Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that occur. We expect $A_N$ to be tightly concentrated around its mean $p$.

Suppose we want to estimate something like $\mathbb{P}(A_N > p^{1/2})$. On the one hand, the multiplicative difference $p^{1/2}/p$ is huge, but on the other hand the additive difference $p^{1/2} - p$ is very small. All the standard concentration inequalities (Azuma-Hoeffding, Chernov, etc.) give an upper bound for the above probability in terms of the additive difference, which gives only a very slow exponential decay rate as $N \to \infty$ for the above probability if $p$ is very small.

My question is: which phenomenon is closer to the truth? Should the event $\{A_N > p^{1/2}\}$ be very rare because $p^{1/2}/p$ is huge, or should it be not so rare because $p^{1/2} - p$ is tiny? If the former, are there any references out there for concentration inequalities that capture that?