Erdős Problem 1096

References:

• erdosproblems.com/1096
• [ErKo98] Erdős, P. and Komornik, V., Developments in non-integer bases. Acta Math. Hungar. (1998), 57--83.
• [Fe16] Feng, D.-J., On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. (2016), 181--193.

theorem Erdos1096.erdos_1096 : True ↔ ∃ ε > 0, ∀ (q : ℝ), 1 < q → q < 1 + ε → ∀ (x : ℕ → ℝ), StrictMono x → Set.range x = {x : ℝ | ∃ (S : Finset ℕ), ∑ i ∈ S, q ^ i = x} → Filter.Tendsto (fun (k : ℕ) => x (k + 1) - x k) Filter.atTop (nhds 0)

Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$.

Is it true that, provided $\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \to 0$?

This was solved affirmatively by Erdős and Komornik [ErKo98], who proved the conclusion whenever $1<q<\sqrt{q_1}$, where $q_1$ is the second Pisot-Vijayaraghavan number.