For any $k$ there exists some $t_k\in (0,1)$ such that the set of $\alpha$
such that the sequence $\lfloor t_k\alpha^n\rfloor$ is complete consists of at least $k$
disjoint line segments.
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theorem exists_t_for_k_disjoint_segments (k : ℕ) :
∃ t ∈ Ioo 0 1, ∃ (ι : Type), k ≤ (Set.univ : Set ι).encard ∧ ∃ I : ι → Set ℝ,
(∀ i, 2 ≤ (I i).encard ∧ (I i).Nonempty ∧ IsConnected (I i)) ∧
Pairwise (Disjoint on I) ∧ (⋃ i, I i) ⊆ {α | α > 0 ∧ IsGoodPair t α}
