crucial that the bond lengths are incommensurate.

We remark that theorem \[thm:weyl\] can also apply to more general functions such as piecewise continuous functions through an argument similar to the one in the following lemma.

\[lem:2\] Theorem \[thm:weyl\] can also be applied to the function $$f({{\mathbf x}}):=\exp\left(\textstyle {{\mathrm i}}\zeta\sum_{j=1}^v\tan x_j\right).$$

[*Proof.*]{}

We treat the real and imaginary parts of $f$ separately. The functions $$\begin{aligned}
f_1({{\mathbf x}})&:=\cos\left(\textstyle \zeta\sum_{j=1}^{v}\tan x_j\right)\\
f_2({{\mathbf x}})&:=\sin\left(\textstyle \zeta\sum_{j=1}^{v}\tan x_j\right)\end{aligned}$$ are smooth everywhere apart from at an essential singularity when $x_i=\pi/2$ for some $i$, which we tame in the following way. Let $\epsilon>0$ . We can construct functions $\phi$ and $\psi$ satisfying the conditions of theorem \[thm:weyl\] such that $$\begin{aligned}
\psi({{\mathbf x}})=-1 & &\mbox{if $|x_i-\pi/2|<\pi\epsilon^{1/v}/8$ for some 
$i=1,\ldots,v$,} \nonumber\\
\phi({{\mathbf x}})=1 & &\mbox{if $|x_i-\pi/2|<\pi\epsilon^{1/v}/8$ for some 
$i=1,\ldots,v$,} \nonumber\\
\psi({{\mathbf x}})=\phi({{\mathbf x}})=f_1({{\mathbf x}})& &\parbox{2.5in}{if 
$\pi\epsilon^{1/v}/4<|x_i-\pi/2|<\pi/2$ for some
$i=1,\ldots,v$,} \nonumber\\
-1\leq\psi({{\mathbf x}})\leq f_1({{\mathbf x}})\leq\phi({{\mathbf x}})\leq 1
& &\mbox{for all ${{\mathbf x}}\in{\mathbb T}^v$.}
\label{eqn:13}\end{aligned}$$ This implies $$\frac{1}{\pi^v}\int_{{\mathbb T}^v} \left(\phi({{\mathbf x}})-\psi({{\mathbf x}})
\right){{\mathrm d}}{{\mathbf x}}
\leq \frac{1}{\pi^v}\left(2\frac{\pi}{2}\epsilon^{1/v}\right)^v =\epsilon.$$ From [(\[eqn:13\])]{} and theorem \[thm:weyl\], $$\begin{aligned}
\frac{1}{\pi^v}\int_{{\mathbb T}^v} \psi({{\mathbf x}}){{\mathrm d}}{{\mathbf x}} \leq
\liminf_{K\to\infty}&\frac{1}{K}\int_0^K f_1(k{{\mathbf L}}){{\mathrm d}}k \\
&\leq\limsup_{K\to\infty}\frac{1}{K}\int_0^K f_1(k{{\mathbf L}}){{\mathrm d}}k \leq
\frac{1}{\pi^v}\int_{{\mathbb T}^v} \phi({{\mathbf x}}){{\mathrm d}}{{\mathbf x}}.\end{aligned}$$ The ends of this inequality 