Preprints in Kakeya and Restriction problems

There are other pages on the web on Kakeya and Restriction; Ben Green, Alex Iosevich, and Izabella Laba all maintain one.  If you are new to this field and want to learn more, I can suggest starting with my Notices article surveying this field.  If you then want to learn more about the Kakeya problem, you could try the El Escorial proceedings survey with Nets Katz, or the Edinburgh lecture notes on the Kakeya problem; if you want to learn more about the restriction problem, I can offer my Park city notes on the Restriction problem.  You can also see my Math 254B home page for a more leisurely-paced introduction, but it is getting a little out of date. If you are more into the algebraic side of things, you can learn about the finite field analogues of these problems in this paper with Gerd Mockenhaupt.  If you like the arithmetic combinatorial side of things, you can start with this short paper with Nets, or my Math 254A home page.  If you are instead interested in the Bochner Riesz or local smoothing problems, you will have to go to my research papers, such as my second paper with Ana Vargas; I do not yet have a good survey of these problems (one should probably go look instead at the home pages of Izabella Laba or of the papers of Tom Wolff), although I mention these problems briefly in the Park city notes.

Math 254B home page - Restriction theorems, Bochner-Riesz, Kakeya, etc. (1999)

Papers, and projects close to completion





Weak-type endpoint bounds for Riesz means


Proc. Amer. Math. Soc. 124 (1996), 2797-2805

dvi ps.Z

The Bochner-Riesz conjecture implies the restriction conjecture


Duke Math. J. 96 (1999), 363-376

dvi ps.Z
Slides: dvi  ps.Z

The weak-type endpoint Bochner-Riesz conjecture and related topics


Indiana U. Math. J. 47 (1998), 1097-1124

Slides: dvi  ps.Z

A bilinear approach to the restriction and Kakeya conjectures

Ana Vargas
Luis Vega

J. Amer. Math. Soc. 11 (1998), 967-1000

Slides: dvi + Figures 12
Summary: dvi

On the Maximal Bochner-Riesz conjecture in the plane, for p<2


Trans. Amer. Math. Soc. 354 (2002), 1947-1959

dvi + Figure 1
Slides: dvi + Figures 1234

A bilinear approach to cone multipliers I. Restriction estimates

Ana Vargas

GAFA 10 (2000), 185-215

dvi + Figures 12345
Summary: dvi

A bilinear approach to cone multipliers II. Applications

Ana Vargas

GAFA10 (2000), 216-258

dvi + Figures 12345

An improved bound on the Minkowski dimension of Besicovitch sets in R^3

Nets Katz
Izabella Laba

Annals of Math. 152 (2000),  383-446

Slides: dvi + Figures 1234567

Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates


Math. Z. 238 (2001), 215-268

Non-endpoint: dvi
Informal: dvi
Slides: dvi + Figures 1234567

Bounds on arithmetic projections, and applications to the Kakeya conjecture

Nets Katz

Math. Res. Letters 6 (1999), 625-630

Slides: dvi + Figures 1234567

An x-ray transform estimate in R^n

Izabella Laba

Revista Mat. Iber. 17 (2001), 375-408

Informal: dvi

Some connections between the Falconer and Furstenburg conjectures

Nets Katz

New York J. Math. 7 (2001), 148-187

Finite field analogues

An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension

Izabella Laba

GAFA 11 (2001), 773-806


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


Notices Amer. Math. Soc. 48 (2001) No 3, 294-303


Kakeya and restriction phenomena for finite fields

Gerd Mockenhaupt

Duke Math. J. 121 (2004), 35-74


New bounds for Kakeya problems

Nets Katz

Journal d'Analyse de Jerusalem,  87 (2002), 231-263

Slides + Figure 1234567

Recent progress on the Kakeya conjecture

Nets Katz

Publicacions Matematiques, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, U. Barcelona 2002, 161-180


Bochner-Riesz summability for analytic functions on the 
m-complex unit sphere and for cylindrically symmetric functions on R^{n-1} \times R

Adam Sikora

to appear, Comm. Anal. Geom.


A new bound for finite field Besicovitch sets in four dimensions


to appear, Pacific J. Math


A sharp bilinear restriction estimate for paraboloids


GAFA 13 (2003), 1359-1384


Some recent progress on the Restriction conjecture


submitted, Proceedings, Fourier Analysis and Convexity Workshop


Recent progress on the Restriction conjecture


submitted, Park City proceedings


Some further papers related to Kakeya and restriction problems can be found on my PDE preprint page.
Some further papers dealing with more general aspects of harmonic analysis can be found here.

Short stories

These are generally very short, toy versions of real results due to other people, and are not publication-quality.  Caveat emptor.  All files other than figures are in dvi format.  Unlike the preprints, these articles are fluid and subject to new developments.  Please let me know if you have any comments, references, etc. on any of them.

Disclaimer: Many of the notes here are based on papers written by other people.  My intention here is not to try to "beat" these authors' work in any way, but rather to isolate the main ingredients of the argument, which are often very beautiful, and try to present them in as simple and brief a context as possible (often sacrificing generality, rigour, and/or details in order to do this).  Certainly I do not view these notes as worthy of publication in a refereed journal, and are definitely inferior to the original article in every single aspect, with the possible exception of brevity.

The Szemeredi-Trotter cell decomposition

Non-endpoint bilinear cone restriction theorems

Informal proof of bilinear cone restriction estimates

Informal proof of x-ray estimates

Sharpness of the Carleson-Sjolin theorem

An informal outline of Wolff's local smoothing estimate

Korner's Besicovitch set construction

Finite field analogues of the Erdos, Falconer, and Furstenburg problems


Back to my preprints page.