Kakeya Sets Research Articles

On Segre's Lemma of Tangents

Segre's lemma of tangents dates back to the 1950's when he used it in the proof of his “arc is a conic” theorem. Since then it has been used as a tool to prove results about various objects including internal nuclei, Kakeya sets, sets with few odd secants and further results on arcs. Here, we survey some of these results and report on how re-formulations of Segre's lemma of tangents are leading to new results.

On the intersections of the Besicovitch sets and the Erdös–Rényi sets

We are interested in two properties of real numbers: the first one is the property of having given digit frequencies in the binary expansion, such as the well known Besicovitch sets, and the second one is the property of having the longest run of heads in the n independent Bernoulli trials, that is the so called Erdos–Renyi sets. In 2013, Chen and Wen (J Math Anal Appl 401:29–37, 2013) considered the intersections of these two kinds of sets by determining the Hausdorff dimension of the sets $$\begin{aligned} \left\{ x\in [0,1):~\liminf \limits _{n\rightarrow \infty }\frac{S_{n}(x)}{n}\ge \alpha ,~ \lim \limits _{n\rightarrow \infty }\frac{R_{n}(x)}{\log _{2}n}=\beta \right\} ,~~0\le \alpha \le 1,~0\le \beta \le +\,\infty , \end{aligned}$$ where $$S_{n}(x)$$ denotes the summation of the first n digits and $$R_{n}(x)$$ is the maximal length of consecutive one digits in the first n terms of the dyadic expansion of $$x\in [0,1)$$ . In the present paper, we complement this result by computing the Hausdorff dimension of the following sets $$\begin{aligned} \left\{ x\in [0,1):~\lim \limits _{n\rightarrow \infty }\frac{S_{n}(x)}{n}=\alpha ,~\lim \limits _{n\rightarrow \infty }\frac{R_{n}(x)}{\log _{2}n}=\beta \right\} , ~~0\le \alpha \le 1,~0\le \beta \le +\,\infty . \end{aligned}$$

A discretized Severi-type theorem with applications to harmonic analysis

In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $\delta^{-2-\varepsilon}$ direction-separated $\delta$-tubes can be contained in the $\delta$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of Guth and the author, this result implies a Kakeya maximal function estimate at dimension $3+1/28$, which is an improvement over the previous bound of $3$ due to Wolff. As a consequence, we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/28$. Recently, Demeter showed that any improvement over Wolff's bound for the Kakeya maximal function yields new bounds on the restriction problem for the paraboloid in $\mathbb{R}^4$.

The Fourier restriction and Kakeya problems over rings of integers modulo N

The Fourier restriction and Kakeya problems over rings of integers modulo N

Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

We formulate the conditional Kolmogorov complexity of x given y at precision r , where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways; (1) We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E . We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. (2)We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim( x | y ) and Dim( x | y ) of x given y , where x and y are points in Euclidean spaces. Intuitively, these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y . We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim( x ) and Dim( x ) and the mutual dimensions mdim( x : y ) and Mdim( x : y ).

Polynomial Wolff axioms and Kakeya-type estimates in R4

We establish new linear and trilinear bounds for collections of tubes in $\mathbb{R}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $\delta$-tubes satisfies the Wolff axioms if not too many tubes can be contained in the $\delta$-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the $\delta$-neighborhood of a low degree algebraic variety. First, we prove that if a set of $\delta^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least $\delta^{1-1/40}$. We also prove a more technical statement which is analogous to a maximal function estimate at dimension $3+1/40$. Second, we prove that if a collection of $\delta^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least $\delta^{3/4}$. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension $3+1/4$. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension $3+1/40$, and in particular this implies that every Kakeya set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/40$. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

