Comment écrire les nombres relatifs dans une base qui n'est pas entière

To each number \(\beta > 1\) we associate a numeration system which allows us to
write any integer \(m\) of \(\mathbb{Z}\) in base \(-\beta\):
\[
m = a_k K_k + \cdots + a_1 K_1 + a_0 K_0
\]
(where \((K_n)_{n \ge 0}\) is a specific sequence which satisfies
\(\lim\limits_{n \to \infty} K_{n+1}/K_n = -\beta\)); the words
\(a_k \ldots a_1 a_0\) that we obtain are the words without leading \(0\) of the
language of a dynamical system, the \(-\beta\) shift. With an adapted order,
the correspondence between the words of the language and \(\mathbb{Z}\)
becomes an increasing bijection.

We also define the corresponding \(-\beta\) substitution.