Discordant sets and ergodic Ramsey theory.

We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemer\'edi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $\bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $\bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $\mathbb{Z}$, $\mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $\bullet$ We show that, for any countable abelian group $G$, any F{\o}lner sequence $\Phi$ in $G$, and any $c \in (0, 1)$, there exists a discordant set $A \subseteq G$ with $d_\Phi(A) = c$. Here $d_\Phi$ denotes density along $\Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.