A Note On Beta Expansion And Numbers Badly Aprroximable

Abstract

Since antiquity, mathematics has been fundamental to advances in science, engineering, and philosophy. Mathematics and humanity itself are nearly at the same age. Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system. It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BCE. We now use the Indu-Arab systems in educations books, integer bases for computer science , binary systems, or others [18]. Among the different possible representations of real numbers in a real (or complex) base 𝜷, the 𝜷 -expansion introduced by Renyi´ is very important for advance in different maths fields. The present paper focuses on 𝜷 -expansion of an algebraic number in an algebraic base 𝜷, with a point of view from Diophantine approximation. We are going to study the point which are called the sets of badly approximable number. They have an important role in classical diophantine approximation [1]. For the set 𝑭(𝒄,𝜷)={𝒙∈𝕊 |𝜷𝒏𝒙≥𝒄 (𝒎𝒐𝒅 𝟏) 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒏≥𝟎 } From now and all this papers beta numbers are Pisot or Salem numbers . The algebraic numbers 𝜷>𝟏 and their conjugate are less or iqual in modul to 1. The expansions of 1 in base 𝜷 turns out to be crucial for characterizing the 𝜷− shift. The closure of the 𝜷− shift is totally determined by the expansions of 1. Some results are to be given on this papers.