Small doubling in cyclic groups

We give a comprehensive description of the sets $A$ in finite cyclic groups such that $\mathrm{\mid }2A\mathrm{\mid }<\frac{9}{4}\mathrm{\mid }A\mathrm{\mid }$; namely, we show that any set with this property is densely contained in a (one-dimensional) coset progression. This improves earlier results of Deshouillers-Freiman and Balasubramanian-Pandey.