Uniformly balanced words with linear complexity and prescribed letter frequencies

TL;DR

This paper presents a method to construct infinite words over a fixed alphabet that are uniformly balanced, have linear complexity, and match prescribed letter frequencies, using mixed multidimensional continued fraction algorithms.

Contribution

It introduces a novel construction approach for such words, combining combinatorics on words with advanced number theoretic algorithms.

Findings

Constructed infinite words with desired properties

Demonstrated linear complexity and uniform balance

Linked letter frequencies to multidimensional continued fractions

Abstract

We consider the following problem. Let us fix a finite alphabet A; for any given d-uple of letter frequencies, how to construct an infinite word u over the alphabet A satisfying the following conditions: u has linear complexity function, u is uniformly balanced, the letter frequencies in u are given by the given d-uple. This paper investigates a construction method for such words based on the use of mixed multidimensional continued fraction algorithms.