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 Rivi\`ere 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-H\"older-homogeneous. From this criterion, we extend a result of McMullen by showing that, for any dimension $d\geq 1$, there exists a threshold of moduli of continuity $\mathcal{M}_d$, including the class of the H\"{o}lder ones, such that for every $\omega\in\mathcal{M}_d$, any two Delone sets within a certain class in $\mathbb{R}^d$ cannot be distinguished under bi-$\omega$-equivalence. Also, we 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 $\mathbb{R}^d$ is equivalent to the standard lattice by a bi-$\omega$-homogeneous map. Finally, we address a problem of continuous nature related to the previous ones about finding solutions to the prescribed volume form equation in intermediate regularity, thereby extending the results of Rivi\`ere and Ye.