For a given irrational number $\alpha>0$ and a real number $\beta$,
the inhomogeneous [Beatty sequence](http://en.wikipedia.org/wiki/Beatty_sequence)
sequence $S\_{\alpha,\beta}$ is the set $\lbrace\lfloor n\alpha+\beta\rfloor:n=1,2,\dots\rbrace$
(the case $\beta=0$ corresponds to a homogeneous Beatty sequence).


If $\beta=0$, the two homogeneous Beatty sequences
$S\_{\alpha\_1,0}$ and $S\_{\alpha\_2,0}$ partition the set of positive integers
iff $1/\alpha\_1+1/\alpha\_2=1$. There is also a similar result for inhomogeneous
$S\_{\alpha\_1,\beta\_1}$ and $S\_{\alpha\_2,\beta\_2}$: assuming that neither
$n\alpha\_1+\beta\_1$ nor $n\alpha\_2+\beta\_2$ is an integer for $n=1,2,\dots$,
the sequences partition $\mathbb Z\_{>0}$ iff $1/\alpha\_1+1/\alpha\_2=1$ and
$\beta\_1/\alpha\_1+\beta\_2/\alpha\_2=0$.


**Question.**
For a given $k\ge3$, what are the conditions on $\alpha\_1,\dots,\alpha\_k$
(and on $\beta\_1,\dots,\beta\_k$ in the inhomogeneous case) to ensure that
the sets $S\_{\alpha\_i,\beta\_i}$, $i=1,\dots,k$, partition the positive integers.


It looks like the book 
*Old and new problems and results in combinatorial number theory*
by P. Erdős and R.L. Graham (which I do not have) mentions a version
of the problem, but I am interested in some (possibly very recent) progress
in the direction. My interest is motivated by the study of functional
equations of the Mahler-type generating functions of the Beatty sequences.



Generalizations of the Rayleigh(-Beatty) theorem
In 1973, [Fraenkel](http://dx.doi.org/10.1016/0097-3165%2873%2990059-9) showed that, for fixed $k \geq 3$, if $\alpha\_i = (2^k - 1)/2^{i-1}$ and $\beta\_i = -2^{k-i} + 1$ for $i = 1, 2, \ldots k$, then the $k$ Beatty sequences $S\_{\alpha\_i,\beta\_i} := \lbrace{\lfloor n\alpha\_i + \beta\_i\rfloor\rbrace}\_{n\geq 1}$ partition the positive integers. Many other cases have been proved by [Simpson](http://dx.doi.org/10.1016/0012-365X%2891%2990293-B) (1991).


Fraenkel also conjectured that any partition of the positive integers into $k \geq 3$ Beatty sequences $S\_{\alpha\_i,\beta\_i}$, with $\alpha\_i$, $\beta\_i$ real and $0 < \alpha\_1 < \alpha\_2 < \cdots < \alpha\_k$, satisfies $\alpha\_i = (2^k - 1)/2^{i-1}$ for $i = 1, 2, \ldots, k$.


To date, Fraenkel's conjecture has been proved for up to $k=7$ sequences. I would recommend taking a look at [this paper](http://www.math.leidenuniv.nl/~tijdeman/tij3.ps) by Tijdeman (2001), who proved the conjecture for $k = 5, 6$ (and for $k = 3$ in an earlier paper). [Altman, Gaujal, Hordijk](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.3158) (1997) proved it for $k = 4$, and more recently, [Barát and Varjú](http://www.sciencedirect.com/science/article/pii/S0019357703900000) (2003) verified the conjecture for $k=7$. It's a tantalising open problem, which I have [dabbled with recently](http://dx.doi.org/10.1016/j.ejc.2008.04.006) too (albeit from a different point of view).


