Optimal expansions in non-integer bases

TL;DR

This paper investigates the existence of specific expansions in non-integer bases that satisfy certain inequalities relative to all other expansions of the same number, revealing new insights into the structure of these expansions.

Contribution

It introduces conditions for the existence of expansions that dominate all other expansions in a specified manner in non-integer bases.

Findings

Almost all numbers in the interval have uncountably many expansions.

The paper establishes criteria for the existence of expansions satisfying inequality constraints.

Results contribute to understanding the structure and ordering of expansions in non-integer bases.

Abstract

For a given positive integer $m$, let $A=\set{0,1,...,m}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 ...$ consisting of elements in $A$ is called an expansion of $x$ if $\sum_{i=1}^{\infty} c_i q^{-i}=x$. It is known that almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many expansions. In this paper we study the existence of expansions $(d_i)$ of $x$ satisfying the inequalities $\sum_{i=1}^n d_iq^{-i} \geq \sum_{i=1}^n c_i q^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$.