  Khanna and Sudan \cite{KS11} studied a natural model of continuous time
channels where signals are corrupted by the effects of both noise and delay,
and showed that, surprisingly, in some cases both are not enough to prevent
such channels from achieving unbounded capacity. Inspired by their work, we
consider channels that model continuous time communication with adversarial
delay errors. The sender is allowed to subdivide time into an arbitrarily large
number $M$ of micro-units in which binary symbols may be sent, but the symbols
are subject to unpredictable delays and may interfere with each other. We model
interference by having symbols that land in the same micro-unit of time be
summed, and we study $k$-interference channels, which allow receivers to
distinguish sums up to the value $k$. We consider both a channel adversary that
has a limit on the maximum number of steps it can delay each symbol, and a more
powerful adversary that only has a bound on the average delay.
  We give precise characterizations of the threshold between finite and
infinite capacity depending on the interference behavior and on the type of
channel adversary: for max-bounded delay, the threshold is at
$D_{\text{max}}=\ThetaM \log\min{k, M}))$, and for average bounded delay the
threshold is at $D_{\text{avg}} = \Theta(\sqrt{M \cdot \min\{k, M\}})$.
