Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II)
Xiumin Ren, Yu-chen Sun, Qingqing Zhang, Rui Zhang
TL;DR
The paper advances Roth-type results for nonlinear diagonal systems in the Piatetski-Shapiro primes by employing a Bourgain-style transference framework. It constructs a majorant on a discretized interval, proves Fourier decay and a restricted (restriction) estimate, and establishes a K-trivial saving to bound the size of $K$-trivial solution sets in $\mathcal{P}^c_x$. Crucially, it reduces the required number of variables to $\bar{s}(d)=2\left\lfloor d^2/2\right\rfloor+1$ and allows $c$ in the range $(1,1+c(d,s))$, yielding bounds of the form $|\mathcal{A}|^c \log^c x \ll x(\log\log\log\log x)^{\frac{2-s}{dc}+\varepsilon}$. This extends Roth-type phenomena to Piatetski-Shapiro primes and refines the interaction between nonlinear Diophantine systems, exponential-sum techniques, and sieve-driven transference methods.
Abstract
We consider the nonlinear system $c_1p_1^d +c_2p_2^d + \dots + c_s p_s^d = 0$ with $c_1, c_2,\dots, c_s\in\mathbb Z$ being nonzero and satisfying $c_1 +c_2 + \dots + c_s = 0$. 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^{\frac1c}}{\log x} $$\left(\log \log \log \log x\right)^{\frac{2-s}{dc}+\varepsilon}$.