For any $\varepsilon > 0$, there exists an infinite sequence $2 \le d_0 < d_1 < \dots$ such
that all sufficiently large integer can be written as $\sum_{i \in I} a_i$ where $a_i$ has only
the digits $0, 1$ when written in base $d_i$,
but $\sum_{i \in I} \frac 1{d_i - 1} \le \varepsilon$.
Proved by Melfi [Me04]
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lemma erdos124.melfi_construction {ε : ℝ} (hε : 0 < ε) :
∃ d : ℕ → ℕ, StrictMono d ∧ ∑' i, (d i - 1 : ℝ)⁻¹ ≤ ε ∧ ∀ᶠ n in atTop,
∃ (I : Finset ℕ) (a : ℕ → ℕ), (∀ i ∈ I, a i ∈ sumsOfDistinctPowers (d i) 0) ∧
∑ i ∈ I, a i = n
