Abstract: We study the number of expansions of a real number x in the form $x=\sum_{i=1}^{\infty}c_i q^{-i}$ where the base $q > 1$ is a fixed real number and $c_i \in \{ 0 , 1\}$, and in the form $x=\sum_{i=1}^{\infty}c_i/F_i$ where (F_i) is the Fibonacci sequence $1, 1, 2, 3, 5, \ldots$. Contrary to the base 2, for $1 < q < 2$ the number of expansions of a given $x$ may be infinite and even uncountable. We report on some joint works with Claudio Baiocchi, Paola Loreti and Marco Pedicini that clarify the situation.
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