Fraenkel’s partition theorem

Fraenkel’s partition theorem is a generalization of Beatty’s Theorem. Set

$$ℬ(\alpha ,{\alpha }^{\prime }):={\left(\lfloor \frac{n-{\alpha }^{\prime }}{\alpha }\rfloor \right)}_{n=1}^{\mathrm{\infty }}.$$

We say that two sequences partition $\mathbb{N}=\{1,2,3,\mathrm{\dots }\}$ if the sequences are disjoint and their union is $\mathbb{N}$.

Fraenkel’s Partition Theorem:The sequences $\mathrm{B}\mathit{}\mathrm{(}\alpha \mathrm{,}{\alpha }^{\mathrm{\prime }}\mathrm{)}$ and $\mathrm{B}\mathit{}\mathrm{(}\beta \mathrm{,}{\beta }^{\mathrm{\prime }}\mathrm{)}$ partition $\mathrm{N}$ if and only if the following five conditions aresatisfied.

1. .

2. $\alpha +\beta =1$.

3. $0\le \alpha +{\alpha }^{\prime }\le 1$.

4. If $\alpha$ is irrational, then ${\alpha }^{\prime }+{\beta }^{\prime }=0$ and $k\alpha +{\alpha }^{\prime }\notin \mathbb{Z}$ 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$\frac{1}{q}\le \alpha +{\alpha }^{\prime }$ and $\lceil q{\alpha }^{\prime }\rceil +\lceil q{\beta }^{\prime }\rceil =1.$

References

[1

] Aviezri S. Fraenkel, The bracket function and complementary sets of integers, Canad. J.Math. 21 (1969), 6–27. http://www.ams.org/mathscinet-getitem?mr=38:3214MR38:3214

[2

] Kevin O’Bryant, Fraenkel’s partition and Brown’s decomposition,http://lanl.arxiv.org/abs/math.NT/0305133arXiv:math.NT/0305133.