The similarity between quantum paths and fractals (Mandelbrot's remarks and references) seems to indicate that fundamental reality itself has a fractal structure.

It seems intuitive that Heisenberg's uncertainty relations suggest that if you zoom to smaller and smaller scales in space you find wilder and wilder fractal fluctuations in velocity. This seems to indicate that the underlying space should be a multifractal with dimension varying between 1 (per ordinary dimension) at very large scales and 2 (per ordinary dimension) at very small scales.

So the fractal dimension of quantum space should be *higher* than that of ordinary space, not lower as you suggest.

Or perhaps there could be smart ways of putting the two pictures together?

EDITED: Yes there is a smart way of putting the two pictures rogehter: the fractal dimension (per ordinary dimension) of space varies between less than 1 (e/3?) at cosmological scales, through 1 at our scales, to 2 at very small scales.