We explore a specific system in the mechanics of thin elastic​ sheets in which geometry and loading conspire to generate fine-scale wrinkling. This system -- a twisted ribbon held with small tension -- was examined experimentally by Chopin and Kudrolli [Phys Rev Lett 111, 174302, 2013]. There is a regime where the ribbon wrinkles near its center. A recent paper by Chopin, Démery, and Davidovitch models this regime using a von Kármán-like variational framework [J Elasticity 119, 137-189, 2015]. Our contribution is to give upper and lower bounds for the minimum energy as the thickness tends to zero. Since the bounds differ by a thickness-independent prefactor, we have determined how the minimum energy scales with thickness. Along the way we find estimates on Sobolev norms of the minimizers, which provide some information on the character of the wrinkling.