A polynomial Roth theorem for corners in $\mathbb{R}^2$ and a related bilinear singular integral operator

TL;DR

This paper establishes a polynomial Roth theorem for corners in the plane, showing that positive measure sets contain polynomial-configured corners and proving boundedness of related bilinear singular integrals.

Contribution

It introduces a quantitative polynomial Roth theorem for corners in  and analyzes a bilinear singular integral operator involving two polynomials, extending prior work.

Findings

Positive measure sets contain polynomial corners with a gap estimate.

Boundedness results for a bilinear singular integral operator involving polynomials.

Sublevel set estimates with explicit exponents and simplified proofs.

Abstract

We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure contains three points $(x,y)$, $(x+P_1(t),y)$, $(x,y+P_2(t))$ with a gap estimate on $t$. We also prove boundedness results for a variant of the triangular Hilbert transform involving two polynomials and its associated maximal function. These results extend some earlier work of Christ, Durcik and Roos. The key of the proof is to establish certain smoothing inequalities involving two polynomials. To accomplish that we give sublevel set estimates with general polynomials, explicit exponents and simplified proofs.