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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 finiteness, periodicity, and purely periodicity.
Further, it is proved that every sufficiently small positive rational numbers has purely
periodic greedy expansion in Pisot unit base under a certain finiteness condition.
1. Introduction
Let be the fixed real number greater than 1 and x be a positive real number.
Then the expansion of the form x =

N0i ai-i
is said to be a greedy expansion
if
(G) |x -
N0iN
ai-i
| < -N
,
holds for every N and ai is a non negative integer with 0 ai < . It is a natural

  

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

 

Collections: Mathematics