Erdős Problem 124

References:

• erdosproblems.com/124
• [BEGL96] Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.

theorem Erdos124.erdos_124 : (∀ (k : ℕ) (d : Fin k → ℕ), (∀ (i : Fin k), 3 ≤ d i) → StrictMono d → ∑ i : Fin k, 1 / (↑(d i) - 1) = 1 → ∀ᶠ (n : ℕ) in Filter.atTop, ∃ (c : Fin k → ℕ) (a : Fin k → ℕ), ∀ (i : Fin k), c i ∈ {0, 1} ∧ ∀ (i : Fin k), ((d i).digits (a i)).toFinset ⊆ {0, 1} ∧ n = ∑ i : Fin k, c i * a i) ↔ sorry

Let $3\leq d_1 < d_2 < \cdots < d_k$ be integers such that $$\sum_{1\leq i\leq k}\frac{1}{d_i-1}\geq 1.$$ Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0, 1\}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$?

Conjectured by Burr, Erdős, Graham, and Li [BEGL96]