# Fraenkel’s partition theorem

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

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 ${\cal B}(\alpha,{\alpha^{\prime}})$ and ${\cal B}(\beta,\beta^{\prime})$ partition $\mathbb{N}$ if and only if the following five conditions are satisfied.

1. 1.

$0<\alpha<1$.

2. 2.

$\alpha+\beta=1$.

3. 3.

$0\leq\alpha+{\alpha^{\prime}}\leq 1$.

4. 4.

If $\alpha$ is irrational, then ${\alpha^{\prime}}+\beta^{\prime}=0$ and $k\alpha+{\alpha^{\prime}}\not\in\mathbb{Z}$ for $2\leq k\in\mathbb{N}$.

5. 5.

If $\alpha$ is rational (say $q\in\mathbb{N}$$q\alpha\in\mathbb{N}$), then $\frac{1}{q}\leq\alpha+{\alpha^{\prime}}$ and $\left\lceil q{\alpha^{\prime}}\right\rceil+\left\lceil q\beta^{\prime}\right% \rceil=1.$

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=38:3214MR 38:3214

[2

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

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