On a family of sparse exponential sums

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.