Local well-posedness of the KdV equation with quasi periodic initial data

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.