Two bipolynomial Roth theorems in $\mathbb{R}$

TL;DR

This paper establishes two Roth theorems involving polynomial configurations in real sets of positive density or fractional dimension, using Fourier analysis techniques.

Contribution

It introduces two new Roth theorems for nonlinear polynomial configurations in real sets, expanding classical results to more complex polynomial patterns.

Findings

Proves Roth theorems for sets of positive density in $ eal$.

Extends Roth-type results to sets with fractional dimensions.

Utilizes Fourier analysis to establish the theorems.

Abstract

We give two Roth theorems, related to the nonlinear configuration $x$, $x+P_1(t)$, $x+P_2(t)$ involving two polynomials, for sets in $\mathbb{R}$ of positive density and of fractional dimensions. The proof uses Fourier analysis.