Optimal expansions in non-integer bases

Abstract

For a given positive integer $m$, let $A=\{0,1,...,m\}$ and $q\in (m,m+1)$. A sequence $(c_i)=c_1{c}_2...$ 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 ${\sum }_{i=1}^n{d}_i{q}^{-i}\ge {\sum }_{i=1}^n{c}_i{q}^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$.

11 pages, 0 figures, to appear in Proc. Amer. Math. Soc