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: 
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278 Citations
Highly Influential Citations
28
Background Citations
121
Methods Citations
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Results Citations
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278 Citations

The Finite Field Kakeya Problem

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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…

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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…

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

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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