The \emph{Gowers uniformity norms} $\|f\|_{U^k(G)}$ of a function $f: G \to
\C$ on a finite additive group $G$, together with the slight variant
$\|f\|_{U^k([N])}$ defined for functions on a discrete interval $[N] :=
\{1,...,N\}$, are of importance in the modern theory of counting additive
patterns (such as arithmetic progressions) inside large sets. Closely related
to these norms are the \emph{Gowers-Host-Kra seminorms} $\|f\|_{U^k(X)}$ of a
measurable function $f: X \to \C$ on a measure-preserving system $X = (X,
{\mathcal X}, \mu, T)$.