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. 
Adnene Besbes, Michael Boshernitzan, Daniel Lenz
math.CO