Nonvanishing Hankel Determinants of Morphic Sequences

Definitions and mathematical statement

Let $I=\{0,1,2,\ldots\}$. For an integer sequence $b=(b_k)_{k\in I}$, a starting position $s\in I$, and a positive integer $n$, define the Hankel determinant \[ H_n^{(s)}(b)=\det\left(b_{s+i+j}\right)_{0\leq i,j<n}\in{\mathbb{Z}}. \] The questions below interpret "every Hankel determinant" as quantifying over every starting position and every positive order. The supplied assessment mentions all positions, but the statement does not explicitly define its Hankel indexing convention; this interpretation requires review against the original source.

Let $\varphi$ be the morphism on finite words over $\{1,2,3,4\}$, extended by concatenation from \[ \varphi(1)=12,\qquad \varphi(2)=23,\qquad \varphi(3)=14,\qquad \varphi(4)=32. \] Since $\varphi(1)$ begins with $1$, its iterates determine the infinite word \[ a=\varphi^\omega(1)=\lim_{r\to\infty}\varphi^r(1), \] where the limit means that each finite prefix eventually stabilizes. Regard its letters as integers and index them as $a=(a_k)_{k\in I}$.

Question 1. Is it true that \[ \forall s\in I\;\forall n\geq1,\qquad H_n^{(s)}(a)\neq0 \quad\text{in }{\mathbb{Z}}? \]

Question 2. The source introduces the following question with "More generally": is it true that \[ \begin{gathered} \forall m\in{\mathbb{Z}},\ m\geq2,\quad \forall A\subseteq{\mathbb{Z}}\text{ finite and nonempty},\quad \forall b\in A^I,\\ \exists s\in I\;\exists n\geq1,\qquad H_n^{(s)}(b)\equiv0\pmod m? \end{gathered} \] Equivalently, for each integer $m\geq2$, no infinite sequence taking values in a finite subset of ${\mathbb{Z}}$ has all its Hankel determinants nonzero modulo $m$. Here nonzero modulo $m$ means not congruent to zero; it does not require invertibility modulo a composite $m$. Question 1 concerns nonvanishing in ${\mathbb{Z}}$, whereas Question 2 concerns modular nonvanishing; no implication between the two formulations is asserted here.

Short English statement

The first question asks whether every positive-order Hankel determinant at every starting position of the specified morphic sequence is a nonzero integer. The second asks whether, for each modulus $m\geq2$, every integer sequence with finite range has some Hankel determinant divisible by $m$. The all-position indexing convention requires source review. Both are supplied as open questions, not independently verified claims.

Sources

• Local UnsolvedMath registry, permanent ID $30001560$, title: Nonvanishing Hankel Determinants of Morphic Sequences; exact source problem code: catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection and lists proposed year $2010$; no proposer is identified.
• Supplied original workshop reference: Mini-Workshop: Combinatorics on Words (2011), Oberwolfach Reports, pp. 2195--2244, DOI 10.4171/owr/2010/37. https://doi.org/10.4171/owr/2010/37. The report is described as documenting workshop discussions and their open problems. The supplied title year $2011$ and proposed year $2010$ are retained without reconciliation.
• Supplied literature source, labeled "current open-problems survey": https://cs.uwaterloo.ca/~shallit/Talks/bc4.pdf. No author, title, or publication date for this survey is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the specified morphism and its all-position nonzero-Hankel-determinant property remain listed as open. It explicitly cautions that finite computation does not constitute a proof. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.