A quantitative bound on Furstenberg–Sárközy patterns with shifted prime power common differences in primes

Let \(k\ge 1\) $k\ge 1$ be a fixed integer and \({\mathscr {P}}_N\) ${\mathcal{P}}_N$ be the set of primes no more than N. We prove that if a set \(\mathscr {A}\subset {\mathscr {P}}_N\) $\mathcal{A}\subset {\mathcal{P}}_N$ contains no patterns \(p_1,p_1+(p_2-1)^k\) $p_1,p_1+{(p_2-1)}^k$ , where \(p_1,p_2\) $p_1,p_2$ are prime numbers, then \(\begin{aligned} \frac{|{\mathscr {A}}|}{|{\mathscr {P}}_N|}\ll \bigl ( \log \log N \bigr )^{-\frac{1}{4k^3+23k^2}}. \end{aligned}\) $$\begin{array}{c}\hfill \frac{|\mathcal{A}|}{|{\mathcal{P}}_N|}\ll (loglogN{)}^{-\frac{1}{4k^3+23k^2}}.\end{array}$$