  Planck-scale dynamical dimensional reduction is attracting more and more
interest in the quantum-gravity literature since it seems to be a model
independent effect. However different studies base their results on different
concepts of spacetime dimensionality. Most of them rely on the
\textit{spectral} dimension, others refer to the \textit{Hausdorff} dimension
and, very recently, it has been introduced also the \textit{thermal} dimension.
We here show that all these distinct definitions of dimension give the same
outcome in the case of the effective regime of Loop Quantum Gravity (LQG). This
is achieved by deriving a modified dispersion relation from the
hypersurface-deformation algebra with quantum corrections. Moreover we also
observe that the number of UV dimensions can be used to constrain the
ambiguities in the choice of these LQG-based modifications of the Dirac
spacetime algebra. In this regard, introducing the \textit{polymerization} of
connections i.e. $K \rightarrow \frac{\sin(\delta K)}{\delta}$, we find that
the leading quantum correction gives $d_{UV} = 2.5$. This result may indicate
that the running to the expected value of two dimensions is ongoing, but it has
not been completed yet. Finding $d_{UV}$ at ultra-short distances would require
to go beyond the effective approach we here present.

General Relativity and Quantum Cosmology