Discrete bilinear Radon transforms along arithmetic functions with many common values

Abstract

We prove that for a large class of functions $P$ and $Q$, there exists $d\in (0,1)$ such that the discrete bilinear Radon transform $$B_{P,Q}^{\mathrm{dis}}(f,g)(n)=\sum _{m\in \mathbb{Z}\setminus \{0\}}f(n-P(m))g(n-Q(m))\frac{1}{m}$$ is bounded from $l^2\times l^2$ into $l^{1+\epsilon }$ for any $\epsilon \in (d,1)$. In particular, the boundedness holds for any $\epsilon \in (0,1)$ when $P$ (or $Q$) is the Euler totient function $\varphi (|m|)$ or the prime counting function $\pi (|m|)$.