A polynomial Roth theorem on the real line

Durcik, Polona; Guo, Shaoming; Roos, Joris

doi:10.1090/tran/7574

A polynomial Roth theorem on the real line
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by Polona Durcik, Shaoming Guo and Joris Roos PDF

Trans. Amer. Math. Soc. 371 (2019), 6973-6993 Request permission

Abstract:

For a polynomial $P$ of degree greater than $1$ 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.

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