The Finite Field Kakeya Conjecture
نویسنده
چکیده
In this paper we introduce the celebrated Kakeya Conjecture in the original real case setting and discuss the proof of its finite field analogue. The story begins in 1917 when, with the hope of figuring out a mathematical formalization of the movement of a samurai during battle, Sochi Kakeya proposed the following problem: What is the least area in the plane required to continuously rotate a needle of unit length and zero thickness around completely (i.e. by 360◦)? Clearly, the circle of radius 1 2 is a set where you can perform the rotation. Thus, this least area should be less than the area of this circle, i.e. less than π4 . In the same paper, Kakeya noticed that the three-cornered hypocycloid inscribed in a circle of radius 14 also works. This is less trivial to justify, but we can see this, for example, by noticing that the tangent line at any point of the hypocycloid meets the hypocycloid at two points, which are at unit distance from each other. The area of the hypocycloid is 2π ( 1 4 )2 = π8 < π4 , and for a while this was conjectured to be the least area that can be attained. Surprinsingly, however, it turns out there exists sets with arbitrarily small area that satisfy Kakeya’s condition. This was proven by Besicovitch in 1928 in [3], where he also gave a first explicit construction of a so-called Kakeya set with (Lebesgue) measure zero. Alternative constructions can be found in [19]. It was subsequently conjectured that a Kakeya set K in Rn (a compact set in Rn containing a line segment in every direction or more formally, a set so that for every x ∈ Sn−1, there exists y = f(x) ∈ K such that {y + tx |t ∈ [0, 1]} ⊂ K) has Hausdorff dimension n. (We won’t get into facts about Hausdorff dimension here, but we refer to [13, pp. 323-380] and [16] for the definition, properties, and several computations). Note that if K ′ ⊂ R2 is a Kakeya set, then K := K ′ × [0, 1]n−2 is a Kakeya set in Rn, and moreover, if K ′ has measure zero, then K also has measure zero, so Kakeya sets, even when considered in Rn, can be of arbitrarily small measure, given Besicovitch’s result. Thus, it is natural to ask if there’s any difference between small Kakeya sets of different (Euclidean) dimensions. And indeed, such a difference is believed to exist. They can be of zero measure, but they have Hausdorff dimension equal to the Euclidean dimension of the space they lie in. We record this claim below once again for reference purposes. Conjecture 1 (Kakeya Conjecture). If K ⊂ Rn is a Kakeya set, then its Hausdorff dimension is n. For n = 1, this is trivially true. For n = 2, this was confirmed by Davies in [5], with a relatively short proof. The proof is also available in [13], which we mentioned above. For n ≥ 3, however, the claim turns out significantly more diffcult to approach, being still open today, after almost a century from its naissance in literature. In 1995, in [17], Thomas Wolff, using purely geometric methods, proved the first important partial result, showing that the Hausdorff dimension of a Kakeya set in Rn must be at least 12(n + 2). For n = 3 and n = 4, this still almost represents the best known result so far. For larger n, the lower bound has been improved to 13 25n+ 12 15 by Bourgain in 2000, using additive combinatorics ideas, and afterwards, in 2002, to (2 − √ 2)(n − 4) + 4 by Katz and Tao, who refined Bourgain’s approach. We refer to [9] for the exposition of this line of thought. Given the incredibly numerous connections the Kakeya Conjecture has with areas of mathematics such as number theory, combinatorics, analysis, and PDE’s, there have been numerous attempts to consider different analogoues of the question instead, with the hope of getting some indirect information about this mysterious Hausdorff dimension that turns so difficult to establish for Kakeya sets. More popular surveys about such connections are the one by Wolff [19] and the one by Tao [14]. In this paper, we will focus only on one particular analogue which Wolff proposed in 1999 in [18]: the finite field version of the Kakeya problem. The setting is very simple and it is extremely convenient, since