An x-ray estimate in $R^n$
@article{Laba1999AnXE,
title={An x-ray estimate in \$R^n\$},
author={Izabella Laba and Terence Tao},
journal={arXiv: Classical Analysis and ODEs},
year={1999}
}We prove an x-ray estimate in general dimension which is stronger than the Kakeya estimates of Wolff. This generalizes an x-ray estimate in three dimensions which is also due to Wolff.
12 Citations
Two bounds for the X-ray transform
- 2006
Mathematics
We use the arithmetic-combinatorial method of Katz and Tao to give mixed-norm estimates for the X-ray transform on $${\mathbb {R}^d}$$ when d ≥ 4.
An incidence bound for $k$-planes in $F^n$ and a planar variant of the Kakeya maximal function
- 2006
Mathematics
We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is…
New bounds for Kakeya problems
- 2001
Mathematics
We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.
An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension
- 2000
Mathematics
Abstract. We use geometrical combinatorics arguments, including the "hairbrush" argument of Wolff [W1], the x-ray estimates in [W2], [LT], and the sticky/plany/grainy analysis of [KLT], to show that…
Recent progress on the Kakeya conjecture
- 2000
Mathematics
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].
An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$
- 2000
Mathematics
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorfi dimensions of such a set must be greater than or equal to 5= 2i n 3 . In…
An improved bound on the Minkowski dimension of Besicovitch sets in R^3
- 1999
Mathematics
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In…
Bounds for Kakeya-type maximal operators associated with $k$-planes
- 2005
Mathematics
A $(d,k)$ set is a subset of $\rea^d$ containing a translate of every $k$-dimensional plane. Bourgain showed that for $k \geq \kcrit(d)$, where $\kcrit(d)$ solves $2^{\kcrit-1}+\kcrit = d$, every…
A new bound for finite field besicovitch sets in four dimensions
- 2002
Mathematics
Let F be a finite field with characteristic greater than two. A Besicovitch set in F 4 is a set P ⊆ F 4 containing a line in every direction. The Kakeya conjecture asserts that P and F 4 have roughly…
A Bound For The (n; k) Maximal Function Via Incidence Combinatorics
- 2006
Mathematics
of the dissertation A Bound For The (n, k) Maximal Function Via Incidence Combinatorics by John Joseph Bueti Doctor of Philosophy in Mathematics University of California, Los Angeles, 2006 Professor…
12 References
An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$
- 2000
Mathematics
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorfi dimensions of such a set must be greater than or equal to 5= 2i n 3 . In…
An improved bound on the Minkowski dimension of Besicovitch sets in R^3
- 1999
Mathematics
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In…
On the Dimension of Kakeya Sets and Related Maximal Inequalities
- 1999
Mathematics, Geography
Abstract. ((Without Abstract)).
A bilinear approach to the restriction and Kakeya conjectures
- 1998
Mathematics
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…