# $X$, $Y$, $U$, $V$ are independent on $\{0,1,\ldots,n-1\}$ with $X$, $Y$ uniform and $X+Y \sim U+V$

$$X$$, $$Y$$, $$U$$, $$V$$ are independent and valued on $$\{0,1,\ldots,n-1\}$$ such that $$X$$ and $$Y$$ are uniform and such that $$X+Y\sim U+V$$. On the site recently, somebody raised the question 'Are $$U$$ and $$V$$ necessarily uniform?'. Somebody else said 'yes' without details. I asked for them, but the question disappeared. Can you help me to find it again?

The question is quite interesting, since it asks essentially for the existence of non trivial polynomials $$P$$ and $$Q$$ with non negative coefficients and of degree $$n-1$$ such that

$$P(s)Q(s)=\frac{(1-s^n)^2}{(1-s)^2}.$$

For instance for $$n=6$$, $$P(s)=(1+s^3)^2$$ and $$Q(s)=(1+s+s^2)^2$$ do not fit since their degree is not $$5$$. For giving a counter example we have to distribute the roots of unity in a clever way and I have not been successful yet.

• why can't one just take $U=0$ and $V$ distributed like $X+Y$ and independent from $X,Y$ as a counterexample? Edit: it is not possible, because $V$ has to have values in $\{0,1,...,n-1\}$. Commented Jun 22 at 8:06
• @dialegou you cannot do that because $V$ is valued on $\{0,1,\cdots,n-1\}$
– Will
Commented Jun 22 at 8:07