Kakeya Sets Research Articles

Hausdorff dimension of Besicovitch sets of Cantor graphs

AbstractWe consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets Γ, that is, sets that contain a rotated copy of Γ in each direction. We show that for a large class of Cantor sets C and Cantor‐graphs Γ built on C, the Hausdorff dimension of any Γ‐Besicovitch set must be at least , where .

The Kakeya Set conjecture over Z/NZ for general N

The Kakeya Set conjecture over Z/NZ for general N

A note on Kakeya sets of horizontal and SL(2)$SL(2)$ lines

AbstractWe consider unions of lines in . These are lines of the form where . We show that if is a Kakeya set of lines, then the union has Hausdorff dimension 3. This answers a question of Wang and Zahl. The lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point‐line duality principle between horizontal and conical lines in , combined with recent work on restricted families of projections to planes, due to Gan, Guo, Guth, Harris, Maldague and Wang. Our result also has a corollary for Nikodym sets associated with horizontal lines, which answers a special case of a question of Kim.

Nonstandard approximations to Besicovitch sets

A nonstandard approximation to a Besicovitch set in R2 is constructed. It is shown that this approximation easily yields the correct upper Minkowski dimension in the two-dimensional case through a direct computation. By constructing an approximation to a Besicovitch set in three dimensions, we use a theorem of Guth and Katz to obtain an upper bound on the number of intersections of line segments in nonstandard terms.

On the dimension of Kakeya sets in the first Heisenberg group

We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the { x o y } \{xoy\} -plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

On the Fourier dimension of (d, k)-sets and Kakeya sets with restricted directions

AbstractA (d, k)-set is a subset of$$\mathbb {R}^d$$Rdcontaining ak-dimensional unit ball of all possible orientations. Using an approach of D. Oberlin we prove various Fourier dimension estimates for compact (d, k)-sets. Our main interest is in restricted (d, k)-sets, where the set only contains unit balls with a restricted set of possible orientations$$\Gamma $$Γ. In this setting our estimates depend on the Hausdorff dimension of$$\Gamma $$Γand can sometimes be improved if additional geometric properties of$$\Gamma $$Γare assumed. We are led to consider cones and prove that the cone in$$\mathbb {R}^{d+1}$$Rd+1has Fourier dimension$$d-1$$d-1, which may be of interest in its own right.

Proof of the Kakeya set conjecture over rings of integers modulo square-free N

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

A note on large Kakeya sets

Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q 2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q 2 – q + q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.

On the Intersection Distribution of Degree Three Polynomials and Related Topics

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed by Li and Pott [\emph{Finite Fields and Their Applications, 66 (2020)}], which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in\mathbb{F}_q\}$. The intersection distribution has an underlying geometric interpretation, which indicates the intersection pattern between the graph of $f$ and the lines in the affine plane $AG(2,q)$. When $q$ is even, the long-standing open problem of classifying o-polynomials can be rephrased in a simple way, namely, classifying all polynomials which have the same intersection distribution as $x^2$. Inspired by this connection, we proceed to consider the next simplest case and derive the intersection distribution for all degree three polynomials over $\mathbb{F}_q$ with $q$ both odd and even. Moreover, we initiate to classify all monomials having the same intersection distribution as $x^3$, where some characterizations of such monomials are obtained and a conjecture is proposed. In addition, two applications of the intersection distributions of degree three polynomials are presented. The first one is the construction of nonisomorphic Steiner triple systems and the second one produces infinite families of Kakeya sets in affine planes with previously unknown sizes.

A promenade through correct test sequences I: Degree of constructible sets, Bézout's Inequality and density

The notion of correct test sequence was introduced in [27]. It has been widely used to design probabilistic algorithms for Polynomial Identity Testing (PIT). In this manuscript we study foundations and generalizations of this notion. We show that correct test sequences are almost omnipresent in the mathematical literature: As hitting sets in PIT, in Function Identity Testing, as norming sets ([1]) or in Reproducing Kernel Hilbert Spaces context. We generalize the main statement of [27] proving that short correct test sequences for lists of polynomials are densely distributed in any constructible set of accurate co–dimension and degree. In order to prove this result we introduce and develop the theory of degree of constructible sets, generalizing [25] and proving two Bezout's Inequalities for two different notions of degree. We exhibit the strength and limitations of correct test sequences with a randomized efficient algorithm for the “Suite Sécante” Problem. Our algorithm generalizes [27], admitting a bigger class of sampling sets, also proving that PIT is in RPK. We reformulate, prove and generalize two results of the Polynomial Method: Dvir's exponential lower bounds for Kakeya sets and Alon's Combinatorial Nullstellensatz.

A Kakeya maximal function estimate in four dimensions using planebrushes

We obtain an improved Kakeya maximal function estimate in \mathbb R^4 using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff’s hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth–Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in \mathbb R^4 at dimension 3.059. As a consequence, every Besicovitch set in \mathbb R^4 must have Hausdorff dimension at least 3.059.

