  For a polynomial $P$ of degree greater than one, we show the existence of
patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive
density subsets of the reals. This is an extension of an earlier result of
Bourgain. Our proof is a combination of Bourgain's approach and more recent
methods that were originally developed for the study of the bilinear Hilbert
transform along curves.
