Expansions of real numbers in non-integer bases and charaterisation of Lazy expansion of 1

TL;DR

This paper explores the representation of real numbers in non-integer bases between 1 and 2, focusing on expansion algorithms, coefficient sequences, and characterizing the lazy expansion of 1, including original solutions to open problems.

Contribution

It introduces new methods for generating expansions in non-integer bases and provides an original characterization of the lazy expansion of 1.

Findings

Defined expansions as sums of fractions with binary coefficients and powers of β.

Analyzed sequences of 0s and 1s generated by these expansions.

Provided an original characterization of the lazy expansion of 1.

Abstract

In this paper, our main focus is expressing real numbers on the non-integer bases. We denote those bases as $\beta$'s, which is also a real number and $\beta \in (1,2)$. This project has 3 main parts. The study of expansions of real numbers in such bases and algorithms for generating them will contribute to the first part of the paper. In this part, firstly, we will define those expansions as the sums of fractions with $1$'s or $0$'s in the nominator and powers of $\beta$ in the denominator. Then we will focus on the sequences of $1$'s and $0$'s generated by the nominators of in the sums we mentioned above. Such sequences will be called \textit{coefficient sequences} throughout the paper. In the second half, we will study the results in the first chapter of \cite{erdos1990characterization}, namely the greedy and lazy $\beta $-expansions . The last part of the paper will be on the characterisation of lazy expansion of 1, which was the first open question at the end of \textit{Erdos and Komornik}. I still don't know if that problem has been solved already. However, the solution that was presented here is the original work of mine.