SETS OF INTEGERS

Title: SETS OF INTEGERS

Research Question: How can we characterize sets of integers that have "simple" representations within a given numeration system?

Methodology: The study uses the concept of recognizable sets, which are sets of non-negative integers that have "simple" algorithms (finite automatons) for testing membership. The researchers focus on star-free sets of integers, which are sets whose representations form star-free regular languages.

Results: The researchers provide a complete logical characterization of the U-star-free sets of integers for suitable numeration systems related to a Pisot number and integer base systems. They also study the problem of base dependence and investigate the case of k-adic systems.

Implications: This research has implications for the understanding of numeration systems and the representation of integers. It provides insights into which sets of integers can be represented "simply" within a given numeration system and how this depends on the base of the system. The results can be applied to various fields, including computer science, mathematics, and engineering.

Link to Article: https://arxiv.org/abs/0111057v1 Authors: arXiv ID: 0111057v1