On the Stern–Brocot expansion of real numbers
Christophe Reutenauer · Journal de Théorie des Nombres de Bordeaux · 2020
The Stern–Brocot expansion of a real number is a finite or infinite sequence of symbols r , l , meaning “right” and “left”, which represents the path in the Stern–Brocot tree determined by this number. It is shown that the expansion is periodic if and only if the number is positive quadratic with a negative conjugate; in this case the conjugate opposite’s expansion is obtained by reversal. The slopes of morphic Sturmian sequences are these quadratic numbers. Two numbers have ultimately the same exapansion if and only they are SL 2 ( ℤ ) -equivalent. A related neighbouring relation for indefinite binary quadratic forms leads to a variant of the Gauss theory of cycles. A bijection is obtained between the set of binary Lyndon words and SL 2 ( ℤ ) -equivalence of these quadratic forms.