Regularity conjecture for k-abelian complexity of automatic sequences

Determine whether, for every integer ℓ ≥ 2 and every ℓ-automatic sequence x over a finite alphabet, and for every integer k ≥ 1, the k-abelian complexity function n ↦ #1{k}{x}(n) is ℓ-regular.

Background

The paper studies generalized abelian complexities of infinite sequences, with a focus on substitutive sequences of Pisot type and effective computation using automata-based methods. Prior works have proposed that the k-abelian complexity of an ℓ-automatic sequence should itself be ℓ-regular, but this has not been established in general.

The authors reinvestigate this conjecture and provide two effective methods to construct DFAOs for k-abelian complexities under specific assumptions (uniform factor-balancedness and ultimately Pisot substitutions). While they verify the conjecture for many concrete instances, the general statement remains unresolved and is restated here explicitly.