Generic regularity of free boundaries for the thin obstacle problem

The free boundary for the Signorini problem in $\mathbb{R}^{n+1}$ is smooth outside of a degenerate set, which can have the same dimension ($n-1$) as the free boundary itself. In [FR21] it was shown that generically, the set where the free boundary is not smooth is at most $(n-2)$-dimensional. Our main result establishes that, in fact, the degenerate set has zero $\mathcal{H}^{n-3-\alpha_0}$ measure for a generic solution. As a by-product, we obtain that, for $n+1 \leq 4$, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in $\mathbb{R}^3$ and $\mathbb{R}^4$ for the thin obstacle problem.