Tangency properties of sets with finite geometric curvature energies

Abstract

We investigate inverse thickness $1/\Delta$ and the integral Menger curvature energies $\mathcal{U}_{p}^{\alpha}$, $\mathcal{I}_{p}^{\alpha}$ and $\mathcal{M}_{p}^{\alpha}$, to find that finite $1/\Delta$ or $\mathcal{U}_{p}^{\alpha}$ implies the existence of an approximate $\alpha$-tangent at all points of the set, when $p\geq \alpha$ and that finite $\mathcal{I}_{p}^{\alpha}$ or $\mathcal{M}_{p}^{\alpha}$ implies the existence of a weak approximate $\alpha$-tangent at every point of the set for $p\geq 2\alpha$ or $p\geq 3\alpha$, respectively, if some additional density properties hold. This includes the scale invariant case $p=2$ for $\mathcal{I}_{p}^{1}$ and $p=3$ for $\mathcal{M}_{p}^{1}$, for which, to the best of our knowledge, no regularity properties are established up to now. Furthermore we prove that for $\alpha=1$ these exponents are sharp, i.e., that if $p$ lies below the threshold value of scale innvariance, then there exists a set containing points with no (weak) approximate 1-tangent, but such that the corresponding energy is still finite. For $\mathcal{I}_{p}^{1}$ and $\mathcal{M}_{p}^{1}$ we give an example of a set which possesses a point that has no approximate 1-tangent, but finite energy for all $p\in (0,\infty)$ and thus show that the existence of weak approximate 1-tangents is the most we can expect, in other words our results are also optimal in this respect.