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… Expand

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.Expand

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… Expand

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

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

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

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

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.… Expand