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

426 49- PDF

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

66 14- PDF

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

87 13- PDF

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

83 9- PDF

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

38 7- PDF

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

100 7- PDF

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.

68 6- PDF

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

20 5- PDF

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

67 3- PDF