Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

===== Take α=0.15\alpha=0.15α=0.15. Then =====

ρ0=0.853.15≈0.26984,ρ0≈0.519.\rho_0 = \frac{0.85}{3.15}\approx 0.26984,
\qquad
\sqrt{\rho_0}\approx 0.519.ρ0=3.150.85≈0.26984,ρ0≈0.519.
Now choose ρ=ρ0+10−6\rho=\rho_0+10^{-6}ρ=ρ0+10−6. Then

γ≈0.15⋅3.151.15⋅1.414⋅10−6≈2.9×10−7≪0.519≈ρ.\gamma
\approx \frac{0.15\cdot 3.15}{1.15\cdot 1.414}\cdot 10^{-6}
\approx 2.9\times 10^{-7}
\ll 0.519 \approx \sqrt{\rho}.γ≈1.15⋅1.4140.15⋅3.15⋅10−6≈2.9×10−7≪0.519≈ρ.
So this path graph gives you a very explicit, closed-form “bad” case where

γ<ρ\gamma < \sqrt{\rho}γ<ρ
by an arbitrarily large margin, while the true support is tiny (∣S⋆∣=1≤1/ρ|S^\star|=1\le 1/\rho∣S⋆∣=1≤1/ρ).