A precise local limit theorem for the multinomial distribution

We develop a precise local limit theorem for the multinomial distribution where the error terms are explicit up to an order smaller than previous known results by a factor of $N^{1/2}$. We show how it can be used to approximate multinomial probabilities on most subsets of $\mathbb{R}^d$ and we also describe potential applications related to asymptotic properties of Bernstein estimators on the simplex, bounds for the deficiency distance with multivariate normal experiments and finely tuned continuity corrections.