Fix a vector $x\in\{0,1\}^n$, and let $a$ be a random vector in $\mathbb{Z}^n_q$ for some prime $q$.
Consider $y=ax$, and $S=\{x'\mid ax'=y\}$. I want to compute the probability that $\lvert S \rvert\geq k$ for some $k$ (when the probability is over the possible values of $a$) and I'm not sure how.
If $a=0$ for example, we would have $2^{n}$ solutions, while if $a=(1,2,...,2^n)$ $x$ must be unique. But I'm not sure how to characterize in general the values of $a$ for which there are many solutions (or a few solutions).