  This article is concerned with the construction and analysis of new time
discretizations for the KdV equation on a torus for low-regularity solutions
below $H^1$. New harmonic analysis tools, including new averaging
approximations to the exponential phase functions, new frequency decomposition
techniques, and new trilinear estimates of the KdV operator, are established
for the construction and analysis of time discretizations with higher
convergence orders under low-regularity conditions. In addition, new techniques
are introduced to establish stability estimates of time discretizations under
low-regularity conditions without using filters when the energy techniques
fail. The proposed method is proved to be convergent with order $\gamma$ (up to
a logarithmic factor) in $L^2$ under the regularity condition $u\in
C([0,T];H^\gamma)$ for $\gamma\in(0,1]$.
