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Hankel determinants of weighted binary sums of digits

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Pith's one-line read A recursion computes every Hankel determinant of weighted binary digit sums, and for ordinary sum-of-digits it yields the closed form H(⌈2^{k+2}/3⌉)=(±)(k+1).

desk verdict Solid exact-evaluation paper: a usable recursion for weighted sum-of-digits Hankel determinants plus an infinite closed-form family that partially answers Allouche–Shallit Problem 4. read the letter →

arxiv 2607.09376 v1 pith:RFUS2BU4 submitted 2026-07-10 math.NT math.CO

Bartosz Sobolewski , Maciej Ulas This is my paper · ORCID
classification math.NTmath.CO MSC 11B8505A1511B8315B05
keywords Hankeldeterminantssumofdigitsweightedbinaryexpansionsperiod-doublingsequencerootsunityautomaticsequences
topics The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Hankel determinants built from the binary sum-of-digits function (and its weighted variants) are hard to evaluate exactly; most prior work only controls them modulo an integer. This paper produces a complete recursive scheme that reduces every such determinant of size n to determinants of strictly smaller size, using only the 2-adic structure of the first differences of the sum-of-digits. Specializing the weights to 1 recovers an infinite explicit family, most strikingly H(⌈2^{k+2}/3⌉)=(-1)^{(k+2)(k+3)/2}(k+1), which partially answers a problem of Allouche and Shallit. The same recursion yields product formulas for the first-difference determinants (recovering and correcting the period-doubling case) and a detailed vanishing theory for the polynomial family with weights t^j, showing that when t=2ζ for a root of unity the determinants vanish on long structured blocks of indices while the non-vanishing set remains infinite but sparse.

What carries the argument

The block-matrix recursion of Theorem 4.2 (and its first-difference companion Proposition 3.2), driven by the 2-adic valuation formula for the first difference d_w(m) and elementary row/column operations that preserve the Hankel structure under weight shifts.

What would settle it

Compute the Hankel determinants of ordinary binary sum-of-digits at the indices n_k=⌈2^{k+2}/3⌉ for several large k by direct linear algebra and check whether they equal exactly (-1)^{(k+2)(k+3)/2}(k+1).

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Extended reading notes

Core claim

There is a uniform recursion (Theorem 4.2) that expresses every Hankel determinant H_w(n) of a weighted binary sum-of-digits sequence in terms of two smaller determinants of the same type (or of the first-difference type). For ordinary sum-of-digits the recursion collapses on the indices n_k=⌈2^{k+2}/3⌉ to the elementary closed form (-1)^{(k+2)(k+3)/2}(k+1), giving an infinite explicit family.

Load-bearing premise

The recursion assumes that the 2-adic formula for the first difference of the weighted sum-of-digits survives the elementary matrix operations and the weight shifts that appear at each reduction step.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies Hankel determinants H_w(n) = det[s_w(i+j)]_{0≤i,j<n} for the weighted binary sum-of-digits function s_w associated to arbitrary complex weights w. It derives a general recursion (Theorem 4.2) that computes every such determinant from smaller ones via auxiliary first-difference determinants. Specializing to constant weights w_j=1 produces closed forms for several infinite families of indices, including the simple evaluation H(⌈2^{k+2}/3⌉)=(-1)^{(k+2)(k+3)/2}(k+1) (Theorem 4.4), partially solving Allouche–Shallit Problem 4. For polynomial weights w_j=t^j the determinants become polynomials H(n,t); the paper determines their degrees, leading coefficients, factorizations and signs, and proves structured vanishing on long intervals about multiples of high powers of 2 when t=2ζ for a root of unity ζ (Theorems 5.4–5.5), while the complementary non-vanishing set is sparse yet infinite (Theorem 5.10). An explicit product formula for the first-difference Hankel determinants G_w(n) is also obtained (Theorem 3.4), recovering and correcting the period-doubling evaluations of Fokkink–Kraaikamp–Shallit.

