A polynomial Roth theorem on the real line

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 me...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-05, Vol.371 (10), p.6973-6993
Main Authors: DURCIK, POLONA, GUO, SHAOMING, ROOS, JORIS
Format: Article
Language:English
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7574