  We study low regularity local well-posedness of the nonlinear Schr\"odinger
equation (NLS) with the quadratic nonlinearity $\overline{u}^2$, posed on
one-dimensional and two-dimensional tori. While the relevant bilinear estimate
with respect to the $X^{s, b}$-space is known to fail when the regularity $s$
is below some threshold value, we establish local well-posedness for such low
regularity by introducing modifications on the $X^{s, b}$-space.
