Self ane tiling and Pisot numeration system Shigeki Akiyama Summary: Self aĆne tiling and Pisot numeration system  Shigeki Akiyama 1 Introduction Let > 1 be a real number. Consider an expansion of positive real number x: x = aN N + aN 1 N 1 + aN 2 N 2 +    ; with a i 2 Z \ [0; ). A greedy expansion of x in base is such expansion with jx N X M a i i j < M (1) for any M . By using greedy algorithm, such an expansion always exists for any x. This is a natural extension of decimal or binary expansion to a real base. An expansion of x is admissible, if (1) holds for all M . Hereafter, we use a notation x = aN aN 1 aN 2    : A Pisot number is an algebraic integer greater than 1 whose conjugates other than itself have modulus smaller than 1. We have a particular interest in the case when > 1 is a Pisot number. Surprisingly, one can nd many similar phenomena with binary or decimal expansion. See [1]. We use a term 'Pisot numeration system' to call this method to represent real numbers in a power series in Pisot number base. Collections: Mathematics