THE POLYNOMIAL METHOD LECTURE 1 THE KAKEYA CONJECTURE IN FINITE FIELDS MARCO VITTURI Abstract. This series of notes is intended to provide an introduction to the polynomial method through examples of its successful application - mainly in Harmonic Analysis, but other elds will also be considered (e.g. combinatorics, transcendence theory, etc). In this rst set of notes we introduce the Kakeya problem and then the analogous Kakeya problem in the nite elds. This introduction will necessarily be brief and won't do justice to this vast and beautiful eld. For a more comprehensive introduction we refer the reader to e.g. [KT02]. We then give the explicit construction of a Besicovitch set. Finally we present Dvir's solution to the Kakeya conjecture in the nite elds by use of the polynomial method. Contents 1. The Kakeya conjecture 1 2. The construction of Besicovitch 9 3. Dvir's proof 11 Appendix A. Proof of lemma 2 14 Appendix B. Multiplicity 15 References 16 Notation. throughout these notes, |E| will denote the cardinality of the set E when this is discrete and nite, and its Lebesgue measure otherwise. Context will usually suce to determine which is the case. The expression X À Y will mean that there is a constant C ¡ 0 s.t. X ¤ CY . We might specify the dependence of this constant on additional parameters e.g. α by writing X Àα Y when this is the case. By X Y we will mean X À Y and Y À X. 1. The Kakeya conjecture In 1917 Soichi Kakeya asked what was the minimal area of a region in which a unit segment could be turned around continuously by 360¥. This question became known as the Kakeya needle problem, and an answer came from a construction of Besicovitch which is described in section 2 (originally appearing in [Bes19]). Namely, for every ¡ 0 there exists a set that contains a unit segment in every direction but has area smaller than , and therefore there is no minimal area to such a region - a result that was somewhat surprising for the time1. A generalization of the denitions leads to2 1It was believed a deltoid curve was a sharp example. 2We drop the requirement that the unit segment can be turned continuously because it won't be relevant to us. 1 2 MARCO VITTURI Denition 1. A Kakeya set is a set E Rn that contains a unit segment in every direction. Thus the construction of Besicovitch shows that for n 2 these sets can have ar- bitrarily small Lebesgue measure - and indeed there are Kakeya sets with Lebesgue measure exactly zero3. Moreover, by taking a solid of revolution with cross section a Besicovitch set it can be seen that we can build such sets with Lebesgue measure zero in dimensions n ¡ 2 too. A question arises then: what's the dimension of these sets? this question can of course be asked with respect to various notions of dimension - e.g. Hausdor, upper/lower Minkowski, packing, etc. In 1971 Davies proved in [Dav71] that, al- though the Lebesgue measure can be zero, when n 2 a Kakeya set must always have Hausdor dimension 2. This result led to the conjecture that the (Hausdor or Minkowski) dimension would still be n for any Kakeya set: Conjecture 1 (The Kakeya Conjecture (for Hausdor dimension)). Let K be a Kakeya set in n. Then p q . R dimHaus K n Conjecture 2 (The Kakeya Conjecture (for Minkowski dimension)). Let K be a Kakeya set in n. Then p q . R dimMink K n There is an analogous conjecture with upper Minkowski dimension, of course. The conjectures are still wide open for n ¡ 2, although a huge number of partial results exist in the literature. We don't attempt to survey them in here. See [KT02] for a summary up to the year 2000 (thus somewhat outdated, as it doesn't contain the result of Dvir presented in these notes). It is still open even in the weaker case4 of upper or lower Minkowski dimension. Recall that the upper Minkowski dimension is dened as log |Eδ| dimMinkpEq : lim sup n ¡ ; δÑ0 log δ where Eδ is the δ-neighbourhood of E, and analogously the lower Minkowski di- mension is dened as | | p q ¡ log Eδ dimMink E : lim inf n ; δÑ0 log δ when they coincide, they are commonly referred to as the Minkowski dimension of E. Here we assume that 0 δ ! 