# A polynomial Roth theorem on the real line

## TL;DR

This paper extends Bourgain's result by proving the existence of polynomial-pattern configurations in positive density subsets of the real line, using advanced harmonic analysis techniques.

## Contribution

It introduces a novel combination of Bourgain's approach with modern methods for bilinear Hilbert transforms to establish polynomial pattern existence.

## Key findings

- Existence of polynomial patterns in positive density sets
- Extension of Bourgain's earlier results
- Application of bilinear Hilbert transform techniques

## 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.