Linear equations in Piatetski-Shapiro primes

We establish discorrelation estimates between the Piatetski-Shapiro prime set $${\mathcal{P}}_{\gamma }:=\{p\text{ is prime and }p=\lfloor n^{1\mathrm{/}\gamma }\rfloor \text{ for some }n\in \mathbb{N}\}$$ and arbitrary nilsequences when $\gamma \in (0,1)$ is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in ${\mathcal{P}}_{\gamma }$ to any “finite-complexity” system of linear equations, including for the number of $k$-term arithmetic progressions in ${\mathcal{P}}_{\gamma }$ up to a threshold $N$ for any given $k\ge 3$. Furthermore, we show that there exists an absolute constant $C>0$ such that if $$1-2^{-Ck}<\gamma <1,$$ then the Piatetski-Shapiro primes ${\mathcal{P}}_{\gamma }$ contain infinitely many non-trivial $k$-term arithmetic progressions. This significantly improves upon the previous range of $\gamma$ obtained by Li and Pan, which is of triple exponential type.