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We extend a discrepancy bound of Lagarias and Pleasants for local weightdistributions on linearly repetitive Delone sets and show that a similar boundholds also for the more general case of Delone sets without finite localcomplexity if linear repetitivity is replaced by $\varepsilon$-linearrepetitivity. As a result we establish that Delone sets that are$\varepsilon$-linear repetitive for some sufficiently small $\varepsilon$ arerectifiable, and that incommensurable multiscale substitution tilings are neveralmost linearly repetitive.

Discrepancy and rectifiability of almost linearly repetitive Delone sets