This is related to Problem $4.6$ in ``The Probabilistic Method'' by Alon and Spencer, where one essentially has to prove the following:
Let $p$ be a prime, and $A$ be any subset of $\mathbb{F}_p$. Then for a uniformly chosen $z \in \mathbb{F}_p$, we have that the dilate $z \cdot A$ intersects every interval of width $\frac{2p}{\sqrt{|A|}}$ with probability at least $0.1$. Here, `intervals' are with respect to the (cyclic) additive group.
My question: What is the correct order of $t$ such that $z\cdot A$ almost surely intersects every interval of width $t$?
For concreteness, let $A$ be any subset of $\mathbb{F}_p$ of size $p^{0.5}$. Then, for a uniformly chosen $z \in \mathbb{F}_p$, does $z \cdot A$ intersect every interval of width $p^{0.5 + \epsilon}$ with high probability?
Here, high probability means probability approaching $1$ as $p$ goes to infinity.
Of course, $p^{0.5}$ is arbitrary. A general statement would say something for intervals of width $\frac{p}{|A|} \cdot \omega(1)$, for some slow growing $\omega(1)$.
