  We consider the $N$ particle classical Riesz gas confined in a
one-dimensional external harmonic potential with power law interaction of the
form $1/r^k$ where $r$ is the separation between particles. As special limits
it contains several systems such as Dyson's log-gas ($k\to 0^+$),
Calogero-Moser model ($k=2$), 1d one component plasma ($k=-1$) and the hard-rod
gas ($k\to \infty$). Despite its growing importance, only large-$N$ field
theory and average density profile are known for general $k$. In this Letter,
we study the fluctuations in the system by looking at the statistics of the gap
between successive particles. This quantity is analogous to the well-known
level spacing statistics which is ubiquitous in several branches of physics. We
show that the variance goes as $N^{-b_k}$ and we find the $k$ dependence of
$b_k$ via direct Monte Carlo simulations. We provide supporting arguments based
on microscopic Hessian calculation and a quadratic field theory approach. We
compute the gap distribution and study its system size scaling. Except in the
range $-1<k<0$, we find scaling for all $k>-2$ with both Gaussian and
non-Gaussian scaling forms.
