\omega) \right| \right\},
        \sup_{||{\ensuremath{\mathbf k}}||>K}\left\{\rho(K)\right\}
        \Big\}.$$

This follows directly from (\[eqn:sigmaN\]).

\[thm:diaphony\] Let ${\mathcal{F}}$, $\rho$ and $F_N^{(\alpha)}(\omega)$ be as in Definition \[definition:generalspectraltest\] and suppose that $$\sum_{{\ensuremath{\mathbf k}}\in \Lambda^*} \rho({\ensuremath{\mathbf k}})^\alpha< \infty.$$ Then sequence $\omega$ is uniformly distributed in $[0,1)^s$ if and only if $$\lim_{N\to \infty} F_N^{(\alpha)}(\omega)= 0.$$

We adapt the proof of [@Hel10c Theorem 2] and the splitting technique used in the proof of Theorem \[thm:spectraltest\] to the case of diaphony.

Examples {#s:examples}
========

Examples I: integration lattices {#ss:spectraltest_examples}
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Definition \[definition:generalspectraltest\] generalizes various known notions of the spectral test and of diaphony.

The spectral test has its origin in pseudorandom number generation.It measures the “coarseness” of lattices that can be associated with certain linear types of generators. We refer to [@Knu98a Ch. 3.3.4.]) for a seminal discussion, and to the surveys [@Lec97a; @Nie92a] as well as to [@Hel98f].

We recall some notions from the theory of lattices (for details see [@Nie92a; @Slo94a]). An $s$-dimensional *lattice* is a discrete subset of ${\ensuremath{\mathbb{R}}}^s$ which is closed under addition and subtraction. For any $s$-dimensional lattice $L$, there exists a lattice basis, by which we understand $s$ independent vectors ${\ensuremath{\mathbf g}}_1, \ldots, {\ensuremath{\mathbf g}}_s$ in ${\ensuremath{\mathbb{R}}}^s$ such that $$L= \left\{ \sum_{i=1}^s t_i {\ensuremath{\mathbf 