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 ∈ ( 0 , 1 ) such that the discrete bilinear Radon transformB P , Q dis( f , g ) ( n ) = ∑ m ∈ Z ∖ { 0 } f ( n − P ( m ) ) g ( n − Q ( m ) ) 1 mis bounded froml 2 × l 2into l 1 + εfor any ε ∈ ( d , 1 ) . In particular, the boundedness holds for any ε ∈ ( 0 , 1 ) when P (or Q ) is the Euler totient function ϕ ( | m | ) or the prime counting function π ( | m | ) .