999]J.Bourgain,“On the dimension of Kakeya sets and related maximal inequalities”, Geom.Funct.Anal.9:2(1999),256–282.MR2000b:42013Zbl0930.43005 [Christ1998]M.Christ,“Convolution,curvature,and combinatorics:a case study”,Internat.Math. Res.Notices19(1998),1033–1048.MR2000a:42026Zbl0927.42008 [Gowers1998]W.T.Gowers,“A new proof of Szemerédi’s theorem for arithmetic progressions of length four”,Geom.Funct.Anal.8:3(1998),529–551.MR2000d:11019Zbl0907.11005 [Harris1992]J.Harris,Algebraic geometry,Graduate Texts in Mathematics133,Springer,New York,1992.MR93j:14001Zbl0779.14001 [Katz and Tao1999]N.H.Katz and T.Tao,“Bounds on arithmetic projections,and applications to the Kakeya conjecture”,Math.Res.Lett.6:6(1999),625–630.MR2000m:28006Zbl0980.42013 [Katz and Tao2002a]N.Katz and T.Tao,“Recent progress on the Kakeya conjecture”,Publ.Mat. V ol.Extra(2002),161–179.MR2003m:42036Zbl1024.42010 [Katz and Tao2002b]N.H.Katz and T.Tao,“New bounds for Kakeya problems”,J.Anal.Math. 87(2002),231–263.MR2003i:28006Zbl1027.42014 [Katz et al.2000]N.H.Katz,I.Łaba,and T.Tao,“An improved bound on the Minkowski di-mension of Besicovitch sets in R3”,Ann.of Math.(2)152:2(2000),383–446.MR2002i:28006 Zbl0980.42014 [Łaba and Tao2001a]I.Łaba and T.Tao,“An improved bound for the Minkowski dimension of Be-sicovitch sets in medium dimension”,Geom.Funct.Anal.11:4(2001),773–806.MR2003b:28006 Zbl1005.42009 [Łaba and Tao2001b]I.Łaba and T.Tao,“An x-ray transform estimate inޒn”,Rev.Mat.Ibero-americana17:2(2001),375–407.MR2003a:44003Zbl1024.44002 [Mockenhaupt and Tao2004]G.Mockenhaupt and T.Tao,“Restriction and Kakeya phenomena for finitefields”,Duke Math.J.121:1(2004),35–74.MR2004m:11200Zbl02103577 [Rogers2001]K.M.Rogers,“Thefinitefield Kakeya problem”,Amer.Math.Monthly108:8(2001), 756–759.MR2002g:11175Zbl1028.43007 [Schlag1998]W.Schlag,“A geometric inequality with applications to the Kakeya problem in three dimensions”,Geom.Funct.Anal.8:3(1998),606–625.MR99g:42025Zbl0939.42012[Tao and Wright2003]T.Tao and J.Wright,“L p improving bounds for averages along curves”,J. Amer.Math.Soc.16:3(2003),605–638.MR2004j:42005Zbl016622 [Tao et al.1998]T.Tao,A.Vargas,and L.Vega,“A bilinear approach to the restriction and Kakeya conjectures”,J.Amer.Math.Soc.11:4(1998),967–1000.MR99f:42026Zbl0924.42008 [Wolff1995]T.Wolff,“An improved bound for Kakeya type maximal functions”,Rev.Mat.Ibero-americana11:3(1995),651–674.MR96m:42034Zbl0848.42015 [Wolff1998a]T.Wolff,“A mixed norm estimate for the X-ray transform”,Rev.Mat.Iberoamericana 14:3(1998),561–600.MR2000j:44006Zbl0927.44002 [Wolff1998b]T.Wolff,“Recent work connected with the Kakeya problem”,pp.129–162in Pros-pects in mathematics(Princeton,NJ,1996),edited by H.Rossi,Amer.Math.Soc.,Providence,RI, 1998.MR2000d:42010Zbl0934.42014 Received April19,2002.Revised September10,2002. T ERENCE T AO D EPARTMENT OF M ATHEMATICS UCLA L OS A NGELES CA90095-1555 tao@math.ucla.edu http://www.math.ucla.edu/~tao下载本文
