The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n 4/3 ) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let La and L b be two sets of t lines in the plane and let P = { a ∩ b : a ∈ La, b ∈ L b } be the set of intersection points between La and L b . We say that (P, La ∪ L b ) forms a natural t × t grid if |P | = t 2 , and conv(P ) does not contain the intersection point of some two lines in La and does not contain the intersection point of some two lines in L b . For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t × t grid determines O(n 4 3 −ε ) incidences, where ε = ε(t) > 0. We also provide a construction of n points and n lines in the plane that does not contain a natural 2 × 2 grid and determines at least Ω(n 1+ 1 14 ) incidences.

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Two sets A and B of points in the plane are mutually avoiding if no line generated by any two points in A intersects the convex hull of B, and vice versa. In 1994, Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman showed that every set of n points in the plane in general position contains a pair of mutually avoiding sets each of size at least n/12. As a corollary, their result implies that for every set of n points in the plane in general position one can find at least n/12 segments, each joining two of the points, such that these segments are pairwise crossing.In this note, we prove a fractional version of their theorem: for every k > 0 there is a constant ε k > 0 such that any sufficiently large point set P in the plane contains 2k subsets A 1 , . . . , A k , B 1 , . . . , B k , each of size at least ε k |P |, such that every pair of sets A = {a 1 , . . . , a k } and B = {b 1 , . . . , b k }, with a i ∈ A i and b i ∈ B i , are mutually avoiding. Moreover, we show that ε k = Ω(1/k 4 ). Similar results are obtained in higher dimensions.

In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.

The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n 4/3 ) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let L 1 and L 2 be two sets of t lines in the plane and let P = { 1 ∩ 2 : 1 ∈ L 1 , 2 ∈ L 2 } be the set of intersection points between L 1 and L 2 . We say that (P, L 1 ∪ L 2 ) forms a natural t × t grid if |P | = t 2 , and conv(P ) does not contain the intersection point of some two lines in L i , for i = 1, 2. For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t × t grid determines O(n 4 3 −ε ) incidences, where ε = ε(t). We also provide a construction of n points and n lines in the plane that does not contain a natural 2 × 2 grid and determines at least Ω(n 1+ 1 14 ) incidences.