  After recalling the concept of the Hausdorff dimension, we study the fractal
properties of a quantum particle path. As a novelty we consider the possibility
for the space where the particle propagates to be endowed with a
quantum-gravity-induced minimal length. We show that the Hausdorff dimension
accounts for both the quantum mechanics uncertainty and manifold fluctuations.
In addition the presence of a minimal length breaks the self-similarity
property of the erratic path of the quantum particle. Finally we establish a
universal property of the Hausdorff dimension as well as the spectral
dimension: They both depend on the amount of resolution loss which affects both
the path and the manifold when quantum gravity fluctuations occur.
