  This is the first part of a two-paper series studying nonlinear Schr\"odinger
equations with quasi-periodic initial data. In this paper, we consider the
standard nonlinear Schr\"odinger equation. Under the assumption that the
Fourier coefficients of the initial data obey a power-law upper bound, we
establish local existence of a solution that retains quasi-periodicity in space
with a slightly weaker Fourier decay. Moreover, the solution is shown to be
unique within this class of quasi-periodic functions. In addition, for the
nonlinear Schr\"odinger equation with small nonlinearity, within the time
scale, as the small parameter of nonlinearity tends to zero, we prove that the
nonlinear solution converges asymptotically to the linear solution with respect
to both the sup-norm $\|\cdot\|_{L_x^\infty(\mathbb R)}$ and the Sobolev-norm
$\|\cdot\|_{H^s_x(\mathbb R)}$.
  The proof proceeds via a consideration of an associated infinite system of
coupled ordinary differential equations for the Fourier coefficients and a
combinatorial analysis of the resulting tree expansion of the coefficients. For
this purpose, we introduce a Feynman diagram for the Picard iteration and
$\ast^{[\cdot]}$ to denote the complex conjugate label.
