A hybrid of two theorems of Piatetski-Shapiro

Let $c>1$ and $0<\gamma <1$ be real, with $c\mathrm{\notin }\mathbb{N}$. We study the solubility of the Diophantine inequality $$\mid p_1^c+p_2^c+\cdots +p_s^c-N\mid <\epsilon$$ in Piatetski-Shapiro primes $p_1, p_2, \dots, p_s$ofindex$\gamma$---thatis,primesoftheform$p = \lfloor m^{1/\gamma} \rfloor$forsome$m \in \mathbb N$.