Consider a dice $D$ with sides $\{1,\ldots,6\}$ and let $p_i$ be the (constant) probability of each side). Let further be $A_n$ the event that any side shows up more than $p_i + h(n)$ for $h(n) = \omega(\sqrt{n})$ times. Show that for $n \rightarrow \infty$ follows $\mathbb{P}[A_n] \rightarrow 0$.
This reminds me of Chernoff's bound, which states that for a Binomial trial $X$ and a constant $t > 0$ we may write
$$\mathbb{P}[X \ge \mathbb{E}[X] + t] \le \exp\left(- \frac{t^2}{2(\mathbb{E}[X] + t/3)} \right).$$
I know that $\mathbb{E}[X] = np = \omega(\sqrt{n})$, but I am not sure if this helps here. Could you please give me a hint?