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 $\A \subset \Zp$ is not too large (with respect to $p$), then $|\A+t\cdot \A|>(2+ \vartheta_t)|\A|-w(t)$ for some constant $w(t)$ depending only on $t$ and for some explicit real number $\vartheta_t >0$ (except in the case $|t|=3$). In the important case $|t|=2$, we may for instance take $\vartheta_2=0.08$.