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 famous theorem due to Lord Rayleigh (later rediscovered by Beatty) states that 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 $\gamma_1/\alpha_1+\gamma_2/\alpha_2=0$.

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.

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.