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Transactions of the American Mathematical Society

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A polynomial Roth theorem on the real line


Authors: Polona Durcik, Shaoming Guo and Joris Roos
Journal: Trans. Amer. Math. Soc. 371 (2019), 6973-6993
MSC (2010): Primary 05D10, 42B20
DOI: https://doi.org/10.1090/tran/7574
Published electronically: January 16, 2019
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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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Additional Information

Polona Durcik
Affiliation: University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: durcik@math.uni-bonn.de

Shaoming Guo
Affiliation: Indiana University Bloomington, 831 East Third Street, Bloomington, Indiana 47405
Address at time of publication: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China
Email: shaomingguo2018@gmail.com

Joris Roos
Affiliation: University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: jroos@math.uni-bonn.de

DOI: https://doi.org/10.1090/tran/7574
Received by editor(s): April 12, 2017
Received by editor(s) in revised form: January 30, 2018
Published electronically: January 16, 2019
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the spring semester of 2017.
The first author is supported by the Hausdorff Center for Mathematics.
The third author is supported by the Hausdorff Center for Mathematics and the German National Academic Foundation.
Article copyright: © Copyright 2019 American Mathematical Society