theorem FurstenbergSet.orponen_shmerkin_2021 (s t : ℝ) (hs : 0 < s) (hst : s < t) : ∃ (ε : ℝ), 0 < ε ∧ ∀ (C : ℝ), 1 ≤ C → ∃ (c : ℝ), 0 < c ∧ ∀ (cfg : FurstenbergSets.FurstenbergConfig), cfg.s = s → cfg.t = t → cfg.C = C → c * cfg.δ ^ (-(2 * s + ε)) ≤ ↑cfg.totalTubes

Orponen–Shmerkin (2021) sharp lower bound towards the Furstenberg conjecture: for parameters $0 < s < t$ there exists $\varepsilon > 0$ such that for any constant $C \ge 1$ there is $c > 0$ so that for every Furstenberg configuration with parameters $(s, t, C)$, the total tube count satisfies $|\mathbb{T}| \ge c\, \delta^{-(2s + \varepsilon)}$.