On rectifiability of Delone sets in intermediate regularity

Abstract

In this work, we deal with Delone sets and their rectifiability under different classes of regularity. By pursuing techniques developed by Rivière and Ye, and Aliste-Prieto, Coronel and Gambaudo, we give sufficient conditions for a specific Delone set to be equivalent to the standard lattice by bijections having regularity in between bi-Lipschitz and bi-Hölder-homogeneous. From this criterion, we extend a result of McMullen by showing that, for any dimension $d\ge 1$, there exists a threshold of moduli of continuity ${\mathcal{M}}_d$, including the class of the Hölder ones, such that for every $\omega \in {\mathcal{M}}_d$, any two Delone sets in ${\mathbb{R}}^d$ within a certain class cannot be distinguished under bi-$\omega$-equivalence. This class accounts for the decreasing rate of the density deviation of a Delone set, with respect to a limit density. We also extend a result due to Aliste, Coronel, and Gambaudo, which establishes that every linearly repetitive Delone set in ${\mathbb{R}}^d$ is rectifiable, by extending it to a broader class of repetitive behaviors. Moreover, we show that for the modulus of continuity $\omega (t)=t(\log (1/t){)}^{1/d}$, every $\omega$-repetitive Delone set in this class is equivalent to the standard lattice by a bi-$\omega$-homogeneous map. Finally, we address a continuous problem related to the previous ones about finding solutions to the prescribed volume form equation in intermediate regularity, thereby extending the results of Rivière and Ye. Some interesting research directions are highlighted.

22 pages --> 26 pages. Comments are welcome. Comments V4: Proofs of Theorems B and C were fixed. We added Lemma 2.4 to estimate m_j