  Consider a (multiple-access) wireless communication system where users are
connected to a unique base station over a shared-spectrum radio links. Each
user has a fixed number $k$ of bits to send to the base station, and his signal
gets attenuated by a random channel gain (quasi-static fading). In this paper
we consider the many-user asymptotics of Chen-Chen-Guo'2017, where the number
of users grows linearly with the blocklength. Differently, though, we adopt a
per-user probability of error (PUPE) criterion (as opposed to classical
joint-error probability criterion). Under PUPE the finite energy-per-bit
communication is possible, and we are able to derive bounds on the tradeoff
between energy and spectral efficiencies. We reconfirm the curious behaviour
(previously observed for non-fading MAC) of the possibility of almost perfect
multi-user interference (MUI) cancellation for user densities below a critical
threshold. Further, we demonstrate the suboptimality of standard solutions such
as orthogonalization (i.e., TDMA/FDMA) and treating interference as noise (i.e.
pseudo-random CDMA without multi-user detection). Notably, the problem treated
here can be seen as a variant of support recovery in compressed sensing for the
unusual definition of sparsity with one non-zero entry per each contiguous
section of $2^k$ coordinates. This identifies our problem with that of the
sparse regression codes (SPARCs) and hence our results can be equivalently
understood in the context of SPARCs with sections of length $2^{100}$. Finally,
we discuss the relation of the almost perfect MUI cancellation property and the
replica-method predictions.
