Sums of dilates in groups of prime order

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or-1 and if is not too large (with respect to p), then | + t | is significantly larger than 2|| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | + t | 2.08 ||-2.