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The Hankel determinants of an automatic sequence need not be automatic

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorycs.FL — Formal Languages and Automata Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

Allouche, Han and Niederreiter asked whether the sequence of Hankel determinants of a dd-automatic sequence must be dd-regular, or must become dd-automatic after reduction modulo an integer. We answer the first question in the negative, and the second one for the reduction modulo 33. Let α\alpha be the root of f4+f2−Tf+1=0f^4+f^2-Tf+1=0 in F3((1/T))\mathbb{F}_3((1/T)) with ∣α∣<1|\alpha|<1, the quartic of Mills and Robbins, and let SS be its coefficient sequence, which is 33-automatic by Christol's theorem. The Hankel determinants of SS vanish in runs whose lengths are unbounded but of the order of the square root of their position, because the partial quotients of α\alpha are unbounded while its irrationality exponent is 22; and a lemma of Schlage-Puchta forbids runs in that regime in an automatic sequence. Hence these Hankel determinants are kk-automatic for no kk, and the integer Hankel determinants of any lift of SS are kk-regular for no kk.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 02, 2026