As we know, Gauss wrote that \begin{equation} \lim_{n \rightarrow \infty} \lambda \left(\tau^n \leq x\right) = \frac{\log(1+x)}{\log2}, \quad 0 \leq x < 1, \end{equation} with $\lambda$ is Lebesgue measure and the map $\tau : [0, 1) \rightarrow [0, 1)$, the so-called regular continued fraction or Gauss transformation, is defined by \begin{equation} \tau (x) = \frac{1}{x}-\left\lfloor \displaystyle \frac{1}{x} \right\rfloor, x \neq 0 \end{equation} and $\tau (x) = 0$, where $\left\lfloor \cdot \right\rfloor$ denotes the floor (or entire) function.

My question is "Why this theorem is important?" I am interested of a non-trivial and interesting explanation.

Thank you!