Roth-type Theorem for High-power System in Piatetski-Shapiro primes

Let c₁, …, c_s be non-zero integers satisfying c₁ + ⋯ + c_s = 0. We consider the system \(c_{1}x_{1}^{d}+\cdots+c_{s}x_{s}^{d}=0\) $$c_1{x}_1^d+\cdots +c_s{x}_s^d=0$$ with d ≥ 3, where x_i are restricted in subset \(\cal{A}\) $$\mathcal{A}$$ of Piatetski-Shapiro primes not exceeding x and corresponding to c. We show that for s > S(d)+2 and \(c\in(1,1+\tilde{c}(d,s))\) $$c\in (1,1+\tilde{c}(d,s))$$ , if the system has only K-trivial solutions in \(\cal{A}\) $$\mathcal{A}$$ , then \(\vert\cal{A}\vert\ll{x^{1/c}\over{\log x}}(\log\log\log\log x)^{{2-s\over{dc}}+\varepsilon}\) $$\mid \mathcal{A}\mid \ll \frac{x^{1/c}}{\log x}(\log \log \log \log x{)}^{\frac{2-s}{dc}+\epsilon }$$ , where S(3) = 8, S(4) = 16, S(d) = d(d + 1) (d ≥ 5), and \(\tilde{c}(d,s)=\min\{{2d\over{(S(d)+1)(3d-4)(3d-2)(3d+2)-d)-d}},{{d\over{S(d)s-d}}}\}\) $$\tilde{c}(d,s)=min\{\frac{2d}{(S(d)+1)(3d-4)(3d-2)(3d+2)-d)-d},\frac{d}{S(d)s-d}\}$$ .