On a paucity result in Incidence Geometry

where the set of points P and the set of lines L belong to F × F with F be a field, say. In our paper the set of points P will be the Cartesian product A × B for some sets A,B ⊆ F. This particular choice of P is very important for the applications see, e.g., [2], [5], [6], [9], [10], [13] because, basically, Cartesian products are naturally connected with arithmetic. In this note we study collinear tuples in A × B. Namely, for any k > 3 we define Ck(A,B) to be the number of collinear k–tuples in A×B. Let Ck(A) = Ck(A,A). A consequence of the famous Szemerédi– Trotter Theorem [12] gives us that for any A ⊂ R one has