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 $$B^{\rm dis}_{P,Q}(f,g)(n)=\sum_{m\in\mathbb{Z}\setminus\{0\}} f(n-P(m))g(n-Q(m))\frac{1}{m}$$ 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 $\phi(|m|)$ or the prime counting function $\pi(|m|)$.