  Intermittency phenomena are known to be among the main reasons why
Kolmogorov's theory of fully developed Turbulence is not in accordance with
several experimental results. This is why some \emph{fractal} statistical
models have been proposed in order to realign the theoretical physical
predictions with the empirical experiments. They indicate that energy
dissipation, and thus singularities, are not space filling for high Reynolds
numbers. This note aims to give a precise mathematical statement on the energy
conservation of such fractal models of Turbulence. We prove that for
$\theta-$H\"older continuous weak solutions of the incompressible Euler
equations energy conservation holds if the upper Minkowski dimension of the
spatial singular set $S \subseteq \T^3$ (possibly also time-dependent) is
small, or more precisely if $\overline{\dim}_{\mathcal{M}}(S)<2+3\theta\,.$ In
particular, the spatial singularities of \emph{non-conservative}
$\theta-$H\"older continuous weak solutions of Euler are concentrated on a set
with dimension lower bound $2+3\theta$. This result can be viewed as the
fractal counterpart of the celebrated Onsager conjecture and it matches both
with the prediction given by the $\beta-$model introduced by Frisch, Sulem and
Nelkin in \cite{FSN78} and with other mathematical results in the endpoint
cases.
