# Recent progress on the Kakeya conjecture

@article{Katz2000RecentPO, title={Recent progress on the Kakeya conjecture}, author={Nets Hawk Katz and Terence Tao}, journal={Publicacions Matematiques}, year={2000}, volume={46}, pages={161-179} }

We survey recent developments on the Kakeya problem.
[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

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#### References

SHOWING 1-10 OF 45 REFERENCES

An improved bound for Kakeya type maximal functions

- Mathematics
- 1995

The purpose of this paper is to improve the known results (specifically [1]) concerning Lp boundedness of maximal functions formed using 1 x d x ... x d tubes.

A Sharp Bilinear Cone Restriction Estimate

- Mathematics
- 2001

The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.

A new proof of Szemerédi's theorem

- Mathematics
- 2001

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these… Expand

From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE

- Mathematics
- 2000

We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic… Expand

Maximal Averages and Packing of One Dimensional Sets

- Mathematics
- 1998

We discuss recent work of several authors on the Kakeya needle problem and other related problems involving nonexistence of small sets containing large families of one dimensional objects.

On the Dimension of Kakeya Sets and Related Maximal Inequalities

- Mathematics
- 1999

Abstract. ((Without Abstract)).

A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four

- Mathematics
- 1998

Abstract. ((Without abstract))

BOUNDS ON ARITHMETIC PROJECTIONS, AND APPLICATIONS TO THE KAKEYA CONJECTURE

- Mathematics
- 1999

Let A, B, be finite subsets of a torsion-free abelian group, and let G ⊂ A × B be suchth at # A, #B,#{a + b :( a, b) ∈ G }≤ N. We consider the question of estimating the quantity #{a − b :( a, b) ∈… Expand

A bilinear approach to the restriction and Kakeya conjectures

- Mathematics
- 1998

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… Expand

Some remarks on the Kakeya problem

- Mathematics
- 1971

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem… Expand