We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range D = { ( p 1 , p 2 , p 3 ) 3 : 1 < p 1

We prove that for a large class of functions P and Q, the discrete bilinear operator T P, Q (f, g)(n)= m Z{0} f (n P (m)) g (n Q (m)) 1 m is bounded from l 2 l 2 into l 1+ , for any (0, 1].

We prove that, under certain conditions on the function pair arphi_1 and arphi_1 , bilinear average arphi_1 along curve arphi_1 satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if arphi_1 with arphi_1 are linearly independent polynomials, then for any arphi_1 with arphi_1 , there are arphi_1 triplets arphi_1 . This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.