## 165 Citations

### Kakeya-Nikodym Problems and Geodesic Restriction Estimates for Eigenfunctions

- Mathematics
- 2017

We record work done by the author [29] on the Kakeya-Nikodym problems, and we also record the joint work done by the author and Cheng Zhang [30] on improved geodesic restriction estimates for…

### An incidence bound for $k$-planes in $F^n$ and a planar variant of the Kakeya maximal function

- Mathematics
- 2006

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…

### On the restriction theorem for the paraboloid in ${\mathbb R}^4$

- MathematicsColloquium Mathematicum
- 2019

We prove that recent breaking by Zahl of the $\frac32$ barrier in Wolff's estimate on the Kakeya maximal operator in $\mathbb R^4$ leads to improving the $\frac{14}{5}$ threshold for the restriction…

### An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³

- Computer ScienceJournal of the American Mathematical Society
- 2018

It is proved that every Besicovitch set in <inline-formula content-type="math/mathml" > must have Hausdorff dimension at least at least to satisfy the Wolff axioms.

### Dimension estimates for Kakeya sets defined in an axiomatic setting

- Mathematics
- 2017

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every…

### On Kakeya-Nikodym type maximal inequalities

- Mathematics
- 2015

We show that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $\mathbb R^d$ for $d\ge3$ without the induction on scales argument. The key…

### A refinement of the Strichartz inequality with applications to the linear and nonlinear wave equations

- Mathematics
- 2012

### L2 bounds for a Kakeya‐type maximal operator in ℝ3

- Mathematics, Computer Science
- 2012

We prove that the maximal operator obtained by taking averages at scale 1 along N arbitrary directions on the sphere is bounded in L2(R3) by N1/4log N. When the directions are N−1/2 separated, we…

### The Kakeya Problem

- Mathematics
- 2008

The original Kakeya problem, posed by S. Kakeya in 1917, is to find a planar domain with the smallest area so that a unit line segment (a “needle”) can be rotated by 180 degrees in this domain. It…

### 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…