Expansions in non-integer bases: lower, middle and top orders

Nikita Sidorov

doi:10.1016/j.jnt.2008.11.003

paper

Expansions in non-integer bases: lower, middle and top orders

arXiv:math/0608263 · doi:10.1016/j.jnt.2008.11.003

Abstract

Let $q \in (1, 2)$ ; it is known that each $x \in [0, 1/ (q - 1)]$ has an expansion of the form $x = \sum_{n = 1}^{\infty} a_{n} q^{- n}$ with $a_{n} \in {0, 1}$ . It was shown in \cite{EJK} that if $q < (5 + 1) /2$ , then each $x \in (0, 1/ (q - 1))$ has a continuum of such expansions; however, if $q > (5 + 1) /2$ , then there exist infinitely many $x$ having a unique expansion \cite{GS}. In the present paper we begin the study of parameters $q$ for which there exists $x$ having a fixed finite number $m > 1$ of expansions in base $q$ . In particular, we show that if $q < q_{2} = 1.71...$ , then each $x$ has either 1 or infinitely many expansions, i.e., there are no such $q$ in $((5 + 1) /2, q_{2})$ . On the other hand, for each $m > 1$ there exists $\ga_m>0$ such that for any $q\in(2-\ga_m,2)$, there exists $x$ which has exactly $m$ expansions in base $q$ .

15 pages; to appear in J. Number Theory