Relative positions of points on the real line and balanced parentheses

Abstract

Consider a finite set of positive real numbers $S$. For any real number $\lambda >1$, a Dyck word denoted $⟨⟨S⟩{⟩}_{\lambda }\in \{a,b{\}}^{\ast }$, 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 $⟨⟨S⟩{⟩}_{\lambda }$ for some finite set $S$ of positive real numbers.

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