Discrete bilinear Radon transforms along arithmetic functions with many common values

We prove that for a large class of functions $P$ and $Q$, there exists $d\in (0,1)$suchthatthediscretebilinearRadontransform$$B^{\rm dis}{P,Q}(f,g)(n)=\sum{m\in$\mathbb{Z}$\setminus{0}} f(n-P(m))g(n-Q(m))\frac{1}{m}$$isboundedfrom$l^2\times l^2$into$l^{1+\epsilon}$forany$\epsilon\in (d,1)$. In particular, the boundedness holds for any $ϵ\in (0,1)$ when $P$ (or $Q$) is the Euler totient function $\varphi (\mathrm{\mid }m\mathrm{\mid })$ or the prime counting function $\pi (\mathrm{\mid }m\mathrm{\mid })$.