  The minimum energy, and, more generally, the minimum cost, to transmit one
bit of information has been recently derived for bursty communication when
information is available infrequently at random times at the transmitter.
Furthermore, it has been shown that even if the receiver is constrained to
sample only a fraction $\rho\in (0,1]$ of the channel outputs, there is no
capacity penalty. That is, for any strictly positive sampling rate $\rho>0$,
the asynchronous capacity per unit cost is the same as under full sampling,
i.e., when $\rho=1$. Moreover, there is no penalty in terms of decoding delay.
  The above results are asymptotic in nature, considering the limit as the
number $B$ of bits to be transmitted tends to infinity, while the sampling rate
$\rho$ remains fixed. A natural question is then whether the sampling rate
$\rho(B)$ can drop to zero without introducing a capacity (or delay) penalty
compared to full sampling. We answer this question affirmatively. The main
result of this paper is an essentially tight characterization of the minimum
sampling rate. We show that any sampling rate that grows at least as fast as
$\omega(1/B)$ is achievable, while any sampling rate smaller than $o(1/B)$
yields unreliable communication. The key ingredient in our improved
achievability result is a new, multi-phase adaptive sampling scheme for
locating transient changes, which we believe may be of independent interest for
certain change-point detection problems.