The Intersection of Typical Besicovitch Sets with Lines

We show that a typical Besicovitch set \(B\) has intersections of measure zero with every line not contained in it. Moreover, every line in \(B\) intersects the union of all the other lines in \(B\) in a set of measure zero.

Intersection distribution, non-hitting index and Kakeya sets in affine planes

In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set S of size q+1 in the classical projective plane PG(2,q), where the intersection distribution of S indicates the intersection pattern between S and the lines in PG(2,q). The second one relates to a polynomial f over a finite field Fq, where the intersection distribution of f records an overall distribution property of a collection of polynomials {f(x)+cx|c∈Fq}. These two perspectives are closely related, in the sense that each polynomial produces a (q+1)-set in a canonical way and conversely, each (q+1)-set with certain property has a polynomial representation. Indeed, the intersection distribution provides a new angle to distinguish polynomials over finite fields, based on the geometric property of the corresponding (q+1)-sets. Among the intersection distribution, we identify a particularly interesting quantity named non-hitting index. For a point set S, its non-hitting index counts the number of lines in PG(2,q) which do not hit S. For a polynomial f over a finite field Fq, its non-hitting index gives the summation of the sizes of q value sets {f(x)+cx|x∈Fq}, where c∈Fq. We derive lower and upper bounds on the non-hitting index and show that the non-hitting index contains much information about the corresponding set and the polynomial. More precisely, using a geometric approach, we show that the non-hitting index is sufficient to characterize the corresponding point set and the polynomial, when it is very close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the non-hitting index and the intersection distribution of several families of point sets and polynomials. As an application, we consider the determination of the sizes of Kakeya sets in affine planes. The polynomial viewpoint of intersection distributions enable us to compute the size of a few families of Kakeya sets with nice algebraic properties. Finally, we describe the connection between these new concepts and various known results developed in different contexts and propose some future research problems.

The Spherical Kakeya Problem in Finite Fields

We study subsets of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a sphere for each first coordinate of the center. We call such sets radii spherical Kakeya sets and center spherical Kakeya sets, respectively. For $n\ge 4$ we prove a general lower bound on the size of any set containing $q-1$ different spheres which applies to both kinds of spherical Kakeya sets. We provide constructions which meet the main terms of this lower bound. We also give a construction showing that we cannot get a lower bound of order of magnitude $q^n$ if we take lower-dimensional objects, such as circles in $\mathbb{F}_q^3$ instead of spheres, showing that there are significant differences to the line Kakeya problem. Finally, we study the case of dimension $n=1$, which is different and equivalent to the study of sum and difference sets that cover $\mathbb{F}_q$.

Conical Kakeya and Nikodym sets in finite fields

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.)Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

The Kakeya Problem

This research paper concentrates on the Kakeya problem. After the introduction of historical issue, we provide a thorough presentation of the results of Kakeya problem with some examples of the early solutions as well as the proof of the final outcome of this problem, the solution of which is known as Besicovitch Set. We give 3 different construction of Besicovitch set as well as the intuition of construction, which is related to iterated integral of 2-variable real function. We also give the Cunningham construction in which the area of a simply connected Kakeya set can also tend to 0. Furthermore, we generalize the process of generating a Kakeya set into a Kakeya dynamic. The definition of multiplicity enables us to estimate the area of a Kakeya set. In following discussion we provided a conjecture related to the solution in particular range. Finally, the derivation of the Kakeya problem is presented.

On the arithmetic Kakeya conjecture of Katz and Tao

The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5–6):625–630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in {mathbf {R}}^n is n. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.

Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q-1, our first data structure relies on (d+1)^{n+2} tabulated values of P to produce the value of P at any of the q^n points using O(nqd^2) arithmetic operations in the finite field. Assuming that s divides d and d / s divides q-1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s+1)^{n+s} tabulated values to produce the value of P at any point using Oleft( nq^ssqright) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (Duke Math J 121(1):35–74, 2004), Saraf and Sudan (Anal PDE 1(3):375–379, 2008) and Dvir (Incidence theorems and their applications, 2012. arXiv:1208.5073) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for computing integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (Partition functions of strongly correlated electron systems as fermionants, 2011. arXiv:1108.2461v1) that captures numerous fundamental algebraic and combinatorial functions such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. In particular, a corollary of our main theorem for fermionants is that the permanent of an mtimes m integer matrix with entries bounded in absolute value by a constant can be computed in time 2^{m-Omega left( sqrt{m/log log m}right) }, improving an earlier algorithm of Björklund (in: Proceedings of the 15th SWAT, vol 17, pp 1–11, 2016) that runs in time 2^{m-Omega left( sqrt{m/log m}right) }.

Almost all non-archimedean Kakeya sets have measure zero

We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.

An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³

We prove that every Besicovitch set in R 3 \mathbb {R}^3 must have Hausdorff dimension at least 5 / 2 + ϵ 0 5/2+\epsilon _0 for some small constant ϵ 0 > 0 \epsilon _0>0 . This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the SL 2 \operatorname {SL}_2 example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension 5 / 2 5/2 . We believe this example may be an interesting object for future study.