Fraenkel's conjecture answers your question in the case when the $\alpha_i$ are distinct. This conjecture, which has remained unsolved (in general) for over 35 years
In 1973, states Fraenkel showed thatevery partition of the positive integers into , 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 $\alpha_i$ and \alpha_i$,$\beta_i$real ,$\alpha_i > 0$, and$\alpha_1 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$, and more recently, Barát and Varjú (2003) proved it verified the conjecture for$k=7$. It's a tantalising open problem, which I have dabbled with recently too (albeit from a different point of view). 1 Fraenkel's conjecture answers your question in the case when the$\alpha_i$are distinct. This conjecture, which has remained unsolved (in general) for over 35 years, states that every partition of the positive integers into$k \geq 3$Beatty sequences$S_{\alpha_i,\beta_i} := \lbrace{\lfloor n\alpha_i + \beta_i\rfloor\rbrace}_{n\geq 1}$, with$\alpha_i$and$\beta_i$real,$\alpha_i > 0$, and$\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$, and more recently, Barát and Varjú (2003) proved it for$k=7\$. It's a tantalising problem, which I have dabbled with recently too (albeit from a different point of view).