  We consider empirical processes associated with high-frequency observations
of a fractional Brownian motion (fBm) $X$ with Hurst parameter $H\in (0,1)$,
and derive conditions under which these processes verify a (possibly uniform)
law of large numbers, as well as a second order (possibly uniform) limit
theorem. We devote specific emphasis to the `zero energy' case, corresponding
to a kernel whose integral on the real line equals zero. Our asymptotic results
are associated with explicit rates of convergence, and are expressed either in
terms of the local time of $X$ or of its \blue{derivatives}: in particular, the
full force of our finding applies to the `rough range' $0< H < 1/3$, on which
the previous literature has been mostly silent. The {\color{black}use of the
derivatives} of local times for studying the fluctuations of high-frequency
observations of a fBm is new, and is the main technological breakthrough of the
present paper. Our results are based on the use of Malliavin calculus and
Fourier analysis, and extend and complete several findings in the literature,
e.g. by Jeganathan (2004, 2006, 2008) and Podolskij and Rosenbaum (2018).
