Erdős Problem #354

declared status 'open'. Formalized: yes. Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset$$\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}$$complete? That is, can all sufficiently large natural numbers $n$ be written as$$n=\sum_{s\in S}\lfloor 2^s\alpha\rfloor+\sum_{t\in T}\lfloor 2^t\beta\rfloor$$for some finite $S,T\subset \mathbb{N}$? What if $2$ is replaced by some $\gamma\in(1,2)$? Current best: Hegyv\'{a}ri [He89] proved that the sequence is not complete if $\alpha\geq 2$ and $\beta =2^k\alpha$ for some $k\geq 0$. Prize: no. Tags: number theory.

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