On the concentration of points of polynomial maps and applications

For a polynomial f ε F _p[X], we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on the Vinogradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve f(x) ≡ y (mod p), where f ε F _p[X] is a polynomial of degree d ≥ 2. We also use some recent results and techniques from arithmetic combinatorics to study the values (x, f (x)) in more general sets.