Work

Complexity functions of infinite sequences

Year: 2026

Type: article

Abstract: We address the inverse problem for complexity functions of infinite words, which asks which functions can occur as complexity functions of infinite words. We survey our recent advances on this problem and several of its ramifications. We describe an asymptotic solution to the inverse problem, realizing every function satisfying two necessary conditions as the complexity function of an infinite word up to asymptotic equivalence. We then turn to phenomena beyond asymptotic equivalence, presenting constructions of infinite words with seemingly contradictory behavior of the discrete derivatives of their complexity functions. These lead to a solution of a 1995 problem of Cassaigne and to applications in symbolic dynamics, including the question of convergence of Rauzy graphs, as well as to the growth and asymptotic invariants of algebraic structures.

Source: Proceedings of the National Academy of Sciences

DOI: 10.1073/pnas.2627766123

Authors: Be’eri Greenfeld, Paul-Henry Leemann, Carlos Gustavo Moreira, Efim Zelmanov

Institutions: City University of New York, Hunter College, Xi’an Jiaotong-Liverpool University, Instituto Nacional de Matemática Pura e Aplicada, Southern University of Science and Technology

Language: English

Topic: semigroups and automata theory

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Funders: Xi’an Jiaotong-Liverpool University, Data Center of Management Science, National Natural Science Foundation of China - Peking University, National Natural Science Foundation of China-Guangdong Joint Fund

OpenAlex ID: W7224476970

FWCI:

Cites: 38

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Related to: 10