Nonsimple closed geodesics with given intersection number on hyperbolic surfaces

We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2{\cosh }^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k>{10}^{13350}$. Our proof is independent of it.