The shifted bilinear Hilbert transform

We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp bilinear H\"ormander multiplier theorem, and a $\log$-Dini theorem for bilinear singular integrals.