The P-Adic Kakeya Conjecture
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As a possible approach to the Kakeya conjecture, Wolff [Wol99] sug- gested an analogous conjecture over finite fields, and this conjecture was proved by Dvir [Dvi09]. As noted by Ellenberg–Oberlin–Tao [EOT10], the analogy between the classical and finite field Kakeya conjectures is imperfect for two reasons. Firstly, there is no non-trivial natural no- tion of distance in finite vector spaces. Secondly, the result of [Dvi09] n is too strong: all Kakeya sets in Fq (where Fq is a finite field) have, in a certain sense, positive measure. In particular, there is no variant in n Fq of the construction of Besicovitch of a Kakeya set of measure zero. Thus Ellenberg–Oberlin–Tao [EOT10] suggested analogues of the Kakeya conjecture in settings that are topologically more similar to n R , namely the Kakeya conjectures over Qp and Fq[[t]]. Both shortcom- n n ings are resolved in these settings. Firstly, Qp and Fq[[t]] have metrics that have multiple scales, just like the metric of Rn. Secondly, there are variants of the Besicovitch construction in both settings: Dummit– n Hablicsek [DH11] constructed a Kakeya set in Fq[[t]] of measure zero, Hickman–Wright [HW18] constructed a Kakeya set in (the closed unit n n ball Zp of) Qp of measure zero, and Fraser [Fra16] constructed more generally a Kakeya set in Kn of measure zero, for any local field K with finite residue field. Dummit–Hablicsek [DH11] further proved the 2 2 analogues of the result of Davies [Dav71] in Zp and Fq[[t]] . All of these theorems suggest that the Kakeya conjectures over Qp and Fq[[t]] could be good models for the classical Kakeya conjecture over R. In this article we prove the Kakeya conjecture over Qp: Theorem 1. Let p be a prime number and n a positive integer. All n Kakeya sets in Qp have Hausdorff and Minkowski dimension n. Let us give a brief explanation of what this theorem means, in terms of where Qp stands in relation to R. The field of p-adic numbers Qp is the completion of Q with respect to the p-adic norm x = p− ordp(x). | |p In fact, the only completions of Q are R and Qp (for varying p). Topo- logically, Qp enjoys many of the properties enjoyed by R (for example, it is separated, locally compact, and closed balls are compact), though there are important differences (for example, the metric is an ultramet- ric, it is totally disconnected, every open ball is closed, and there are countably many open balls). As a topological space, the closed unit ball Zp of Qp is homeomorphic to the Cantor set C [0, 1]. There is a well developed analogue of classical analysis over the⊆p-adic numbers (p-adic analysis), which has historically been of particular interest in number theory. The proof is inspired by ideas of Dvir [Dvi09] and Dhar–Dvir [DD], and the method can be summarized as “a variant of the polynomial THE P -ADIC KAKEYA CONJECTURE 3 method over discrete valuation rings”. In particular, a key component is a discrete valuation variant of the Schwartz–Zippel lemma.
2. Proof Let p be a prime number, n and k be positive integers, and q = pk. Let F = Fp, and R = Z/qZ. Let Qp denote the p-adic numbers, and Zp denote the p-adic integers. That is, Qp is the completion of Q with − ordp(x) respect to the norm x p = p , and Zp is the closed unit ball in | | n Qp centered at the origin Zp = z Qp z p 6 1 Qp. We equip Qp with the maximum product norm{ ∈ || | } ⊆ (x ,...,x ) = max x , for (x ,...,x ) Qn, | 1 n |p i∈{1,...,n} | i|p 1 n ∈ p n and by the Hausdorff and Minkowski dimensions of a subset of Qp we of course mean the Hausdorff and Minkowski dimensions with respect to this norm. Let ζ be a primitive qth root of unity in the algebraic closure of Qp. Let T = Zp(ζ)[t], and let T = F[t] be the reduction of T modulo the maximal ideal m =(ζ 1) of Z [ζ]. Thus T is a subset − p of the discrete valuation ring T (t), whose field of fractions is F(t), and q−1 whose valuation we denote by vt. Let C = 1,..., (1 + t) T , and let C T be the reduction of C modulo m{. } ⊆ ⊆ Definition 2. 1 Let ε, δ [0, 1]. Let us normalize the measures of the ∈ n n closed unit balls Zp Qp and Zp Qp to be equal to 1. ⊆ n ⊆ n An (ε, δ)-Kakeya set in Qp is a bounded subset of Qp containing at least ε of a line segment of unit length in at least δ of directions—that is, K n a bounded subset Qp with the following property: there is a subset D Zn of measure at⊆ least δ such that, for all x D, there are b Qn ⊆ p ∈ x ∈ p and a subset Λx Zp of measure at least ε such that bx + λx K for all λ Λ . ⊆ ∈ ∈ x We obtain theorem 1 as a corollary to the following theorem, which gives a lower bound on the number of closed balls of radii p−k needed n to cover an (ε, δ)-Kakeya set in Zp . Theorem 3. Let p be a prime number, and n and k positive integers. n Let ε, δ [0, 1]. An (ε, δ)-Kakeya set in Zp cannot be covered by fewer than ∈ εδq/pkn+n−1 n closed balls of radii p−k.
