Minimal non-integer alphabets allowing parallel addition

TL;DR

This paper investigates the minimal non-integer alphabets that enable parallel addition in various numeration systems, extending existing theories from integer alphabets and complex bases to more general cases.

Contribution

It establishes necessary conditions and lower bounds for alphabets allowing parallel addition, including non-integer alphabets, broadening the understanding of numeration systems.

Findings

Necessary conditions for alphabets enabling parallel addition

Lower bounds on alphabet size for generalized systems

Extension of characterization to non-integer alphabets

Abstract

Parallel addition, i.e., addition with limited carry propagation, has been so far studied for complex bases and integer alphabets. We focus on alphabets consisting of integer combinations of powers of the base. We give necessary conditions on the alphabet allowing parallel addition. Under certain assumptions, we prove the same lower bound on the size of the generalized alphabet that is known for alphabets consisting of consecutive integers. We also extend the characterization of bases allowing parallel addition to numeration systems with non-integer alphabets.