A polynomial Roth theorem on the real line

Abstract

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.

18 pages, to appear in Transactions of the AMS

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• The non-resonant bilinear Hilbert--Carleson operator
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