1. If K is a Kakeya set in Rn then its δ-neighbourhood contains a δ-tube in every direction, where by δ-tube we mean exactly the δ- neighbourhood of a unit segment. Since the tubes have cross section of radius δ it doesn't make sense to distinguish directions that dier by an angle less than δ (they become indistinguishable); therefore we might restrict ourselves to sets that are unions of N distinct δ-tubes that have δ-separated directions, in the sense that 1 1 ¡ 1 whenever T;T are δ-tubes with distinct directions !; ! P Sn 1 then =!; ! Á δ. Notice the δ-separation condition forces N to be Opδ1¡nq, and so we can assume - and we will do so from now on - that the collection is critical in the sense that N δ1¡n. With this setup we have that conjecture 2 is equivalent to 3They can be built by specializing the construction in section 2 to nested collections of paral- lelograms, for example, and then taking intersections. 4Weaker because p q ¤ p q ¤ p q when they are all dened. dimHaus E dimMink E dimMink E THE KAKEYA PROBLEM 3 Conjecture 2 (The Kakeya Conjecture (for Minkowski dimension)). Let t uN Tj j1 be a collection of δ-tubes in Rn with δ-separated directions. Then for every " ¡ 0 § ¤N § § § " Tj Á" δ : (1) j1 Remark 1. At the heuristical level, conjecture 1 is saying that the δ-separated δ-tubes are essentially disjoint, up to a logarithmic error. Indeed, if they were exactly disjoint, then we'd have ¤N n¡1 | Tj| Nδ 1: j1 We can now sketch the proof of Davies for the weaker conjecture 1. With a little more eort it can be turned in a proof for the Hausdor dimension. Observe that ¸ ¸ 2 | X | χTj L2 Tj Tk ; j j;k let the direction of tube be given by P 1, then Tj !j S ¸ Op¸δ¡1q ¸ |Tj X Tk| |Tj X Tk|: j;k k1 j;k : |=!j ;!k|kδ Now, it's a simple geometric fact that if the angle between Tj and Tk is θ then δ2 |T X T | À ; j k |θ| so that Opδ¡1q Opδ¡1q Opδ¡1q ¸ ¸ ¸ δ2 ¸ 1 |T X T | À N log δ¡1: j k kδ k k1 j;k : k1 k1 |=!j ;!k|kδ Thus by Cauchy-Schwarz ¸ ¸ § ¤ § 2 2 2 1 χ ¤ χ χ À log δ¡1§ T §; Tj L1 Tj L2 j Tj L2 j j j j which gives the desired inequality. There is also a stronger variant of the Kakeya conjecture(s) that is more directly relevant to harmonic analysts, namely the Kakeya maximal function conjecture. This can be stated in the language of δ-tubes too: Conjecture 3 (The Kakeya maximal function conjecture). Let t uN be a col- Tj j1 lection of δ-tubes in Rn with δ-separated directions (as above). Then for every " ¡ 0 ¸N À ¡" (2) χTj n{pn¡1qp nq " δ : L R j1 The conjecture can also be put in the more suggestive form ¸N ¸N À ¡" | |qpn¡1q{n χTj n{pn¡1qp nq " δ Tj ; L R j1 j1 4 MARCO VITTURI which is another assertion of essential disjointness of the tubes. A few observations are in order. Firstly, for 1 d ¤ n (not necessarily integer) notice that (2) implies the inequality ¸N ¡ n ¡ À 1 d " (3) χTj d{pd¡1qp nq " δ : L R j1 Indeed, we have the (sharp) trivial inequality ¸N À 1¡n χTj 8p nq δ ; L R j1 thus if P r s is such that d¡1 p ¡ q n¡1 1 , then by logarithmic Hölder δ 0; 1 d 1 θ n θ 8 inequality ¸N ¡ ¡ ¡ ¡ p q ¡ n À p "q1 θp 1 nqθ O " 1 d χTj d{pd¡1qp nq " δ δ δ δ : L R j1 Notice that n d for , so n is the endpoint of a family of estimates n¡1 d¡1 d n n¡1 really. Secondly, it is stronger than conjecture 1 because it implies it; notice in particular that the expression on the left in the conjectured inequality is not just measuring the Lebesgue measure of the union but is taking into account the multiplicity of overlapping too, so this is to be expected. The implication goes as follows (im- plicitely already used above): by Hölder ¸N ¸N § ¤N § ¡"§ §1{n ¤ j À χTj 1p nq χTj n{pn¡1qp nq χ T np nq " δ Tj ; L R L R j L R j1 j1 j1 but then ¸N ¸N | | χTj 1p nq Tj 1; L R j1 j1 and thus (1) is implied. As it turns out, conjecture 3 also implies the Kakeya conjecture for Hausdor dimension, but we won't comment further on this here. Thirdly, the name of the conjecture refers to an actual maximal function that is hidden in the statement through duality. The Kakeya maximal function is given by » ¦p q 1 | p q| fδ ! : sup | | f y dy; T δ¡tube s.t.