THE P -ADIC KAKEYA CONJECTURE BODAN ARSOVSKI n Abstract. We prove that all bounded subsets of Qp containing a line segment of unit length in every direction have Hausdorff and Minkowski dimension n. This is the analogue of the classical Kakeya conjecture with R replaced by Qp. 1. Introduction In 1917, Kakeya posed the Kakeya needle problem, asking about the minimum area of a region in the plane in which a needle of unit length can be rotated around by 360◦. The problem reduces to asking about the minimum area of a Kakeya set in R2—that is, a bounded subset of R2 containing a line segment of unit length in every direction. Besicovitch [Bes63] showed that, in a certain sense, the answer is “arbi- trarily small”, by constructing a Kakeya set in R2 of Lebesgue measure zero. On the other hand, Davies [Dav71] showed that, in a different sense, the answer is “large”: a Kakeya set in R2 must have Minkowski dimension 2. The construction of [Bes63] extends immediately to Rn with n > 3. The higher-dimensional analogue of the result of [Dav71] is much more difficult: it is the Minkowski dimension version of the notorious Kakeya conjecture, which is one of the most important open problems in geometric measure theory, and analysis in general. Conjecture A (Kakeya). Let n be a positive integer. All Kakeya sets n arXiv:2108.03750v2 [math.NT] 13 Sep 2021 in R have Hausdorff and Minkowski dimension n. The Kakeya conjecture has deep connections with harmonic analysis, PDEs, and combinatorics among other fields, and it is open for n > 3. For n 3, 4 , the state of the art are the results of Wolff [Wol95], Katz–Łaba–Tao∈{ } [KŁT00], and Łaba–Tao [ŁT01] that all Kakeya sets in Rn have Hausdorff dimension at least (n + 2)/2 and Minkowski di- mension at least (n + 2)/2+ ε, for some ε> 0. For n > 5, the state of the art is the result of Katz–Tao [KT02] that all Kakeya sets in Rn have Hausdorff dimension at least (2 √2)(n 4) + 3 and Minkowski dimension at least (n 1)/α + 1, where− α 1−.675 is the largest root of the polynomial z3 −4z + 2. ≈ − 1 2 BODANARSOVSKI 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.