Sums of dilates over groups of prime order

For $p$ prime, $A\subseteq \mathbb{Z}\mathrm{/}p\mathbb{Z}$ and $\lambda \in \mathbb{Z}$, the sum of dilates $A+\lambda \cdot A$ is defined by $$A+\lambda \cdot A=\{a+\lambda a^{\mathrm{\prime }}:a,a^{\mathrm{\prime }}\in A\}\mathrm{.}$$ The basic problem on such sums of dilates asks for the minimum size of $\mathrm{\mid }A+\lambda \cdot A\mathrm{\mid }$ for given $\lambda$, $A$ of given density $\alpha$, and $p$ tending to infinity. We investigate this problem for $\alpha$ fixed and $\lambda$ tending to infinity, proving near-optimal bounds in this case.