Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II)

Abstract

We consider the nonlinear system $c_1{p}_1^d+c_2{p}_2^d+\cdots +c_s{p}_s^d=0$ with $c_1,c_2,\dots ,c_s\in \mathbb{Z}$ being nonzero and satisfying $c_1+c_2+\cdots +c_s=0$. We show that for $s\ge 2\lfloor \frac{d^2}{2}\rfloor +1$ and $c\in \left(1,1+c(d,s)\right)$, if the system has only $K$-trivial solutions in subset $\mathcal{A}$ of Piatetski-Shapiro primes up to $x$ and corresponding to $c$, then $|\mathcal{A}|\ll \frac{x^{\frac{1}{c}}}{\log x}$${\left(\log \log \log \log x\right)}^{\frac{2-s}{dc}+\epsilon }$.

14 pages