Complexity functions of infinite sequences.

An asymptotic inverse solution shows that every function meeting two necessary conditions can occur as the complexity function of an infinite word, up to asymptotic equivalence.

• Why it matters: The work tackles which complexity-growth profiles are realizable by infinite words, and extends beyond asymptotic equivalence to resolve related open questions and enable applications.
• What they did: The authors develop an asymptotic characterization for the inverse problem and construct infinite words whose complexity-function discrete derivatives show seemingly contradictory behavior.
• The result: The constructions solve a 1995 problem of Cassaigne and yield applications in symbolic dynamics, including Rauzy graph convergence, plus results on growth and asymptotic invariants of algebraic structures.

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