in the sense that its discrepancy is low.

The conditions $\gcd(a_i,N)=1$, $1\le i\le s$, ensure that all points in $\omega_{\ensuremath{\mathbf a}}$ and in its projections to lower dimensions are distinct.

We may view $\omega_{\ensuremath{\mathbf a}}$ as the node set of an $s$-dimensional $N$-point lattice rule. Let $L(\omega_{\ensuremath{\mathbf a}})$ denote the integration lattice associated with $\omega_{\ensuremath{\mathbf a}}$ defined by (\[eqn:integrationlattice\]). The dual lattice is given by $L(\omega_{\ensuremath{\mathbf a}})^\bot = \left\{ {\ensuremath{\mathbf k}}\in{\ensuremath{\mathbb{Z}}}^s: {\ensuremath{\mathbf k}}\cdot {\ensuremath{\mathbf a}}\equiv 0 \pmod{N}  \right\}$.

With an integer point ${\ensuremath{\mathbf a}}\in {\ensuremath{\mathbb{Z}}}^s$ subject to the conditions of Definition \[def:GLP\], we may associate several figures of merit. In view of the spectral test for lattices, define $\sigma({\ensuremath{\mathbf a}},N)=\sigma (L(\omega_{\ensuremath{\mathbf a}}))$. Then $$\label{eqn:classicalspectraltest2}
\sigma({\ensuremath{\mathbf a}},N)= 1/
        \min \left\{ ||\mathbf{k}||_2: \mathbf{k} \in
        \mathbb{Z}^s\setminus \{ \mathbf{0} \},\
        \mathbf{k}\cdot \mathbf{a} \equiv 0 \pmod{N}  \right\}.$$

For a real number $\alpha> 1$, define $$\label{eqn:Palpha}
P_\alpha(\mathbf{a},N) = \sum_{\scriptstyle
\mathbf{k} \neq \mathbf{0}
\atop \scriptstyle \mathbf{k}\cdot \mathbf{a} \equiv 0 \pmod{N}}
\frac{1}{r(\mathbf{k})^\alpha},$$ where summation is over integer vectors ${\ensuremath{\mathbf k}}$, and $$r(\mathbf{k})= \prod_{i=1}^s \max\{1, |k_i| \},\quad
\mathbf{k} = (k_1, \ldots, k_s) \in \mathbb{Z}^s.$$ Another important figure of merit is the *Babenko-Zaremba index* $$\label{eqn:BZindex}
\varkappa (\mathbf{a},N) = 1 / \min \left\{ r(\mathbf{k}): \mathbf{k} \in
        \mathbb{Z}^s\setminus \{ \mathbf{0} \},\
         \mathbf{k}\cdot 