  Many natural notions of additive and multiplicative largeness arise from
results in Ramsey theory. In this paper, we explain the relationships between
these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic
structures. We show that multiplicative largeness begets additive largeness in
three ways and give a collection of examples demonstrating the optimality of
these results. We also give a variety of applications arising from the
connection between additive and multiplicative largeness. For example, we show
that given any $n, k \in \mathbb{N}$, any finite set with fewer than $n$
elements in a sufficiently large finite field can be translated so that each of
its elements becomes a non-zero $k^{\text{th}}$ power. We also prove a theorem
concerning Diophantine approximation along multiplicatively syndetic subsets of
$\mathbb{N}$ and a theorem showing that subsets of positive upper Banach
density in certain multiplicative sub-semigroups of $\mathbb{N}$ of zero
density contain arbitrarily long arithmetic progressions. Along the way, we
develop a new characterization of upper Banach density in a wide class of
amenable semigroups and make explicit the uniformity in recurrence theorems
from measure theoretic and topological dynamics. This in turn leads to
strengthened forms of classical theorems of Szemer\'edi and van der Waerden on
arithmetic progressions.
