Asymptotic behavior of beta-integers

TL;DR

This paper investigates the asymptotic properties of beta-integers, especially for Parry numbers, providing rigorous theorems that clarify their behavior and relation to ordinary integers in quasicrystalline contexts.

Contribution

It establishes four theorems that precisely describe the asymptotic behavior of Parry beta-integers, advancing understanding of their structure and properties.

Findings

Proves four theorems characterizing asymptotic behavior

Shows Parry beta-integers resemble ordinary integers asymptotically

Provides mathematical foundation for quasicrystalline applications

Abstract

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic'' ordinary integers. When the number $\beta$ is a Parry number, the corresponding $\beta$-integers realize only a finite number of distances between consecutive elements and somewhat appear like ordinary integers, mainly in an asymptotic sense. In this letter we make precise this asymptotic behavior by proving four theorems concerning Parry $\beta$-integers.