  We exhibit simple lattice systems, motivated by recently proposed cold atom
experiments, whose continuum limits interpolate between real and $p$-adic
smoothness as a spectral exponent is varied. A real spatial dimension emerges
in the continuum limit if the spectral exponent is negative, while a $p$-adic
extra dimension emerges if the spectral exponent is positive. We demonstrate
Holder continuity conditions, both in momentum space and in position space,
which quantify how smooth or ragged the two-point Green's function is as a
function of the spectral exponent. The underlying discrete dynamics of our
model is defined in terms of a Gaussian partition function as a classical
statistical mechanical lattice model. The couplings between lattice sites are
sparse in the sense that as the number of sites becomes large, a vanishing
fraction of them couple to one another. This sparseness property is useful for
possible experimental realizations of related systems.
