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