Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group (Q6107814)

In this paper, the classical continued fraction expansion is extended, in a non-obvious way, to the ring \(\mathbb{R}[X]/(X^2)\) of dual numbers, where it doesn't give a representation of elements, but rather a map \(a+bX\mapsto a+b_SX\), where \(b_S\) is called the \textit{shadow} of \(b\). By the ambiguity of continued fraction representations, rational numbers have two different shadows, whereas the shadow of an irrational is uniqely determined. Classical results, like the representation of continued fractions via Euler's Continuants and the Farey tree, are transferred to the current setting. As the authors say themselves, this paper is only the beginning of an investigation and many new questions arise, like for continuity of the shadow map on the irrationals.