that type has on the size of the discontinuous term. Theorem \[the2\] also gives an explicit formula for the discontinuous term, up to a remainder of order $n^{-1}$. Sections \[sec2.5\] and \[sec2.6\] show how to bound the discontinuous remainder term, for two different approaches to defining that quantity, and show how the effects of irrationals of different types can be teased from the remainder. Applications to the bootstrap are straightforward, and in fact Section \[sec2.7\] outlines a bootstrap version of Theorem \[the2\] and discusses its implications.

We do not treat in any detail cases where the differences between two lattice distributions arise mainly in terms of their centres, rather than their lattice edge widths. For example, if two independent sample means ${\bar X}_j$, for $j=1,2$, are respectively averages of $n_j$ independent variables and are distributed on lattices $x_j+\nu n_j^{-1}
e_j$; and if the difference $x_1-x_2$ between the lattice translations equals an irrational multiple of the ratio $\rho_{12}=(e_2 n_1)/(e_1
n_2)$; then the distribution of ${\bar X}_1+{\bar X}_2$ is non-lattice. While this problem and its implications are of mathematical interest, they do not enjoy the practical motivation of problems where, say, $x_1=x_2$ and $\rho_{12}$ can be almost arbitrary. For example, $x_1=x_2$ in 