(Not sure if this belongs on stack-exchange or overflow; let me know if I should switch it).

Given a sum of $n$ IID random variables $\{X_i\}_{i=1}^n$, each uniform on the integers $0,1,...,r$ for some (fixed) $r$, I would like to estimate $$ \mathbb{P}[\sum_{i=1}^n X_i = k]$$ for $k$ of order $n$. If $k$ happened to be the mean of the sum, then I know how to handle this via the Local Central Limit Theorem. But, when $k$ is far from the mean of the sum, the error term in the LCLT is larger than the first-order term.

This seems like it should be a very standard exercise in large-deviations, but I am not very familiar with that field and am having trouble finding the right tool. Could someone help point me to a theorem (and ideally an example calculation) that might help?