Significance. Exact closed-form evaluations of Hankel determinants for digital sequences remain rare; most prior work addresses only modular behaviour or automaticity. The recursion of Theorem 4.2 together with the infinite family of explicit values in Theorem 4.4 therefore constitutes a genuine advance on the Allouche–Shallit problem and supplies a practical computational tool. The product formula for G_w and the vanishing theorems for doubled roots of unity give precise structural information that is directly relevant to Padé approximation and irrationality-exponent applications. The proofs are self-contained inductive arguments based on block-matrix row/column operations and Laplace expansions; they include an independent sign correction of earlier literature and are consistent with low-order direct checks. These features make the contribution solid and reusable.

minor comments (4)
  1. Several typographical errors appear in the manuscript text (e.g., “V anishing”, “prodduct formula”, “eGis proved”, missing spaces after periods). A careful proof-reading pass is needed before final production.
  2. Section 2 introduces a compact block-matrix notation that suppresses indices and writes block heights to the right; while efficient, a short explicit example of the expansion used in the proofs of Propositions 3.2 and Theorem 4.2 would improve readability for non-specialists.
  3. The base cases of the recursions (n < 5 for Theorem 4.2, small k for Theorems 3.3–3.4) are asserted to hold by direct computation; listing the explicit 2×2, 3×3 and 4×4 matrices (or their determinants) in an appendix or remark would make the induction completely self-contained.
  4. Conjectures 4.6, 4.7 and 5.7 are clearly labelled as open; it would be helpful to indicate, even briefly, which of them appear most accessible to the same recursive methods already developed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: recursions and closed forms are self-contained matrix identities and inductions.

full rationale

The paper derives Hankel determinants for weighted binary sum-of-digits via elementary row/column operations on block matrices (Props. 3.1–3.2, Thm. 4.2) that track the 2-adic valuation formula (3) for the first difference d_w. The resulting recursions are proved by induction and specialize to an explicit product formula for G_w(n) (Thm. 3.4) and to closed forms such as H(n_k,t) (Thm. 4.4). These identities are verified by direct low-order checks and recover (with a sign correction) the known period-doubling determinants of Fokkink–Kraaikamp–Shallit; the correction is independent of the new recursion. No parameter is fitted to data and then re-presented as a prediction, no uniqueness theorem is imported from the authors’ prior work, and no ansatz is smuggled via self-citation. The vanishing results for t=2ζ (Thms. 5.4–5.5, 5.10) likewise rest on explicit kernel constructions from the same block identities. The derivation chain is therefore self-contained against external benchmarks and exhibits no circular reduction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Pure-mathematics paper resting entirely on standard linear algebra, 2-adic valuations and induction; no free parameters, no ad-hoc physical entities, and only the usual background axioms of complex numbers and matrix determinants.

assumptions (2)
  • standard math Standard determinant identities (Laplace expansion, elementary row/column operations, Kronecker-product multiplicativity) hold over C.
    Invoked throughout Sections 3–4 for every block-matrix reduction.
  • domain assumption The 2-adic valuation formula d_w(m)=∑_{i=0}^{ u_2(m+1)} u_i is valid for the first-difference sequence of any weighted digit sum.
    Equation (3); used as the starting point for all subsequent matrix identities.

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Pith. "Pith review of Hankel determinants of weighted binary sums of digits." pith.science (2026). https://pith.science/paper/RFUS2BU4

@misc{pith2026260709376,
  author       = {Pith},
  title        = {Pith review of: Hankel determinants of weighted binary sums of digits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RFUS2BU4}},
  note         = {Machine review of arXiv:2607.09376}
}
abstract

Let $s_\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\mathbf{w}=(w_j)_{j\geq 0}$. We investigate Hankel determinants $\mathcal{H}_\mathbf{w}(n) = \det [s_{\mathbf{w}}(i+j)]_{0\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\mathcal{H}_\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\mathcal{H}_\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \mathcal{H}_\mathbf{w}(\lceil 2^{k+2}/3\rceil)= (-1)^{\frac{(k+2)(k+3)}{2}}(k+1). $$ This yields an infinite family of explicit evaluations, giving a partial solution to a problem posed by Allouche and Shallit. Moreover, we closely study the specialization $w_j=t^j$, where the determinants become polynomials in $t$, and investigate their vanishing. For $t=2\zeta$, where $\zeta$ is a root of unity, we show that the determinants vanish on a large structured set of indices, while the complementary is sparse but infinite. In addition to $\mathcal{H}_\mathbf{w}(n)$, we consider Hankel determinants associated with the first difference of $s_{\mathbf{w}}$, obtaining an explicit product formula. This generalizes the results by Fokkink, Kraaikamp, and Shallit concerning Hankel determinants for the period-doubling sequence.

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