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 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 improve a result of McMullen by showing that, for any dimension , there exists a threshold of moduli of continuity , including the class of the H\"{o}lder ones, such that for every , any two Delone sets in cannot be distinguished under bi--equivalence. Also, we extend a result due to Aliste, Coronel, and Gambaudo, which establishes that every linearly repetitive Delone set in is rectifiable by extending it to a broader class of repetitive behaviors. Moreover, we show that for the modulus of continuity , every -repetitive Delone set in is equivalent to the standard lattice by a bi--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.