Linearly repetitive Delone sets Delone sets with finite local complexity: Linear repetitivity versus positivity of weights

We consider Delone sets with finite local complexity. We characterize validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In this context, we show that linear repetitivity is equivalent to positivity of weights combined with a certain balancedness of the shape of return patterns.