Fraenkel's partition theorem is a generalization of Beatty's Theorem. Set $$ {\cal B}(\alpha,\alpha^\prime) := \left( \floor{\frac{n-\alpha^\prime}{\alpha}} \right)_{n=1}^\infty. $$ We say that two sequences partition $\mathbb{N}=\{1,2,3,\ldots\}$ if the sequences are disjoint and their union is $\mathbb{N}$ .

Fraenkel's Partition Theorem:The sequences ${\cal B}(\alpha,\alpr)$ and ${\cal B}(\beta,\bepr)$ partition $\mathbb{N}$ if and only if the following five conditions are satisfied.

1. $0<\alpha<1$ .
2. $\alpha+\beta=1$ .
3. $0\le\alpha+\alpr \le 1$ .
4. If $\alpha$ is irrational, then $\alpr+\bepr=0$ and $k\alpha+\alpr\not\in\mathbb{Z}$ for $2\le k\in \mathbb{N}$ .
5. If $\alpha$ is rational (say $q\in \mathbb{N}$ is minimal with $q\alpha \in \mathbb{N}$ ), then $\frac1q \le \alpha+\alpr$ and $\ceiling{q\alpr}+\ceiling{q\bepr}=1.$

References

[1]

Aviezri S. Fraenkel, The bracket function and complementary sets of integers, Canad. J. Math. 21 (1969), 6-27. MR 38:3214

[2]

Kevin O'Bryant, Fraenkel's partition and Brown's decomposition, arXiv:math.NT/0305133.