A Non-Linear Roth Theorem for Sets of Positive Density

Abstract: Suppose that $A \subset \mathbb{R}$ has positive upper density, [ \limsup_{|I| \to \infty} \frac{|A \cap I|}{|I|} = \delta > 0,] and $P(t) \in \mathbb{R}[t]$ is a polynomial with no constant or linear term, or more generally a non-flat curve: a locally differentiable curve which doesn't "resemble a line" near $0$ or $\infty$. Then for any $R_0 \leq R$ sufficiently large, there exists some $x_R \in A$ so that [ \inf_{R_0 \leq T \leq R} \frac{|{ 0 \leq t < T : x_R - t \in A, \ x_R - P(t) \in A }|}{T} \geq c_P \cdot \delta^2 ] for some absolute constant $c_P > 0$, that depends only on $P$.