Sturmian sequences provide prototypical mathematical models for one dimensional quasicrystals. Regularity properties of these sequences are well understood, thanks mostly to foundational results of Morse and Hedlund, and physicists have used this understanding to study random Schrodinger operators and lattice gas models for one dimensional quasicrystals. A fact which plays an important role in these problems is the existence of a subadditive ergodic theorem, which is guaranteed when the corresponding point set is linearly repetitive. In this talk we will explain an effort to extend the one dimensional model to cut and project sets, which generalize Sturmian sequences in higher dimensions, and which are frequently used in mathematical and physical literature as models for higher dimensional quasicrystals. Most of the above concepts and terminology will be defined in the talk, only basic knowledge of analysis and dynamical systems will be supposed.