Relative positions of points on the real line and balanced parentheses

Consider a finite set of positive real numbers $S$. For any real number $\lambda > 1$, a Dyck word denoted $\langle\! \langle S \rangle\! \rangle_{\lambda} \in \{a,b\}^{\ast}$, was defined in [CaballeroWords2017] in order to compute Hooley's $\Delta$-function and its generalization. The aim of this paper is to prove that, given a real number $\lambda > 1$, any Dyck word can be expressed as $\langle\! \langle S \rangle\! \rangle_{\lambda}$ for some finite set $S$ of positive real numbers.