Exponential Sums with Sparse Polynomials and Distribution of the Power Generator

We obtain new bounds on complete rational exponential sums with sparse polynomials modulo a prime, under some mild conditions on the degrees of the monomials of such polynomials. These bounds, when they apply, give explicit versions of a result of J. Bourgain (2005). In turn, as an application, we also obtain an explicit version of a result of J. Bourgain (2010) on national exponential sums with sparse polynomials modulo an arbitrary composite number. We then use one of these bounds to study the multidimensional distribution of the classical power generator of pseudorandom numbers, which has not been possible within previously known results.