  We report an investigation of the Snyder noncommutative spacetime and of some
of its most natural generalizations, also looking at them as a powerful tool
for comparing different notions of dimensionality of a quantum spacetime. It is
known that (generalized-)Snyder noncommutativity, while having rich off-shell
implications (kinematical Hilbert space), does not affect on-shell particles
(physical Hilbert space), and we argue that physically meaningful notions of
dimensionality should describe such spacetimes as trivially four-dimensional,
without any running with scales. By studying the thermodynamics of a gas of
massless particles living on these spacetimes, we find that indeed the Snyder
model and its generalizations have constant thermal dimension of four. We also
compute the spectral dimension of the Snyder model and its generalizations,
finding that, as a result of its sensitivity to off-shell properties, it runs
from the standard value of four in the infrared towards lower values in the
ultraviolet limit.
