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PISOT NUMBERS AND GREEDY ALGORITHM Shigeki Akiyama
 

Summary: PISOT NUMBERS AND GREEDY ALGORITHM
Shigeki Akiyama
Niigata University
Abstract. We study the greedy expansion of real numbers in Pisot number base.
We will show a certain criterions of niteness, periodicity, and purely periodicity.
Further, it is proved that every suĂciently small positive rational numbers has purely
periodic greedy expansion in Pisot unit base under a certain niteness condition.
1. Introduction
Let be the xed real number greater than 1 and x be a positive real number.
Then the expansion of the form x =
P 1
N0i a i i is said to be a greedy expansion
if
(G) jx
X
N0iN
a i i j < N ;
holds for every N and a i is a non negative integer with 0  a i < . It is a natural
generalization of binary or decimal expansion to the expansion in real bases. In
this note, we say ` -expansion' by the algorithm:

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics