theorem BeckLemma.beck_bootstrap_epsilon_improvement (u s : ℝ) (hu : 0 < u) (hs : 0 < s) (hs_min : s < min u 1) : ∃ (ε : ℝ) (η : ℝ), ε > 0 ∧ 0 < η ∧ η < 1 ∧ ∀ (Δ : ℝ) (m : ℕ) (C : NNReal) (E : Set (EuclideanSpace ℝ (Fin 2))) (Lx : EuclideanSpace ℝ (Fin 2) → Set (EuclideanSpace ℝ (Fin 1))), 0 < Δ → Δ < 1 → ∀ G ⊆ E, ENNReal.ofReal (1 - η) * ↑(BourgainUniform.meshCount (Δ ^ m) E) ≤ ↑(BourgainUniform.meshCount (Δ ^ m) G) → (∀ x ∈ G, BourgainUniform.IsUniform Δ m (Lx x)) → (∀ x ∈ G, BourgainUniform.IsRegularSet (Δ ^ m) s C (Lx x)) → ∃ (C' : NNReal), ∃ G' ⊆ E, ENNReal.ofReal (1 - η) * ↑(BourgainUniform.meshCount (Δ ^ m) E) ≤ ↑(BourgainUniform.meshCount (Δ ^ m) G') ∧ ∀ x ∈ G', BourgainUniform.IsRegularSet (Δ ^ m) (s + ε) C' (Lx x)

ε-bootstrap step for Continuum Beck's Theorem. If 0 < s < min(u, 1) then there exist ε > 0 and η ∈ (0, 1) such that the following holds: for any scale Δ ∈ (0, 1), any m, any planar set E with a uniform "good" subset G ⊆ E of proportional Δ^m-mesh count at least 1 − η, where each Lx (the line-set through x ∈ G) is (Δ^m, s, C)-regular, one can produce a possibly smaller good set G' ⊆ E of the same proportional size on which the line-set is (Δ^m, s + ε, C')- regular. This is the bootstrap mechanism behind the lemma "if a typical L_{x,E} is (δ, s, C) then a typical L_{x,E} is (δ, s + ε, C')".