Sums and products in sets of positive density

We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish two main results. First, we show that if $Q(1) = 0$, then any set of natural numbers with positive upper logarithmic density contains a pair of the form $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N} \setminus \{1\}$. Second, we prove that if $Q(0) = 0$, then any set of natural numbers with positive density relative to a new multiplicative notion of density, which arises naturally in the context of such problems, contains $\{x + Q(y), xy\}$ for some $x, y \in \mathbb{N}$.