  By assuming a self-similar structure for Kelvin waves along vortex loops with
successive smaller scale features, we model the fractal dimension of a
superfluid vortex tangle in the zero temperature limit. Our model assumes that
at each step the total energy of the vortices is conserved, but the total
length can change. We obtain a relation between the fractal dimension and the
exponent describing how the vortex energy per unit length changes with the
length scale. This relation does not depend on the specific model, and shows
that if smaller length scales make a decreasing relative contribution to the
energy per unit length of vortex lines, the fractal dimension will be higher
than unity. Finally, for the sake of more concrete illustration, we relate the
fractal dimension of the tangle to the scaling exponents of amplitude and
wavelength of a cascade of Kelvin waves.