Dimension estimates for Kakeya sets defined in an axiomatic setting

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying certain axioms. These allow us to prove some lower bounds for the Hausdorff dimension of generalized Kakeya sets using two methods introduced in the Euclidean context by Bourgain and Wolff. With this abstract setup we can deal with many special cases in a unified way, recovering some known results and proving new ones. In the second part we present various applications. We recover some of the known estimates for the classical Kakeya and Nikodym sets and for curved Kakeya sets. Moreover, we prove lower bounds for the dimension of sets containing a segment in a line through every point of a hyperplane and of an (n-1)-rectifiable set. We then show dimension estimates for Furstenberg type sets (already known in the plane) and for the classical Kakeya sets with respect to a metric that is homogeneous under non-isotropic dilations and in which balls are rectangular boxes with sides parallel to the coordinate axis. Finally, we prove lower bounds for the classical bounded Kakeya sets and a natural modification of them in Carnot groups of step two whose second layer has dimension one, such as the Heisenberg group. On the other hand, if the dimension is bigger than one we show that we cannot use this approach.

Some Results in Support of the Kakeya Conjecture

A Besicovitch set is a subset of R d that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least (d + 1)/2 (better estimates are available in the literature). We also study the ‘generic validity’ of the Kakeya conjecture in the setting of Baire Category and prove that typical Besicovitch sets have full upper box-counting dimension. We also study a weaker version of the Kakeya problem where unit line segments are replaced by half-infinite lines. We prove that such ‘half-extended Besicovitch sets’ have full Assouad dimension. This can be viewed as full resolution of a (much weakened) version of the Kakeya problem.

Quasisymmetric Maps on Kakeya Sets

I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of a.e. line segment inside $K$ has dimension at most $\min\{2n/(n + 2),7n/(4n + 3)\}$. The Kakeya maximal function conjecture implies that the bounds can be improved to $n$ and $1$, respectively.

Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem

We prove the discrete analogue of Kakeya conjecture over $$\mathbb {R}^n$$ . This result suggests that a (hypothetically) low-dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all dimensions. The main tool of the proof is a theorem of Wongkew (Pac J Math 159:177–184, 2003), which states that a low-degree polynomial cannot have its zero set being too dense inside the unit cube, coupled with Dvir-type polynomial arguments (Dvir in J Am Math Soc 22(4):1093–1097, 2009). From the viewpoint of the proofs, we also state a conjecture that is stronger than and almost equivalent to the (lower) Minkowski version of the Kakeya conjecture and prove some results toward it. We also present our own version of the proof of the theorem in Wongkew (Pac J Math 159:177–184, 2003). Our proof shows that this theorem follows from a combination of properties of zero sets of polynomials and a general proposition about hypersurfaces which might be of independent interest. Finally, we discuss how to generalize Bourgain’s conjecture to high dimensions, along the way providing a counterexample to the most naive generalization.

Are lines much bigger than line segments?

We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in R^n has Hausdorff dimension at least n-1 and (upper) Minkowski dimension n. We also prove that conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.

The small Kakeya sets in T2*(C), C a conic

A Kakeya set in the linear representation , a nonsingular conic, is the point set covered by a set of lines, one through each point of . In this article, we classify the small Kakeya sets in . The smallest Kakeya sets have size , and all Kakeya sets with weight less than are classified: there are approximately types.

Kakeya-Type Sets Over Cantor Sets of Directions in $$\mathbb {R}^{d+1}$$ R d + 1

Given a Cantor-type subset \(\Omega \) of a smooth curve in \(\mathbb R^{d+1}\), we construct examples of sets that contain unit line segments with directions from \(\Omega \) and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small \((d+1)\)-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of \(\Omega \), and extends to higher dimensions an analogous planar result of Bateman and Katz (Math Res Lett 15(1):73–81, 2008). In contrast to the planar situation, a significant aspect of our analysis is the classification of intersecting tube tuples relative to their location, and the deduction of intersection probabilities of such tubes generated by a random mechanism. The existence of these Kakeya-type sets implies that the directional maximal operator associated with the direction set \(\Omega \) is unbounded on \(L^p(\mathbb {R}^{d+1})\) for all \(1\le p<\infty \).

