(\[eq2.19\]) holds.

The pair $(n_1,n_2)$ can be chosen from tables of, or formulae for, convergents for commonly arising irrationals of type 1. See, for example, Griffiths [@Gri04] and references therein, and note that $e$ and any algebraic irrational is of type 1.

Alternative formula for $K_n$, and derivation of (2.16) from (2.14) when eta=1 {#sec2.5}
------------------------------------------------------------------------------

Part (i) of Theorem \[the2\] can be stated for a version of $K_n(x)$ simpler than that at (\[eq2.10\]): $$\label{eq2.20} K_n(x)={\gamma}\sum_\nu\phi
\biggl(\frac{x}{ c_1} -\frac{e_2 n_1^{1/2}}{{\sigma}_1 n_2} \nu \biggr) \phi \bigl
\{e_2 \bigl({\sigma}_2 n_2^{1/2}
\bigr)^{-1}\nu \bigr\} \psi \biggl\{\xi_n(x)-
\frac{e_2 n_1}{ e_1 n_2} \nu \biggr\}.$$ Indeed, the $K_n(x)$ at (\[eq2.20\]) is just ${\gamma}I_4(x)$, where $I_4(x)$ is as defined at (\[eq4.17\]) in the proof of Theorem \[the1\], and in fact that formula provides a convenient point of access to a proof of (\[eq2.14\]) with $K_n(x)$ as at (\[eq2.20\]). However, in the case $\eta>1$ it is not straightforward to pass from (\[eq2.20\]) to (\[eq2.16\]), and that is why we used the definition of $K_n(x)$ at (\[eq2.10\]).

To appreciate that (\[eq2.16\]) follows from (\[eq2.20\]) when $\eta=1$, note that the definition of $K_n(x)$ at (\[eq2.20\]) is equivalent to: $$\label{eq2.21} {\gamma}^{-1}K_n(x)=\sum
_\nu\Psi(x,\nu) \psi \biggl\{\xi_n(x)-
\frac{e_2 n_1}{ e_1 n_2} \nu \biggr\},$$ where $$\Psi(x,\nu)=\phi 