  In this article, we study the smallest gaps of the log-gas $\beta$-ensemble
on the unit circle (C$\beta$E), where $\beta$ is any positive integer. The main
result is that the smallest gaps, after being normalized by $n^{\frac
{\beta+2}{\beta+1}}$, will converge in distribution to a Poisson point process
with some explicit intensity. And thus one can derive the limiting density of
the $k$-th smallest gap, which is proportional to
$x^{k(\beta+1)-1}e^{-x^{\beta+1}}$. In particular, the result applies to the
classical COE, CUE and CSE in random matrix theory. The essential part of the
proof is to derive several identities and inequalities regarding the Selberg
integral, which should have their own interest.
