Recent work connected with the Kakeya problem

  title={Recent work connected with the Kakeya problem},
  author={Thomas H. Wolff},
where Sn−1 is the unit sphere in R. This paper will be mainly concerned with the following issue, which is still poorly understood: what metric restrictions does the property (1) put on the set E? The original Kakeya problem was essentially whether a Kakeya set as defined above must have positive measure, and as is well-known, a counterexample was given by Besicovitch in 1920. A current form of the problem is as follows: 

The Finite Field Kakeya Problem

A Besicovitch set in AG(n, q) is a set of points containing a line in every direction. The Kakeya problem is to determine the minimal size of such a set. We solve the Kakeya problem in the plane, and

The reverse Kakeya problem

Abstract We prove a generalization of Pál's conjecture from 1921: if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously

The restriction and Kakeya conjectures

We survey the progress made on the restriction problem since it was first conjectured in the 1960s by E. M. Stein, in particular the oscillatory-integral approach which culminated in the Tomas-Stein

On the Size of δ−SEPARATED δ−TUBES

In this preprint we will prove the surprising result that lim sup of the size of the Kakeya tube-sets are bounded below by a constant depending on the dimension. This implies that the δ−tubes are not

Overlapping self-affine sets of Kakeya type

Abstract We compute the Minkowski dimension for a family of self-affine sets on ℝ2. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of

On the multilinear restriction and Kakeya conjectures

We prove d-linear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of

eb 2 00 5 Kakeya sets of curves

In this paper we investigate an analogue for curves of the famous Kakeya conjecture about straight lines. The simplest version of the latter asks whether a set in R that includes a unit line segment

On the size of Kakeya sets in finite fields

The motivation for studying Kakeya sets over finite fields is to try to better understand the more complicated questions regarding Kakeya sets in W1. A Kakeya set K C Rn is a compact set containing a

Minimal Kakeya Sets

Blokhuis and Mazzocca (A. Blokhuis and F. Mazzocca, The finite field Kakeya problem (English summary). Building bridges. Bolyai Soc Math Stud 19 (2008) 205–218) provide a strong answer to the finite


We show that given α ∈ (0, 1) there is a constant c = c(α) > 0 such that any planar (α, 2α)-Furstenberg set has Hausdorff dimension at least 2α+ c. This improves several previous bounds, in



A generalization of Bourgain’s circular maximal theorem

where dσr is the normalized surface measure on r S . It is easy to see that M is not bounded on L (see Example 1.1 below). A well-known result of Bourgain [1] asserts that M is bounded on L for 2 < p

A Kakeya-type problem for circles

We prove full Hausdorff dimension in a variant of the Kakeya problem involving circles in the plane, and also sharp estimates for the relevant maximal function. These results can also be formulated

On the Hausdorff dimensions of distance sets

If E is a subset of ℝn (n ≥ 1) we define the distance set of E asThe best known result on distance sets is due to Steinhaus [11], namely, that, if E ⊂ ℝn is measurable with positive n-dimensional

Hausdorff dimension and distance sets

According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝn has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than

Extremal problems in discrete geometry

Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.

Remarks on Maximal Operators Over Arbitrary Sets of Directions

Throughout this paper, we shall let Σ be a subset of [0, 1] having cardinality N. We shall consider Σ to be a set of slopes, and for any s ∈ Σ, we shall let es be the unit vector of slope s in R2.

A bilinear approach to the restriction and Kakeya conjectures

The purpose of this paper is to investigate bilinear variants of the restriction and Kakeya conjectures, to relate them to the standard formulations of these conjectures, and to give applications of

Restriction implies Bochner–Riesz for paraboloids

  • A. Carbery
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1992
Let Σ ⊆ ℝn be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝn (U open patches in ℝn−1). The restriction problem for Σ is the


obvious cases: n = 1 or p =2; and also Fefferman [4] proved that TX is bounded on L P (Rn) provided that p (X) (n - 1)/4. This result has been sharpened by Tomas [15] to X > (n - 1)/2(n + 1). Finally


is bounded on L(R) if p > n/(n − 1). He also showed that no such result can hold for p ≤ n/(n − 1) if n ≥ 2. Thus, the 2-dimensional case is more complicated since the circular maximal operator