# Recent work connected with the Kakeya problem

@inproceedings{Wolff2007RecentWC, title={Recent work connected with the Kakeya problem}, author={Thomas H. Wolff}, year={2007} }

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:

## 278 Citations

### The Finite Field Kakeya Problem

- Mathematics
- 2008

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

- MathematicsSoCG
- 2018

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

- Mathematics
- 2014

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

- Mathematics
- 2020

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

- MathematicsErgodic Theory and Dynamical Systems
- 2009

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

- Mathematics
- 2005

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

- Mathematics
- 2008

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

- Mathematics
- 2008

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

- Mathematics
- 2014

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…

### AN IMPROVED BOUND FOR THE DIMENSION OF (α, 2α)-FURSTENBERG SETS

- Mathematics
- 2020

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…

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