A Set of Sequences of Complexity $2n+1$

TL;DR

This paper constructs ternary sequences with complexity $2n+1$ for any rationally independent letter frequencies using a specific substitution process linked to a Multidimensional Continued Fraction algorithm, related to the Selmer algorithm.

Contribution

It demonstrates the existence of such sequences for any frequency vector and connects the algorithm to the Selmer algorithm, providing a new method for sequence construction.

Findings

Sequences have complexity $2n+1$ for any frequency vector

The algorithm is conjugate to the Selmer algorithm

Experimental evidence suggests finite balance properties

Abstract

We prove the existence of a ternary sequence of factor complexity $2n+1$ for any given vector of rationally independent letter frequencies. Such sequences are constructed from an infinite product of two substitutions according to a particular Multidimensional Continued Fraction algorithm. We show that this algorithm is conjugate to a well-known one, the Selmer algorithm. Experimentations (Baldwin, 1992) suggest that their second Lyapunov exponent is negative which presages finite balance properties.