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 $\mathrm{\Delta }$-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 $⟨\text{\hspace{0.17em}⁣}⟨S⟩\text{\hspace{0.17em}⁣}{⟩}_{\lambda }$ for some finite set $S$ of positive real numbers.