Consider two identically distributed independent random vectors $X$ and $Y$ of dimension $n$ (assume $n$ is large). Both $X$ and $Y$ have only non-negative integer entries less than or equal to $n$ and $H(X)= H(Y) = n$.
What is the maximum possible value of $H(X + Y)$?
In particular, how close can it get to $2n$?
My working so far
If $X$ and $Y$ are random $0/1$ vectors then $H(X+Y) = 3n/2$ I believe. This is the highest value for $H(X+Y)$ I have managed to find so far.