Journal of the
Korean Mathematical Society JKMS

The works of Erd\"{o}s et al. about expansions of $1$ with respect to a non-integer base $q$, referred to as $q$-expansions, are investigated to determine how far they continue to hold when the number $1$ is replaced by a positive number $x$. It is found that most results about $q$-expansions for real numbers greater than or equal to $1$ are in somewhat opposite direction to those for real numbers less than or equal to $1$. The situation when a real number has a unique $q$-expansion, and when it has exactly two $q$-expansions are studied. The smallest base number $q$ yielding a unique $q$-expansion is determined and a particular sequence is shown, in certain sense, to be the smallest sequence whose corresponding base number $q$ yields exactly two $q$-expansions.

Keywords: expansions of numbers, non-integer bases

MSC numbers: 11A67, 11B83