is assumed that all points of support in the distribution (of which ${\theta}$ is the mean) have the form $x_0+\nu e_0$ for an integer $\nu$.

Contributions of this paper {#sec1.2}
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We show that in multi-sample problems, where ${\hat{\theta}}$ is a sum of several independent means, the discontinuous term can be ignored if sample sizes are chosen judiciously. For example, if there are just two sample sizes (as in the case of a sum or difference of two binomial proportions), and if the lattice edge widths are identical (this simplifies our discussion here, but is not essential), then it is sufficient that the ratio of the two sample sizes converge to an irrational number, or converge sufficiently slowly to a rational number. These results are corollaries of Theorem \[the1\] in Section \[sec2.1\], and they and other properties are discussed in Section \[sec2.2\].

We also show that the discontinuous term can be replaced by $O(n^{\delta-1})$, for all $\delta>0$, provided that four moments are finite and the ratio of the two sample sizes converges sufficiently quickly to an irrational number of “type” 1. (See Section \[sec2.3\] for a definition of the type of an irrational number.) More generally, we explore the effect 