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On the Erdős distinct distances problem in the plane
In this paper, we prove that a set of N points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os. We follow the setup of Elekes and Sharir…
A sum-product estimate in finite fields, and applications
- J. Bourgain, N. Katz, T. Tao
- Mathematics, Computer Science
- 29 January 2003
A Szemerédi-Trotter type theorem in finite fields is proved, and a new estimate for the Erdös distance problem in finite field, as well as the three-dimensional Kakeya problem in infinite fields is obtained.
On the Erdos distinct distance problem in the plane
In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erd\"os. We follow the set-up of…
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…
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…
A cheap Caffarelli—Kohn—Nirenberg inequality for the Navier—Stokes equation with hyper-dissipation
- N. Katz, N. Pavlović
- 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…
Finite time blow-up for a dyadic model of the Euler equations
- N. Katz, N. Pavlović
- 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…
A slight improvement to Garaev’s sum product estimate
- N. Katz, Chun-Yen Shen
- Mathematics
- 20 March 2007
max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality.…
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