• 1 Institut de Mathématiques de Bordeaux

Let $c>\sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$