  We present theoretical and numerical results for the one-dimensional
stochastically forced Burgers equation decimated on a fractal Fourier set of
dimension $D$. We investigate the robustness of the energy transfer mechanism
and of the small-scale statistical fluctuations by changing $D$. We find that a
very small percentage of mode-reduction ($D \lesssim 1$) is enough to destroy
most of the characteristics of the original non-decimated equation. In
particular, we observe a suppression of intermittent fluctuations for $D <1$
and a quasi-singular transition from the fully intermittent ($D=1$) to the
non-intermittent case for $D \lesssim 1$. Our results indicate that the
existence of strong localized structures (shocks) in the one-dimensional
Burgers equation is the result of highly entangled correlations amongst all
Fourier modes.
