This is a supplement to Jeremy Rouse's nice answer. Let $\alpha$ and $\beta$ be positive irrational numbers. Skolem proved in 1957 (see Theorem 8 here) that the Beatty sequences $[\alpha n]$ and $[\beta n]$ are disjoint if and only if $a/\alpha+b/\beta=1$ holds for some positive integers $a$ and $b$. It follows that the OP's sets $u$ and $v$ are disjoint if and only if $k$ is odd and not a perfect square.

This is a supplement to Jeremy Rouse's nice answer. Let $\alpha$ and $\beta$ be positive irrational numbers. Skolem proved in 1957 (see Theorem 8 in On certain distributions of integers in pairs with given differences) that the Beatty sequences $[\alpha n]$ and $[\beta n]$ are disjoint if and only if $a/\alpha+b/\beta=1$ holds for some positive integers $a$ and $b$. It follows that the OP's sets $u$ and $v$ are disjoint if and only if $k$ is odd and not a perfect square.