(\[eq2.2\]) holds uniformly in $x$. () However, if $\rho_{j_1j_2}$ equals a fixed rational number (not depending on $n$) for each pair $j_1,j_2$, and if the points $x_j$ can all be taken equal, then the expansion at (\[eq2.2\]) fails to hold because it does not include an appropriate discontinuous term of size $n^{-1/2}$.

Circumstances where (2.5) holds {#sec2.2}
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If $\rho_0$ is irrational, then $|\sin(\ell\rho_0\pi)|>0$ for all integers $\ell$. Therefore, (\[eq2.5\]) holds if $\rho_{j_1j_2}$ converges to an irrational number as $n{{\to}\infty}$.

However, in many cases (\[eq2.5\]) holds without the sequence $\rho_{j_1j_2}$ converging. For example, assume for simplicity that the lattice edge widths $e_j$ are all identical, let $\rho_1$ and $\rho_2$ be two distinct irrational numbers, and let the sequence of values of the ratio $n_{j_1}/n_{j_2}$ be a sequence of convergents of $\rho_1$ and $\rho_2$, chosen so that an infinite number of convergents come from each $\rho_j$. (For a definition of convergents of irrational numbers, see, e.g., Leveque [@Lev56], p. 70.) Then (\[eq2.5\]) holds, although the sequence $\rho_{j_1j_2}$ does not converge.

Importantly, (\[eq2.5\]) also holds in many cases where each $\rho_{j_1j_2}$ is close to a rational number, indeed where each $\rho_{j_1j_2}$ converges to a rational number. For example, we claim that (\[eq2.5\]) 