  Studied in this paper is the well-posedness of the Cauchy problem for the
coupled KdV-KdV systems \[ u_t+a_1u_{xxx} =
c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[
v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \quad
v(x,0)=v_0(x)\] posed on the torus $\mathbb{T}$ in the spaces \[ {\cal
H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_2:=H^s_0
(\mathbb{T})\times H^s(\mathbb{T}), \quad {\cal H}^s_3:=H^s (\mathbb{T})\times
H^s_0 (\mathbb{T}), \quad {\cal H}^s_4:=H^s (\mathbb{T})\times H^s
(\mathbb{T}).\] For $k=1,2,3,4$, it is shown that for given $a_1$, $a_2$,
$(c_{ij})$ and $(d_{ij})$, there exists a unique $s^*_k \in (-\infty,
+\infty]$, called the critical index, such that the system is analytically
well-posed in $\cal{H}^s_k$ for $s>s^*_k$ while the bilinear estimate, the key
for the proof of the analytical well-posedness, fails if $s<s^{*}_k$. Viewing
the critical index $s^*_k$ as a function of the coefficients $a_1$, $a_2$,
$(c_{ij})$ and $(d_{ij})$, its range $\cal{C}_k$ is called the critical index
set for the analytical well-posedness of the system in the space $\cal{H}^s_k$.
Invoking some classical results of Diophantine approximation in number theory,
we are able to identify that \[ \mbox{$ {\cal C}_1= \left \{ -\frac12, \infty
\right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \}$ }
\quad\text{and}\quad \mbox{${\cal C}_q= \left \{ -\frac12, -\frac14, \infty
\right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \}$ $\quad$
for $\quad$ $q=2,3,4$.}\] This is in sharp contrast to the $R$ case in which
the critical index set ${\cal C}$ for the analytical well-posedness of in the
space $H^s (R)\times H^s (R)$ consists of exactly four numbers: $ {\cal
C}=\left \{ -\frac{13}{12}, -\frac34, 0, \frac34 \right \}.$
