Exponential sums with polynomials and their applications to primes in sparse sets

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an application, we study the Piatetski-Shapiro sequence of the form $(\lfloor n^c\rfloor )$ where $c>1$ is not an integer. We improve the admissible range of the asymptotic formula for primes in the intersection of Piatetski-Shapiro sequences. We also study the iterated Piatetski-Shapiro sequence and prove an asymptotic formula for the prime counting function.