  A novel investigation of the nature of intermittency in incompressible,
homogeneous and isotropic turbulence is performed by a numerical study of the
Navier-Stokes equations constrained on a fractal Fourier set. The robustness of
the energy transfer and of the vortex stretching mechanisms is tested by
changing the fractal dimension, D, from the original three dimensional case to
a strongly decimated system with D=2.5, where only about $3\%$ of the Fourier
modes interact. This is a unique methodology to probe the statistical
properties of the turbulent energy cascade, without breaking any of the
original symmetries of the equations. While the direct energy cascade persists,
deviations from the Kolmogorov scaling are observed in the kinetic energy
spectra. A model in terms of a correction with a linear dependency on the
co-dimension of the fractal set, $E(k) \sim k^{-5/3 + 3-D}$, explains the
results. At small scales, the intermittency of the vorticity field is observed
to be quasi-singular as a function of the fractal mode reduction, leading to an
almost Gaussian statistics already at $D \sim 2.98$. These effects must be
connected to a genuine modification in the triad-to-triad nonlinear energy
transfer mechanism.
