Discrepancy and rectifiability of almost linearly repetitive Delone sets

We extend a discrepancy bound of Lagarias and Pleasants for local weight distributions on linearly repetitive Delone sets show that similar holds also the more general case without finite complexity if linear repetitivity is replaced by $\varepsilon$-linear repetitivity. As result we establish are some sufficiently small $\varepsilon$ rectifiable, incommensurable multiscale substitution tilings never almost repetitive.