This talks considers a simple ``1-bit'' random multiple access channel
(MAC), where $n$ users communicate simultaneously to a single receiver.
Each user is assigned a single codeword over $m$ degrees of freedom
which it transmits with some probability $\lambda$.  The receiver must
detect which users transmitted.  The problem provides a simple model for
certain short-message wireless random access control message channels.
 
We show that the detection of users in this random MAC is similar to
the problem of detecting the nonzero positions of a sparse vector from
noisy linear measurements.  Using recent and new scaling laws for
sparsity detection, we can estimate the capacity of the random MAC\@.
The sparsity analysis provides insight into the precise role of power
control, multi-user detection and the trade-off between spread spectrum
and orthogonalization.