# The abelian complexity of infinite words and the Frobenius problem

## TL;DR

This paper investigates conditions under which the abelian complexity of infinite words ensures that a semigroup homomorphism applied to their factors covers all but finitely many natural numbers, linking combinatorics and number theory.

## Contribution

It introduces new conditions connecting abelian complexity of infinite words with the Frobenius problem, expanding understanding of factor sets in combinatorics on words.

## Key findings

- Identifies specific conditions on S and abelian complexity for coverage of N
- Analyzes various infinite words with different abelian complexity functions
- Establishes links between combinatorics on words and number theory

## Abstract

We study the following problem, first introduced by Dekking. Consider an infinite word x over an alphabet {0,1,...,k-1} and a semigroup homomorphism S:{0,1,...,k-1}* -> N. Let L_x denote the set of factors of x. What conditions on S and the abelian complexity of x guarantee that S(L_x) contains all but finitely many elements of N? We examine this question for some specific infinite words x having different abelian complexity functions.