FormalConjectures.ErdosProblems.«1096»

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import FormalConjecturesUtil

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.


open Filter

open scoped Topology


namespace Erdos1096

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.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_1096 :
    answer(True) ↔ ∃ ε > 0, ∀ q, 1 < q → q < 1 + ε →
    ∀ x : ℕ → ℝ, StrictMono x → Set.range x = { ∑ i ∈ S, q ^ i | S : Finset ℕ } →
    Tendsto (fun k => x (k + 1) - x k) atTop (𝓝 0) := by⊢ True ↔
  ∃ ε > 0,
    ∀ (q : ℝ),
      1 < q →
        q < 1 + ε →
          ∀ (x : ℕ → ℝ),
            StrictMono x → Set.range x = {x | ∃ S, ∑ i ∈ S, q ^ i = x} → Tendsto (fun k => x (k + 1) - x k) atTop (𝓝 0)
  sorryAll goals completed! 🐙


-- TODO: add variants and additional material from the Erdős Problems page.

end Erdos1096