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In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiablecurves in the plane, using a multiscale sum of what is now known as Jones$\beta$-numbers, numbers measuring flatness in a given scale and location. Thiswork was generalized to R^n by Okikiolu, to Hilbert space by the second author,and has many variants in a variety of metric settings. Notably, in 2005,Hahlomaa gave a sufficient condition for a subset of a metric space to becontained in a rectifiable curve. We prove the sharpest possible converse toHahlomaa's theorem for doubling curves, and then deduce some corollaries forsubsets of metric and Banach spaces, as well as the Heisenberg group.}

A sharp necessary condition for rectifiable curves in metric spaces