Noncanonical number systems in the integers

TL;DR

This paper explores noncanonical number systems in integers, demonstrating the existence of digit sets without zero that allow finite expansions for all integers in various bases, including a complete characterization for base -2.

Contribution

It proves the existence of zero-free digit sets for finite expansions in bases |b|≥4 and b=-2, with a full characterization for base -2.

Findings

Existence of digit sets without zero for bases |b|≥4 and b=-2

Infinitely many such digit sets for each base

Complete characterization of valid digit sets for base -2

Abstract

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b|\ge 4, and for b=-2. For the special case b=-2, we give a full characterisation of all valid digit sets.