Generic regularity of free boundaries for the thin obstacle problem

The free boundary for the Signorini problem in Rn+1 is smooth outside of a degenerate set, which can have the same dimension (n - 1) as the free boundary itself. In [15] 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 Hn-3-alpha 0measure for a generic solution. As a by-pro duct, we obtain that, for n + 1