  In view of promising applications of fractal nanostructures, we analyze the
spectra of quantum particles in the Sierpinski carpet and study the
non-correlated electron gas in this geometry. We show that the spectrum
exhibits scale invariance with almost arbitrary spacing between energy levels,
including large energy gaps at high energies. These features disappear in the
analogous random fractal---where Anderson localization dominates---and in the
regular lattice of equally sized holes---where only two length scales are
present. The fractal structure amplifies microscopic effects, resulting in the
presence of quantum behavior of the electron gas even at high temperatures. Our
results demonstrate the potential of fractal nanostructures to improve the
light-matter interaction at any frequency, with possible applications, e.g., in
the development of solar cells with a wide absorption spectrum, artificial
photosynthesis, or nanometamaterials with tailored Fermi levels and band gaps,
operating in a wide range of frequencies, and with extended operating
temperature range.
