  A cache-aided $K$-user Gaussian broadcast channel (BC) is studied. The
transmitter has a library of $N$ files, from which each user requests one. The
users are equipped with caches of different sizes, which are filled without the
knowledge of the user requests in a centralized manner. Differently from the
literature, it is assumed that each file can be delivered to different users at
different rates, which may correspond to different quality representations of
the underlying content, e.g., scalable coded video segments. Accordingly,
instead of a single achievable rate, the system performance is characterized by
a rate tuple, which corresponds to the vector of rates users' requests can be
delivered at. The goal is to characterize the set of all achievable rate tuples
for a given total cache capacity by designing joint cache and channel coding
schemes together with cache allocation across users. Assuming that the users
are ordered in increasing channel quality, each file is coded into $K$ layers,
and only the first $k$ layers of the requested file are delivered to user $k$,
$k=1,...,K$. Three different coding schemes are proposed, which differ in the
way they deliver the coded contents over the BC; in particular, time-division,
superposition, and dirty paper coding schemes are studied. Corresponding
achievable rate regions are characterized, and compared with a novel outer
bound. To the best of our knowledge, this is the first work studying the
delivery of files at different rates over a cache-aided noisy BC.