1 n The definition of a Kakeya set in Qp is slightly different in [EOT10, HW18, DD] n−1 (only directions in P (Zp) are considered), but it is equivalent to our definition. 4 BODANARSOVSKI
Proof that theorem 3 implies theorem 1. Suppose that 1 1 ε, δ &p,n −k = . logp(1/p ) k K n An (ε, δ)-Kakeya set Qp is bounded, and therefore contained in finitely many disjoint closed⊆ unit balls. Suppose that K is covered by N(K ,p−k) closed balls of radii p−k. By translating and superimposing n these closed unit balls onto the closed unit ball Zp , we end up with an K˜ n K −k (ε, δ)-Kakeya set Zp that is covered by N( ,p ) closed balls of radii p−k, implying,⊆ by theorem 3, that K −k (εδ)npkn pkn N( ,p ) > (pkn)nn! &p,n k3n . The standard dyadic method of Bourgain [Bou99] can then be used to conclude that the Hausdorff dimension of K is n (and therefore so is the Minkowski dimension).
Lemma 4 (Discrete valuation Schwartz–Zippel). Let f T [z1,...,zn] m1 mn ∈ be non-zero, let the coefficient of z1 zn in f be c, and suppose that the coefficient of any monomial that··· is larger than zm1 zmn in the 1 ··· n lexicographic order is zero. Suppose that mi 6 q, for all i 1,...,n , and let ν be a non-negative integer. The number of points∈{s Cn such} that v (f(s)) > v (c)+ np−νq is at most kpν+1(m + + m∈ )qn−1. t t 1 ··· n Proof. We use induction. Let n = 1, and suppose, on the contrary, ν+1 that there is a subset S C such that S > kp m1, on which f has v -valuation at least v⊆ (c)+ p−νq. Let| |S = (1 + t)l l L , so t t { | ∈ } that L 0,...,q 1 and L = S . Then L has a subset L0 that contains⊆{ no pair of− distinct} elements| | | | whose difference is divisible by p−ν−⌈logp k⌉q, and such that L = m +1. Let S be the subset of S | 0| 1 0 that corresponds to L0. By theorem 1.4 of [Cla14] (applied to the field F(t) that contains T ), −1 1= c (s u) f(s). Ps∈S0 Qu∈S0\{s} − Therefore, there is some s S such that v (s u) > p−νq. ∈ 0 t Qu∈S0\{s} − Let L0 = l0,...,lm1 . By using Kummer’s theorem on the p-adic val- { } l uations of binomial coefficients, which implies that w is a unit in F if and only if every p-adic digit of w is at most as large as the correspond- α β ordp(α−β) ing p-adic digit of l, we get that vt (1 + t) (1 + t) = p . Therefore, v (s u) is also equal to− t Qu∈S0\{s} − m1 v (1 + t)li (1 + t)l0 = m1 pordp(li−l0) Pi=1 t − Pi=1 6 ⌈logp m1⌉ p−ν−⌈logp k⌉−jq(pj pj−1) Pj=1 −
n n Proof of theorem 3. The image under the projection Zp R of an n n → (ε, δ)-Kakeya set in Zp is a subset of R with the following property: there is a subset D Rn of size at least δ R n such that, for all x D, there are b Rn and⊆ a subset Λ R |of| size at least εq such∈ that x ∈ x ⊆ bx + λx S for all λ Λx. Our goal is to show that such a subset of Rn has size∈ at least ∈ εδq/pkn+n−1 l = n . Suppose, on the contrary, that there is such a subset S Rn of size less than