Consider the constant-coefficient ordinary differential equation $\dot x(t) = Ax(t)$ where $x(t)\in\mathbb R^n$. Consider the function $y(t)=Bx(t)$. We know that $y(t)$ is a real analytic function. For a finite interval $[0,t_f]$, the analytic property tells us that $y(t)=0$ either for a finite set of intervals or the set at which $y(t)=0$ is countable (see Proposition 4.1 on p. 41 of Cartan, 1995).
Now, consider the "magnitude of Eucledian projection on a convex set" function:
$$ P(y) = \|\arg\min_{z\in\mathcal S}\|y-z\|_2\|_2, $$
where $\mathcal S$ is a convex set. Can we say anything about the zero set of $P(y)$? Can it be guaranteed that the set cannot be nowhere dense but of positive measure (e.g. fat Cantor set)?