On additive complexity of infinite words

TL;DR

This paper investigates the structure of infinite words with bounded additive complexity, exploring how such complexity influences the word's slope, the effects of morphisms, and the existence of recurrent words with specific additive properties.

Contribution

It introduces the concept of bounded additive complexity in infinite words, analyzes its implications, and proves the existence of recurrent words with any given odd additive complexity.

Findings

Bounded additive complexity constrains the slope of infinite words.

Non-erasing morphisms can alter the additive complexity of words.

Recurrent words with any odd positive additive complexity exist.

Abstract

We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded by a constant that depends only on the word. We describe how bounded additive complexity impacts the slope of the word and how a non-erasing morphism may affect the boundedness of a given word's additive complexity. We prove the existence of recurrent words with constant additive complexity equal to any given odd positive integer. In the last section, we discuss a generalization of additive complexity. Our results suggest that words with bounded additive complexity may be viewed as a generalization of balanced words.