Let $X$ and $Y$ be two i.i.d (independent and identically distributed) discrete random variables with distribution $P=(p_0, p_1, \ldots, p_{q-1})$ and support $\{0,1,...,q-1\}$ with $q \geq 2$. Take $$ P_M = \arg\max_{P} H(X-Y) $$ where $H$ is the Shannon entropy function, i.e. $H(X) = \sum_{i=0}^{q-1} -p_i \log p_i$.
For which $P_M$ we achieve the maximum?
For $q=2$ it is not difficult to show that the uniform distribution achieve the maximum. Is it true also for $q>2$?