Small gaps between the Piatetski-Shapiro primes

Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $\{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\}$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).

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