zero at $(a_0,t_0,x_0)$. Therefore the matrix ${\begin{pmatrix}
\ds\frac{\partial P}{\partial t} & \ds\frac{\partial P}{\partial x} \\
\ds\frac{\partial Q}{\partial t} & \ds\frac{\partial Q}{\partial x}
\end{pmatrix}}$ is invertible when the entries are taken at $(a_0,t_0,x_0)$. Hence by Implicit function theorem $t$ and $x$ can be expressed as analytic function of $a$ around $a_0$. The same proof holds for $\gamma_{p/q}^r$.

Angle between the bounding curves of the rational tongue {#sec_angle}
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In this section we prove that there are lines tangents to these boundary curves of rational tongues at $(p/q,0)$. A slight modification in the definition of the boundaries $\gamma_{p/q}^{l}$ and $\gamma_{p/q}^r$ gives the angle between them. The average of the translates of $\p$ by $p/q$ plays a role here.

Let us define the average of the translates of $\p$ by $p/q$ as $\cal A_q(\p)$, i.e. $$\cal A_q(\p)(x)=\frac{1}{q}\sum_{k=0}^{q-1}\p(x+kp/q).$$ Moreover we set $$\cal M(\p)=\int_0^1\p(x)\d x, \quad M_{\cal A}=\max \cal A_q(\p)\quad \text{and}\quad
m_{\cal A}=\min \cal A_q(\p).$$

\[prop\_angle\] We have the following asymptotic expansions for the boundaries of the tongues.

-   For $\alpha\in \R\setminus \Q$, we have $\gamma_{\alpha}(a)=\alpha-\cal M(\p)a+o(a)$ .

-   For $p/q\in \Q$ if $\cal A_q(\p)\equiv \cal M(\p)$ then $\gamma_{p/q}^{l}(a)=p/q-\cal M(\p)a+o(a)$ and $\gamma_{p/q}^{r}(a)=p/q-\cal M(\p)a+o(a)$.

-   For $p/q\in \Q$, we have

    1.  $\gamma_{p/q}^-(a)=p/q-M_{\cal A}a+o(a)$, where $\gamma_{p/q}^-(a)=\left\{\begin{array}{ll} \gamma_{p/q}^l(a)\;\text{for}\; 