the equidistribution behavior of the sequence $\omega$ in ${\ensuremath{\mathcal{O}}}$, because the group operation on ${\ensuremath{\mathcal{S}}}$ is intrinsically related to the [*dual group*]{} $\hat{{\ensuremath{\mathcal{S}}}}$ of ${\ensuremath{\mathcal{S}}}$.

An example of such a suitable match between sequences and function systems based on this *duality principle* is given by Kronecker sequences or, in the discrete version, good lattice points, and the trigonometric functions. Here, ${\ensuremath{\mathcal{S}}}={\ensuremath{\mathcal{O}}}=[0,1)^s$, and the construction method uses [*addition modulo one*]{} on the $s$-torus $[0,1)^s$, $T(s)=s+\boldsymbol{\alpha}$, with $\boldsymbol{\alpha}\in{\ensuremath{\mathbb{R}}}^s$ (see [@Drm97a], [@Nie92a Ch. 5] and [@Slo94a]). The dual of the compact group ${\ensuremath{\mathcal{S}}}$ may be interpreted as the trigonometric function system. For the background of these group rotations in ergodic theory, we refer to the monographs [@Parry04a; @Walters82a].

A second example of this duality principle is given by digital nets and sequences and the Walsh functions. Here, [*addition without carry*]{} of digit vectors comes into play (see [@Nie92a Ch. 4] and [@Dick10a]). For example, for nets and sequences in base 2, the underlying group ${\ensuremath{\mathcal{S}}}$ is the compact group ${{\mathbb{F}}}_2^\infty$ and its dual group is the Walsh function system in base $2$. The same relation holds for general integer bases $b\ge 2$.

An important type of a digital sequence, the Halton sequence in 