  In the present article, we prove the sharp local well-posedness and
ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the
initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and
ill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained
from reduction of the problem into a quadratic nonlinear Schr\"odinger equation
and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof
of the crucial bilinear estimates in these spaces, especially in the lowest
regularity, rely on some bilinear estimates for one dimensional periodic
functions in $X^{s,b}$ spaces, which are generalization of the bilinear
refinement of the $L^4$ Strichartz estimate on $\mathbb{R}$. Our result
improves the known local well-posedness in $H^s(\mathbb{T})$ with $s>-3/8$
given by Oh and Stefanov (2012) to the regularity threshold
$H^{-1/2}(\mathbb{T})$. Similar ideas also establish the sharp local
well-posedness in $H^{-1/2}(\mathbb{R})$ and ill-posedness below $H^{-1/2}$ for
the nonperiodic case, which improves the result of Tsugawa and the author
(2010) in $H^s(\mathbb{R})$ with $s>-1/2$ to the limiting regularity.
