  We suggest a new type of substrates for the Surface Enhanced Raman Scattering
measurements with the geometry based on self-similar fractal space filling
curves. As an example, we have studied theoretically the dielectric response
properties of doped semiconductor nanostructures, where the conducting
electrons are trapped in the effective potential having the geometry of the
Hilbert curve. We have found that the system may exhibit the induced charge
distribution specific for either two dimensional or one dimensional systems,
depending on the frequency of the external applied field. We have demonstrated
that with the increasing of the depth of the trapping potential the resonance
of the system counterintuitively shifts to lower frequencies.
