  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
$P$.
