Linear equations in Piatetski-Shapiro primes

Abstract

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_{\gamma} := \{p \text{ is prime and } p = \lfloor n^{1/\gamma} \rfloor \text{ for some } n \in \mathbb{N}\} \] 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 $k \geq 3$. 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.