  An important probe of quantum geometry is its spectral dimension, defined via
a spatial diffusion process. In this work we study the spectral dimension of a
``spatial hypersurface'' in a manifoldlike causal set using the induced spatial
distance function. In previous work, the diffusion was taken on the full causal
set, where the nearest neighbours are unbounded in number. The resulting
super-diffusion leads to an increase in the spectral dimension at short
diffusion times, in contrast to other approaches to quantum gravity. In the
current work, by using a temporal localisation in the causal set, the number of
nearest spatial neighbours is rendered finite. Using numerical simulations of
causal sets obtained from $d=3$ Minkowski spacetime, we find that for a flat
spatial hypersurface, the spectral dimension agrees with the Hausdorff
dimension at intermediate scales, but shows clear indications of dimensional
reduction at small scales, i.e., in the ultraviolet. The latter is a direct
consequence of ``discrete asymptotic silence'' at small scales in causal sets.
