  We prove the local well-posedness for the Cauchy problem of the Korteweg-de
Vries equation in a quasi periodic function space. The function space contains
functions such that f=f_1+f_2+...+f_N where f_j is in the Sobolev space of
order s>-1/2N of a_j periodic functions. Note that f is not a periodic function
when the ratio of periods a_i/a_j is irrational. The main tool of the proof is
the Fourier restriction norm method introduced by Bourgain. We also prove an
ill-posedness result in the sense that the flow map (if it exists) is not C^2,
which is related to the Diophantine problem.
