arXiv2012
We investigate inverse thickness $1/Δ$ and the integral Menger curvature energies $\mathcal{U}_{p}^α$, $\mathcal{I}_{p}^α$ and $\mathcal{M}_{p}^α$, to find that finite $1/Δ$ or $\mathcal{U}_{p}^α$ implies the existence of an approximate $α$-tangent at all points of the set, when $p\geq α$ and that finite $\mathcal{I}_{p}^α$ or $\mathcal{M}_{p}^α$ implies the existence of a weak approximate $α$-tangent at every point of the set for $p\geq 2α$ or $p\geq 3α$, 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 $α=1$ 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.