  We study relations between subsets of integers that are large, where large
can be interpreted in terms of size (such as a set of positive upper density or
a set with bounded gaps) or in terms of additive structure (such as a Bohr
set). Bohr sets are fundamentally abelian in nature and are linked to Fourier
analysis. Recently it has become apparent that a higher order, non-abelian,
Fourier analysis plays a role in both additive combinatorics and in ergodic
theory. Here we introduce a higher order version of Bohr sets and give various
properties of these objects, generalizing results of Bergelson, Furstenberg,
and Weiss.