it avoids all the technical issues involving the Hausdorff dimension. We will be working over a finite field F with q elements. A Kakeya set in Fn is a set K ⊂ Fn containing a line in every direction, i.e. for all nonzero directions v ∈ Fn there is an a ∈ Fn such that a+ tv ∈ K for all t ∈ F. Will then such sets, no matter how small, always be n-dimensional? Or more precisely formulated, is there a positive constant Cn (depending on n) so that if K ⊂ Fn is a Kakeya set, then |K| ≥ Cnq? Modulo minor technicalities, the progress on answering this question was, until very recently, essentially the same as that of the original Euclidean Kakeya conjecture, with all the lower bounds on the Hausdorff dimension carrying to the finite field case. Nevertheless, in 2009, the finite field analogue was finally settled by Zeev Dvir, using the so-called polynomial method from algebraic extremal combinatorics. The proof is surprisingly short and really beautiful, and so the plan is to cover it in full detail below. Conjecture 2 (Finite Field Kakeya Conjecture). Let K ⊂ Fn be a Kakeya set. Then, K has cardinality at least Cnq. The proof is essentially after Dvir’s original paper [8] with some minor technical simplifications that have appeared afterwards in literature (essentially due to Alon and Tao). Proof of Conjecture 2. The idea is incredibly simple. First, one has to note that if F is any field, and K ⊂ Fn is any ”small” set, then there exists a polynomial P ∈ F[X1, . . . , Xn]−{0}, which has ”low” total degree, and vanishes on all of K. Afterwards, the only thing one has to do is to see that if K is a Kakeya set, then this polynomial P , which vanishes on all of K, must be in fact the zero polynomial. Combining these two facts shows that a Kakeya set cannot be ”small”. We isolate the two preliminary results that lie at the heart of the proof. The first one is a formalization of the first step mentioned above. Lemma 3. Let F be any field (not necessarily finite). If S ⊂ Fn is a finite set, and d is an integer such that |S| < ( n+d n ) , then there exists a nonzero polynomial in F[X1, . . . , Xn]−{0} which has degree at most d, and vanishes on all of S. Note that the n = 1 case of this Lemma is simply that for any subset S ⊂ F, there is a polynomial in F[X]− {0} of degree (at most) |S|, which vanishes on all of S, namely, ∏
منابع مشابه
Finite field Kakeya and Nikodym sets in three dimensions
We give improved lower bounds on the size of Kakeya and Nikodym sets over Fq. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in Fq, and show that this conjecture implies an optimal bound on the size of a Nikodym set. Finally, we study the notion of a weak Nikodym set and give improved, and in some special cases optimal, bounds fo...
متن کاملAlgebraic Methods in Discrete Analogs of the Kakeya Problem
We prove the joints conjecture, showing that for any N lines in R 3 , there are at most O(N 3 2) points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N 3). Both our ...
متن کاملar X iv : m at h / 99 06 09 7 v 2 [ m at h . C O ] 3 1 A ug 1 99 9 BOUNDS ON ARITHMETIC PROJECTIONS , AND APPLICATIONS TO THE KAKEYA CONJECTURE
Let A, B, be finite subsets of an abelian group, and let G ⊂ A × B be such that #A, #B, #{a + b : (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain obtained the bound of N 2− 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N
متن کاملThe Endpoint Case of the Bennett-carbery-tao Multilinear Kakeya Conjecture
We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery, and Tao. The proof uses the polynomial method introduced by Dvir. In [1], Bennett, Carbery, and Tao formulated a multilinear Kakeya conjecture, and they proved the conjecture except for the endpoint case. In this paper, we slightly sharpen their result by proving the endpoint case of the conjecture. Our method ...
متن کاملKakeya-Type Sets in Local Fields with Finite Residue Field
We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space K over a local field with finite residue field. The construction is an adaptation of the ideas appearing in [12] and [13]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one presented in [4] over Fq[[t]].
متن کامل