Solutions to the Thin Obstacle Problem with non-2D frequency

For all odd positive integers $m$, we construct $\mu$-homogeneous solutions to the thin obstacle problem in ${\mathbb{R}}^3,$ with $\mu \in (m,m+1)$. For $m$ large, $\mu -m$ converges to $1$, so $\mu \mathrm{\ne }m+\frac{1}{2}$. The restriction to odd values of $m$ is necessary: we show that, for all $n\ge 2$, there are no $\mu$-homogeneous solutions to the thin obstacle problem in ${\mathbb{R}}^n$ with $\mu \in {\bigcup }_{k\ge 0}(2k,2k+1)$. These examples also apply to $2$-valued $C^{1,1\mathrm{/}2}$ stationary harmonic functions or $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-eigenfunctions of the laplacian on the sphere.