Improvements on exponential sums related to Piatetski-Shapiro primes

We prove a new bound to the exponential sum of the form $$\sum _{h\sim H}{\delta }_h{}_{mn\sim x}{a}_m{b}_n\text{e}(\alpha mn+h(mn+u{)}^{\gamma }),$$ by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form $\lfloor n^c\rfloor$. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in the intersection of multiple Beatty sequences and so on.