Author pages are created from data sourced from our academic publisher partnerships and public sources.
- Publications
- Influence
A sum-product estimate in finite fields, and applications
- J. Bourgain, N. Katz, T. Tao
- Mathematics
- 29 January 2003
AbstractLet A be a subset of a finite field
$$ F := \mathbf{Z}/q\mathbf{Z} $$ for some
prime q. If
$$ |F|^{\delta} < |A| < |F|^{1-\delta} $$
for some δ > 0, then we prove the estimate
$$ |A + A| +… Expand
New Bounds on cap sets
- Michael D. Bateman, N. Katz
- Mathematics
- 31 January 2011
We provide an improvement over Meshulam's bound on cap sets in $F_3^N$. We show that there exist universal $\epsilon>0$ and $C>0$ so that any cap set in $F_3^N$ has size at most $C {3^N \over… Expand
Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type
In this paper we investigate three unsolved conjectures in geomet- ric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We… Expand
Finite time blow-up for a dyadic model of the Euler equations
- N. Katz, N. Pavlovic
- Mathematics
- 12 March 2004
We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the… Expand
Garaev's inequality in Finite Fields not of prime order
- N. Katz, Chun-Yen Shen
- Mathematics
- 22 March 2007
In the present paper, we extend Garaev’s techniques to the set of fields which are not necessarily of prime order. Our goal here is just to find an explicit estimate in the supercritical setting… Expand
A cheap Caffarelli—Kohn—Nirenberg inequality for the Navier—Stokes equation with hyper-dissipation
- N. Katz, N. Pavlovic
- Physics, Mathematics
- 19 April 2001
Abstract. We prove that for the Navier—Stokes equation with dissipation
$ (-\Delta)^\alpha $ where 1 < α < 5 /4, and smooth initial data, the Hausdorff dimension of the singular set at time of first… Expand
New bounds for Kakeya problems
We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.
Kakeya sets in Cantor directions
- Michael D. Bateman, N. Katz
- Mathematics
- 6 September 2006
We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union of… Expand
An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$
- N. Katz, Izabella Laba, T. Tao
- Mathematics
- 29 March 1999
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… Expand