Let $\alpha ,\beta \in {\mathbb{R}}_{>0}$ such that $\alpha /\beta$ is irrational. Is the multiset$$\{\lfloor \alpha \rfloor ,\lfloor 2\alpha \rfloor ,\lfloor 4\alpha \rfloor ,\dots \}\cup \{\lfloor \beta \rfloor ,\lfloor 2\beta \rfloor ,\lfloor 4\beta \rfloor ,\dots \}$$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 $\gamma \in (1,2)$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

For the base $2$ version: this is **open in general**, even under the hypothesis (\alpha/\beta\notin\mathbb Q). In fact, your question is exactly the **Erdős–Graham problem** (often listed as Erdős Problem #354). ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.