On some exponential sums with exponential and rational functions

We study exponential sums of the form ∑t x=1 * exp(2πi(aν^x/p + f(x)/t)) where ν is an integer of multiplicative order t modulo a prime p, f(X) is rational function modulo t and Σ * indicates that the poles of f are excluded. The case of f(X) = bX is well studied and has been considered in a number of works. For f(X) = b/X these sums have recently been estimated by Bourgain and the author. Here we consider the general case of an arbitrary rational function f.