# Can the sum of two non-degenerative random variables ever be uniform?

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

1. 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 $$S$$.
2. 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 $$S$$.
3. 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 $$S \subseteq \mathbb{R}^p$$.
4. 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.

If $$X$$ or $$Y$$ (or both) is discrete then it is possible to find examples that sum to a uniform distribution. For example, let $$X$$ assign a weight of $$1/2$$ to each of $$0$$ and $$1$$, and let $$S$$ be any subset of $$\mathbb R$$ such that $$S$$ and $$S+1$$ are disjoint. Then if the measure of $$Y$$ is the uniform (or counting, if $$S$$ is finite) measure over $$S$$ then $$X+Y$$ is uniformly distributed over $$S\cup (S+1)$$.