of (2.16) from (2.14) when eta&gt;=1 {#sec2.6}
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Our proof of (\[eq2.16\]), in Section \[sec4.2\], will proceed by deriving implicitly a version of (\[eq2.14\]) in the case $k\geq2$, and showing that, if (\[eq2.15\]) holds, then that version of (\[eq2.14\]) entails (\[eq2.16\]). The relative complexity of a form of (\[eq2.14\]) for general $k$ discouraged us from including it in Theorem \[the2\], but it is nevertheless instructive to show how, when $k=2$, one can obtain (\[eq2.16\]) from (\[eq2.14\]). We outline the proof below, highlighting the properties of irrational numbers, particularly the differences between the case of irrationals of type $\eta=1$ and the case of those of larger type, that determine the bound for the remainder in (\[eq2.16\]).

Note that if $q$ is a polynomial function then, applying Koksma’s inequality (see, e.g., Theorem 5.1, p. 143 of [@Kui74]) and the Erdős-Turán inequality (see, e.g., formula (2.42), p. 114 of [@Kui74]), it can be shown that $$\label{eq2.26} \chi(N,q,\tau)\equiv\sup_{-\infty<z<\infty} \Biggl|\sum
_{i=1}^N q(i/N) \psi(z-\tau i)\Biggr | \leq
C_1(q) \Biggl\{\frac{N}{ m} +\sum
_{\ell=1}^m \frac{1}{\ell|\sin(\ell\tau\pi)|} \Biggr\} ,$$ for all integers $m\geq1$. Here $\tau>0$ is permitted to vary with $N$, and the constant $C_1(q)$ depends on the degree and the coefficients of $q$ but not on the positive integer 