Relative positions of points on the real line and balanced parentheses

TL;DR

This paper proves that for any given real number greater than one, every Dyck word can be represented using a finite set of positive real numbers through a specific construction related to Hooley's Δ-function.

Contribution

It establishes a universal representation of Dyck words as $ extless extless S extgreater extgreater_{\lambda}$ for some finite set S, extending previous definitions.

Findings

Any Dyck word can be expressed as $ extless extless S extgreater extgreater_{\lambda}$ for some finite S.

The representation holds for all real numbers $\lambda > 1$.

The result connects combinatorial structures with number-theoretic functions.

Abstract

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.