A Non-Linear Roth Theorem for Sets of Positive Density

Abstract

Suppose that $A\subset \mathbb{R}$ has positive upper density, \[ \limsup_{|I| \to \infty} \frac{|A \cap I|}{|I|} = δ> 0,\] 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 $R_0\le R$ sufficiently large, there exists some $x_R\in A$ so that \[ \inf_{R_0 \leq T \leq R} \frac{|\{ 0 \leq t < T : x_R - t \in A, \ x_R - P(t) \in A \}|}{T} \geq c_P \cdot δ^2 \] for some absolute constant $c_P>0$, that depends only on $P$.

References in corpus (1)

• An optimal version of Sarkozy's theorem

Cited by in corpus (2)

• Two bipolynomial Roth theorems in $\mathbb{R}$
• The non-resonant bilinear Hilbert--Carleson operator