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 $ρ\in (0,1]$ of the channel outputs, there is no capacity penalty. That is, for any strictly positive sampling rate $ρ>0$, the asynchronous capacity per unit cost is the same as under full sampling, i.e., when $ρ=1$. 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 $ρ$ remains fixed. A natural question is then whether the sampling rate $ρ(B)$ 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 $ω(1/B)$ 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.