l. Let ⊆ S = (ζs1 ,...,ζsn ) Z [ζ]n (s ,...,s ) S . C { ∈ p | 1 n ∈ } Then there is a non-zero polynomial f Z [ζ][z ,...,z ] of total degree ∈ p 1 n less than εδq/pkn that vanishes on SC . By normalizing, we may assume that f m[z ,...,z ]. Therefore, for all x D and all λ Λ , 6∈ 1 n ∈ ∈ x (bx)1+λx1 (bx)n+λxn λ f(ζ ,...,ζ )= gx(ζ )=0, for some polynomial gx Zp[ζ][z]. This means that gx(z) is divisible by (z ζλ), implying∈ that the reduction g of g modulo m is Qλ∈Λx x x divisible by− (z 1)⌈εq⌉, in turn implying that − x1 xn vt f((1 + t) ,..., (1 + t) ) > εq. Let n C = ((1 + t)x1 ,..., (1 + t)xn ) C (x ,...,x ) D . D n ⊆ | 1 n ∈ o 6 BODANARSOVSKI
Then, for all s CD, vt f(s) > εq. As f T [z1,...,zn] is non-zero, has total degree∈ less than εδq/pkn 6 q, and∈ has coefficients in F, it follows from lemma 4 with some c F× and η = log (n/ε) that ∈ p δqn 6 D = C < (kn/ε)(εδq/pkn)qn−1 = δqn, | | | D| which is a contradiction. References [Bes63] Abram S. Besicovitch, The Kakeya problem, Am. Math. Mon. 70 (1963), no. 7, 697–706. [Bou99] Jean Bourgain, On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), no. 2, 256–282. [Cla14] Pete L. Clark, The combinatorial nullstellensätze revisited, Electron. J. Combin. 21 (2014), no. 4, P4.15. [Dav71] Roy O. Davies, Some remarks on the Kakeya problem, Math. Proc. Cam- bridge Philos. Soc. 69 (1971), no. 3, 417–421. [DD] Manik Dhar and Zeev Dvir, Proof of the Kakeya set conjecture over rings of integers modulo square-free N, preprint. [DH11] Evan P. Dummit and Márton Hablicsek, Kakeya sets over non- archimedean local rings, Mathematika 59 (2011), no. 2, 257–266. [Dvi09] Zeev Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), 1093–1097. [EOT10] Jordan S. Ellenberg, Richard Oberlin, and Terence Tao, The Kakeya set and maximal conjectures for algebraic varieties over finite fields, Mathe- matika 56 (2010), no. 1, 1–25. [Fra16] Robert Fraser, Kakeya-type sets in local fields with finite residue field, Mathematika 62 (2016), no. 2, 614–629. [HW18] Jonathan Hickman and James Wright, The Fourier restriction and Kakeya problems over rings of integers modulo N, Discrete Anal. 11 (2018), 18pp. [KŁT00] Nets H. Katz, Izabella Łaba, and Terence Tao, An improved bound on the Minkowski dimension of Besicovitch sets in R3, Ann. of Math. (2) 152 (2000), no. 2, 383–446. [KT02] Nets H. Katz and Terence Tao, New bounds for Kakeya problems, J. Anal. Math. 87 (2002), 231–263. [ŁT01] Izabella Łaba and Terence Tao, An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension, Geom. Funct. Anal. 11 (2001), no. 2, 773–806. [Wol95] Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoam. 11 (1995), 651–674. [Wol99] , Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996) (1999), 129–162.
School of Mathematics and Statistics, University of Sheffield Email address: [email protected] URL: bodanarsovski.github.io