On a Theorem of Deshouillers and Freiman

The study of `structure' on subsets of abelian groups, with small `doubling constant', has been well studied in the last fifty years, from the time Freiman initiated the subject. In \cite{DF} Deshouillers and Freiman establish a structure theorem for subsets of \n with small doubling constant. In the current article we provide an alternate proof of one of the main theorem of \cite{DF}. Also our proof leads to slight improvement of the theorems in \cite{DF}

Small doubling in groups with moderate torsion

We determine the structure of a finite subset $A$ of an abelian group given that $|2A|0$; namely, we show that $A$ is contained either in a "small" one-dimensional coset progression, or in a union of fewer than $\epsilon^{-1}$ cosets of a finite subgroup. The bounds $3(1-\epsilon)|A|$ and $\epsilon^{-1}$ are best possible in the sense that none of them can be relaxed without tightened another one, and the estimate obtained for the size of the coset progression containing $A$ is sharp. In the case where the underlying group is infinite cyclic, our result reduces to the well-known Freiman's $(3n-3)$-theorem; the former thus can be considered as an extension of the latter onto arbitrary abelian groups, provided that there is "not too much torsion involved"

Small doubling in cyclic groups

We give a comprehensive description of the sets $A$ in finite cyclic groups such that $|2A|<\frac94|A|$; 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