On rectifiability of Delone sets in intermediate regularity

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 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 $\omega (t)=t(\log (1\mathrm{/}t){)}^{1\mathrm{/}d}$, 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 Rivière and Ye.