\biggl(\frac{x}{ c_1} -\frac{e_2 n_1^{1/2}}{{\sigma}_1 n_2} \nu \biggr) \phi
\bigl\{e_2 \bigl({\sigma}_2 n_2^{1/2}
\bigr)^{-1}\nu \bigr\}.$$ If (\[eq2.15\]) holds with $\eta=1$ then a standard argument for bounding discrepancies of sequences (see p. 123 of Kuipers and Niederreiter [@Kui74]) can be used to prove that for all $\delta>0$, $$\label{eq2.22} \sup_{-\infty<z<\infty} \Biggl|\sum_{\nu=1}^N
\psi \biggl(z-\frac{e_2 n_1}{ e_1 n_2} \nu \biggr) \Biggr| =O \bigl(N^\delta \bigr).$$ Note too that $$\label{eq2.23} \sup_{\nu\geq1} \sup_{-\infty<x<\infty} \bigl|\Psi(x,\nu+1)-\Psi(x,
\nu)\bigr|\leq C n^{-1/2}.$$ Taking $a_\nu=\Psi(x,\nu)$ and $b_\nu=\psi\{\xi_n(x)-(e_2 n_1/e_1
n_2) \nu\}$, and employing Abel’s method of summation, we can write: $$\sum_{\nu=1}^N a_\nu
b_\nu =a_N \sum_{\nu=1}^N
b_\nu-\sum_{\nu=1}^{N-1}
(a_{\nu+1}-a_\nu) \sum_{j=1}^\nu
b_j,$$ which in company with (\[eq2.22\]) and (\[eq2.23\]) allows us to prove that, provided $N=O(n^C)$ for some $C>0$, $$\label{eq2.24} \sup_{-\infty<x<\infty} \Biggl|\sum_{\nu=1}^N
\Psi(x,\nu) \psi \biggl\{\xi_n(x)-\frac{e_2 n_1}{ e_1 n_2} \nu \biggr\}\Biggr |
=O \bigl(N^\delta \bigr)$$ for all $\delta>0$. More simply, if $N\geq n^2$ then $$\begin{aligned}
\label{eq2.25} &&\sup_{-\infty<x<\infty}\Biggl |\sum_{\nu=N+1}^\infty
\Psi(x,\nu) \psi \biggl\{\xi_n(x)-\frac{e_2 n_1}{ e_1 n_2} \nu \biggr\} \Biggr|
\nonumber
\\
&&\hspace*{157pt}\leq\frac{1}{2} \sup_{-\infty<x<\infty} \sum
_{\nu=N+1}^\infty \Psi(x,\nu) =O(1).\end{aligned}$$ Combining (\[eq2.24\]) and (\[eq2.25\]), using a similar argument to treat series where $\nu\leq0$, and noting the definition of $K_n(x)$ at (\[eq2.21\]), we deduce that $\sup_x |K_n(x)|=O(n^\delta)$ for all $\delta>0$. In the case $k=2$, and for $\eta=1$, this gives (\[eq2.16\]) as a corollary of (\[eq2.14\]).

Derivation 