Generalizations of the Rayleigh(-Beatty) theorem

Question

For a given irrational number $\alpha>0$ and a real number $\beta$, the inhomogeneous 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.

Answer 1

In 1973, Fraenkel 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 (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 by Tijdeman (2001), who proved the conjecture for $k = 5, 6$ (and for $k = 3$ in an earlier paper). Altman, Gaujal, Hordijk (1997) proved it for $k = 4$, and more recently, Barát and Varjú (2003) verified the conjecture for $k=7$. It's a tantalising open problem, which I have dabbled with recently too (albeit from a different point of view).