Assume that $E|X_{j1}|^4<\infty$ for $j=1,\ldots,k$; that $X_{j1}$ is distributed on a lattice $x_j+\nu e_j$, for integers $\nu
$, where $e_j$ is the maximal lattice edge width; and that (\[eq2.4\]) holds. Choose $r_0\geq4{\alpha}/(1-2{\alpha})$ in (\[eq2.10\]). () If $k=2$ and $K_n$ is as defined at (\[eq2.10\]), then $$\label{eq2.14}
\hspace*{-5pt}P \biggl\{\frac{S-E(S)}{({\operatorname{var}}S)^{1/2}}\leq x \biggr\} =\Phi(x)+n^{-1/2}{
\frac{1}{6}}{\beta}\bigl(1-x^2 \bigr) \phi(x) +(n_1n_2)^{-1/2}K_n(x)
+O \bigl(n^{-1} \bigr),$$ uniformly in $x$. () If, for some pair $j_1,j_2$ with $1\leq
j_1<j_2\leq k$, the ratio $\rho_{j_1j_2}=(e_{j_2}n_{j_1})/ (e_{j_1}n_{j_2})$ satisfies $$\label{eq2.15} |\rho_{j_1j_2}-\rho_0|=O \bigl(n^{-(1/2) \{1+(1/\eta)+\delta\}}
\bigr)$$ for some $\delta>0$, where $\rho_0$ is an irrational number of type $\eta$, then, for each $\delta>0$, $$\label{eq2.16} P \biggl\{\frac{S-E(S)}{({\operatorname{var}}S)^{1/2}}\leq x \biggr\} =\Phi(x)+n^{-1/2}{
\frac{1}{6}}{\beta}\bigl(1-x^2 \bigr) \phi(x) +O
\bigl(n^{\delta-(1/2)-(1/2\eta)} \bigr),$$ uniformly in $x$.

Result (\[eq2.16\]) is of particular interest in the case $\eta=1$, which encompasses almost all irrational numbers (with respect to Lebesgue measure), including all the algebraic irrationals and some transcendental numbers. When $\eta=1$, $$\label{eq2.17} P \biggl\{\frac{S-E(S)}{({\operatorname{var}}S)^{1/2}}\leq x \biggr\} =\Phi(x)+n^{-1/2}{
\frac{1}{6}}{\beta}\bigl(1-x^2 \bigr) \phi(x) +O
\bigl(n^{\delta-1} \bigr),$$ uniformly in $x$ for each $\delta>0$. Result (\[eq2.17\]) implies that the lattice nature of the distribution of $X_{ji}$ can be ignored, almost up to terms of second order in Edgeworth expansions, when considering the impact of latticeness on the accuracy of normal approximations.

Practical 