Abstract. We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in F 2 , with m 7/8 < n < m 8/7 , determine at most O(m 11/15 n 11/15 ) incidences (where, if F has positive characteristic p, we assume m −2 n 13 ≪ p 15 ). This improves on the previous best known bound, due to Jones.To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned.We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.

Let F ∈ C[x, y, z] be a constant-degree polynomial, and let A, B, C ⊂ C be finite sets of size n. We show that F vanishes on at most O(n 11/6 ) points of the Cartesian product A × B × C, unless F has a special group-related form. This improves a theorem of Elekes and Szabó [4], and generalizes a result of Raz, Sharir, and Solymosi [13]. The same statement holds over R, and a similar statement holds when A, B, C have different sizes (with a more involved bound replacing O(n 11/6 )).This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.

Let S be a set of n points in R 2 contained in an algebraic curve C of degree d. We prove that the number of distinct distances determined by S is at least c d n 4/3 , unless C contains a line or a circle.We also prove the lower bound c d min{m 2/3 n 2/3 , m 2 , n 2 } for the number of distinct distances between m points on one irreducible plane algebraic curve and n points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer, and Solymosi in [18].

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős' conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.