Some spectral results on Kakeya sets

Abstract The finite field Kakeya problem asks both the minimum size of a point set inAG(2, q)which contains a line in every direction, as well as a characterization of the examples. Blokhuis and Mazzocca [2] solved this problem, and a subsequent paper [1] addresses the stability of this solution for even order planes, i.e. the spectrum of sizes near the minimum size of a Kakeya set for which non-minimum Kakeya sets exist. In this paper we provide some computational results in small order planes to determine the full spectrum of sizes of Kakeya sets. We then address some spectrum issues on the upper end of possible sizes, providing some bounds and new constructions.We also address the question of minimality, i.e.whether a given Kakeya set contains any smaller Kakeya set.

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc.As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually ‘thick’ families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension.Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.

Small Kakeya sets in non-prime order planes

The finite field analog of the classical Kakeya problem asks the smallest possible size for a set of points in the Desarguesian affine plane which contains a line in every direction. This problem has been definitively solved by Blokhuis and Mazzocca (2008), who show that in AG(2,s), s a prime power, the smallest possible size of such a set is 12s(s+1). In this paper we examine a new construction of an infinite family of sets in AG(2,s) containing a line in every direction, where s=qe with q a prime power and e an integer with e>1. These sets have size 12q2e+O(q2e−1), which is small in the sense that the lower bound is also 12q2e plus smaller order terms. In addition, we discuss the minimality of our sets, showing that they contain no proper subset containing a line in every direction.

Hunter, Cauchy rabbit, and optimal Kakeya sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Z n \mathbb {Z}_n . A hunter and a rabbit move on the nodes of Z n \mathbb {Z}_n without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al. (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first n n steps of order 1 / log ⁡ n 1/\log n . We show these strategies yield a Kakeya set consisting of 4 n 4n triangles with minimal area (up to constant), namely Θ ( 1 / log ⁡ n ) \Theta (1/\log n) . As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions { B ( s ) : s ≥ 0 } \{B(s): s \ge 0\} and { W ( s ) : s ≥ 0 } \{W(s): s \ge 0\} . Let τ t := min { s ≥ 0 : B ( s ) = t } \tau _t:=\min \{s \ge 0: B(s)=t\} . Then X t = W ( τ t ) X_t=W(\tau _t) is a Cauchy process and K := { ( a , X t + a t ) : a , t ∈ [ 0 , 1 ] } K:=\{(a,X_t+at) : a,t \in [0,1]\} is a Kakeya set of zero area. The area of the ε \varepsilon -neighbourhood of K K is as small as possible, i.e., almost surely of order Θ ( 1 / | log ⁡ ε | ) \Theta (1/|\log \varepsilon |) .

Heisenberg Hausdorff Dimensionof Besicovitch Sets

Abstract We consider (bounded) Besicovitch sets in the Heisenberg group and prove that Lp estimates forthe Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.

Twisted Hilbert transforms vs Kakeya sets of directions

Abstract Given a discrete group G ${\mathrm {G}}$ and an orthogonal action γ : G → O ( n ) ${\gamma : \mathrm {G}\rightarrow {\rm O}(n)}$ we study Lp -convergence of Fourier integrals which are frequency supported on the semidirect product ℝ n ⋊ γ G ${\mathbb {R}^n \rtimes _\gamma \mathrm {G}}$ . Given a unit u ∈ ℝ n ${u \in \mathbb {R}^n}$ and 1 &lt; p ≠ 2 &lt; ∞ ${1 &amp;lt; p \ne 2 &amp;lt; \infty }$ , our main result shows that the twisted (directional) Hilbert transform H u ⋊ γ id G ${H_u \rtimes _\gamma {\rm id}_\mathrm {G}}$ is Lp -bounded iff the orbit 𝒪 γ ( u ) ${\mathcal {O}_\gamma (u)}$ is finite. This is in sharp contrast with twisted Riesz transforms R u ⋊ γ id G ${R_u \rtimes _\gamma {\rm id}_\mathrm {G}}$ , which are always bounded. Our result characterizes Fourier summability in Lp for this class of groups. We also extend de Leeuw's compactification theorem to this setting and obtain stronger estimates for functions with “lacunary” frequency support.

On a variant of the Kakeya problem in $${\mathbb{R}}$$ R

We prove that the sharp lower bounds of the Minkowski and Hausdorff dimensions of circular Kakeya sets in \({\mathbb{R}}\) are 1/2 and 0 respectively.