C_2(r) n^{(1/2)-{\alpha}}.$$ (In more detail, without loss of generality the block ${{\mathcal N}}_0$ is centred at 0, in which case, when bounding the series on the left-hand side of (\[eq2.12\]), ${\bar\nu}_\ell$ can be interpreted as $2
\ell n^{\alpha}$. Consequently the left-hand side of (\[eq2.12\]) is bounded by a constant multiple of $C_1(r)\int\phi(2 u n^{\alpha}/n_1^{1/2}) \,du$, and (\[eq2.12\]) follows.)

More simply, since (a) $|\nu_\ell|\leq n^{\alpha}$, (b) $|{{\mathcal N}}_\ell|\leq
(2 n^{\alpha}+1)$ (where we define $|{{\mathcal N}}_\ell|=\#{{\mathcal N}}_\ell$), and (c) $|\psi|\leq{\frac{1}{2}}$, then $$\label{eq2.13} \sup_{-\infty<x<\infty} \biggl|\sum_{\nu\in{{\mathcal N}}_\ell}
\nu_\ell^r \psi \biggl\{\xi_n(x)-
\frac{e_2 n_1}{ e_1 n_2} \nu \biggr\} \biggr| \leq C_3(r) n^{(r+1) {\alpha}}.$$ Combining (\[eq2.10\]), (\[eq2.12\]) and (\[eq2.13\]), and replacing each summand on the right-hand side of (\[eq2.10\]) by its absolute value, we obtain the bound: $n^{-1/2}|K_n(x)|\leq C_4(r_0)$, uniformly in $x$. This inequality demonstrates the claimed absolute convergence of the series in (\[eq2.10\]).

Recall the definition of $S$ at (\[eq2.1\]), and that $\rho_{j_1j_2}=(e_{j_2}n_{j_1})/(e_{j_1}n_{j_2})$. Part (i) of Theorem \[the2\], below, captures the analogue of the discontinuous term, $d_n(x)$, in a multisample version of (\[eq1.2\]), and part (ii) gives conditions under which the net contribution of that term equals $O(n^{\delta
-(1/2)-(1/2\eta)})$, for all $\delta>0$ when some $\rho_{j_1j_2}$ is sufficiently close to an irrational number of type $\eta$.

\[the2\] 