Optimal expansions in non-integer bases

Abstract:

For a given positive integer $m$, let $A=\{0,1,\ldots ,m\}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 \ldots$ 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 $\sum _{i=1}^n d_iq^{-i} \ge \sum _{i=1}^n c_i q^{-i}$ , $n=1,2,\ldots ,$ for each expansion $(c_i)$ of $x$.