  A major theme in arithmetic combinatorics is proving multiple recurrence
results on semigroups (such as Szemer\'edi's theorem) and this can often be
done using methods of ergodic Ramsey theory. What usually lies at the heart of
such proofs is that, for actions of semigroups, a certain kind of one
recurrence (mixing along a filter) amplifies itself to multiple recurrence.
This amplification is proved using a so-called van der Corput difference lemma
for a suitable filter on the semigroup. Particular instances of this lemma (for
concrete filters) have been proven before (by Furstenberg,
Bergelson--McCutcheon, and others), with a somewhat different proof in each
case. We define a notion of differentiation for subsets of semigroups and
isolate the class of filters that respect this notion. The filters in this
class (call them $\partial$-filters) include all those for which the van der
Corput lemma was known, and our main result is a van der Corput lemma for
$\partial$-filters, which thus generalizes all its previous instances. This is
done via proving a Ramsey theorem for graphs on the semigroup with edges
between the semigroup elements labeled by their ratios.
