Let $X_{1},\ldots,X_{n}$ be i.i.d. random variables, each variable is uniform over the set of integers $\{ 0,\ldots,D-1 \}$. Let $S = \sum_{i=1}^{n}X_{i}$.
By ``small ball probability'', we have that $$\Pr[|S - nD/2| \le \varepsilon] \le C \cdot \frac{\varepsilon + D^{3}}{\sqrt{n}}$$ for some constant $C$. I don't know if the $\sqrt{n}$ can be improved to some higher power in this specific setting.
I am interested in the case where $X_{1},\ldots,X_{n}$ are not completely independent, but only $t$-wise independent (that is, the marginal distribution of every $t$ variables is uniform) for $t = t(n)$. Obviously, the smaller $t$ is the better.
Can I hope to achieve a similar claim? For which power of $n$?
