Density of length ratios on Teichmüller space

We study length ratios of simple closed geodesics on finite-type hyperbolic surfaces of positive genus. We allow cusps and fix the geodesic boundary lengths. Let $d$ be the dimension of the corresponding Teichmüller space ${\mathcal{T}}_{\mathbf{L}}(S)$. We prove that for every positive vector $(r_0,\dots ,r_d)$, the set of surfaces carrying $d+1$ distinct essential interior simple closed geodesics with lengths proportional to this vector is dense in ${\mathcal{T}}_{\mathbf{L}}(S)$. In particular, one-holed tori with three distinct simple closed geodesics of equal length are dense. The proof uses high powers of Dehn twists, asymptotics of length differentials, and the Brouwer fixed-point theorem.