Background
A rather famous problem is regarding if the sum of loaded dice can be uniformly distributed over $\{2, \cdots, 12\}$; the answer is no. We can see that the top answer in the above link can be generalized to an $n$-sided dice for odd $n$. But what about even $n$? This isn't so hard to prove through a different technique; denote $\mathbb{P}(X = k) = p_k$ and $\mathbb{P}(Y=k) = q_k$ for $k \in \{0, \cdots, n-1\}$ (it's more convenient to work with zero indexing). In order for $X+Y$ to be uniform over $\{0, \cdots, 2n-2\}$, we must have \begin{align*} \mathbb{P}(X+Y = 0) = p_0q_0 = \frac{1}{2n-1} \qquad \text{and} \qquad \mathbb{P}(X+Y = 2n-2) = p_{n-1}q_{n-1} = \frac{1}{2n-1} \end{align*} But \begin{align*} \mathbb{P}(X+Y = n-1) &= \sum_{k=0}^{n-1}\mathbb{P}(X = k)\mathbb{P}(Y = n-1-k) \\ &\ge p_{0}q_{n-1} + p_{n-1}q_0 \\ &\ge 2\sqrt{p_{0}q_{n-1}p_{n-1}q_0} \\ &= \frac{2}{2n-1} \\ &> \frac{1}{2n-1} \end{align*} We see that the above strategy would've also worked for the odd case as well.
Generalizations I have the following few problems. Is it possible to construct two (real) independent random variables $X, Y$ such that $X+Y$ is
- uniformly distributed over a finite set $S \subseteq \mathbb{R}$ (e.g. $S = \{1, 1.6162, 2.71828, 3.14159\}$, it doesn't even have to be integers for that matter). In other words, the measure associated with $X+Y$ is a normalized counting measure over $S$.
- uniformly distributed over finite-Lebesgue measure set $S \subseteq \mathbb{R}$ (e.g. Cantor set, it doesn't have to be an interval). In other words, the measure associated with $X+Y$ is a normalized Lebesgue measure over $S$.
- If the above settings are two limited, then we can extend $X$ and $Y$ to be $p$-dimensional random vectors and ask the associated existence question for $S \subseteq \mathbb{R}^p$.
- If still not possible, perhaps ask if it's possible to construct a sequence $\{X_k\}$ and $\{Y_k\}$ with $X_k, Y_k$ independent and converging to non-degenerate $X, Y$ respectively such that the limit of the measure associated with $X_k + Y_k$ converges to either a normalized counting or normalized Lebesgue measure.
I conjecture the answer to the first 3 problems is not possible because, intuitively in the continuous case, the density of $X+Y$ must be ``smooth'' because convolutions generally smooth the functions they are working with, but a uniform density is discontinuous; a similar principle applies to the discrete case. Problem 4 would be interesting to think further.
