#### Event Title

Same Property of Beta-expansion

#### Session

Information Systems and Security

#### Description

Peoples over the ages use different counting systems. Appling that to cryptography, we use to represent numbers with a small number of non-zero digits. The problem of finding representations with minimal numbers of digits has been solved for integer bases. In this paper, we consider numeration systems with respect to a real base which is a Pisot numbers. The theory of beta-expansion creat a link between symbolic dynamics and a part of number theory. On this papers we give a Pisot numers with the finiteness property (F) and with weak finiteness property (W). The set of numbers with finite greedy expansion defined by Frougny and Solomyak is exactly Z[β^{-1}]∩[0,1]. Same examples with finetnes propety is given in the end of this work with beta –expansion of numbers 1.

#### Keywords:

Counting systems, Pisot number, Beta-expansion, Cryptography

#### Session Chair

Naim Preniqi

#### Session Co-Chair

Blerton Abazi

#### Proceedings Editor

Edmond Hajrizi

#### ISBN

978-9951-437-54-7

#### Location

Durres, Albania

#### Start Date

28-10-2017 4:00 PM

#### End Date

28-10-2017 5:30 PM

#### DOI

10.33107/ubt-ic.2017.197

#### Recommended Citation

Baushi, Arben; Zaka, Orgest; and Xhoxhi, Olsi, "Same Property of Beta-expansion" (2017). *UBT International Conference*. 197.

https://knowledgecenter.ubt-uni.net/conference/2017/all-events/197

Same Property of Beta-expansion

Durres, Albania

Peoples over the ages use different counting systems. Appling that to cryptography, we use to represent numbers with a small number of non-zero digits. The problem of finding representations with minimal numbers of digits has been solved for integer bases. In this paper, we consider numeration systems with respect to a real base which is a Pisot numbers. The theory of beta-expansion creat a link between symbolic dynamics and a part of number theory. On this papers we give a Pisot numers with the finiteness property (F) and with weak finiteness property (W). The set of numbers with finite greedy expansion defined by Frougny and Solomyak is exactly Z[β^{-1}]∩[0,1]. Same examples with finetnes propety is given in the end of this work with beta –expansion of numbers 1.