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Hankel determinants of weighted binary sums of digits

This paper establishes a general recursion for computing Hankel determinants of weighted binary sum-of-digits functions, deriving explicit closed-form evaluations for the ordinary case to partially solve a problem by Allouche and Shallit, and analyzing the vanishing behavior of these determinants for polynomial weights and their first differences.

Original authors: Bartosz Sobolewski, Maciej Ulas

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bartosz Sobolewski, Maciej Ulas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical calculator that doesn't just add numbers, but looks at how many "1"s are hiding in a number's secret binary code (like a secret language of only zeros and ones). Let's call this the "digit-sum." Now, imagine you take a whole bunch of these digit-sums and arrange them into a giant, square grid, where every cell holds a specific sum based on its row and column. This grid is called a Hankel matrix.

The big question the authors, Bartosz Sobolewski and Maciej Ulas, are asking is: If you crunch this grid down into a single number (called a determinant), what do you get?

Usually, these numbers are messy, unpredictable, and hard to calculate. But this paper is like finding a master key that unlocks a hidden pattern in the chaos.

The Magic Key: A Recursive Recipe

The authors discovered a general "recipe" (a recursion) that lets you calculate these giant grid numbers without having to build the whole grid every time. It's like realizing that instead of counting every single grain of sand on a beach, you can just look at a few specific dunes and use a formula to know the total count.

They tested this recipe on the most basic version of the problem: the standard binary digit-sum. And guess what? They found some incredibly simple answers for specific, infinite families of grid sizes.

For example, if you pick a grid size based on the formula $n = \lceil 2^{k+2}/3 \rceil$ (which just means a specific sequence of growing numbers), the result is shockingly simple:
$\text{Result} = (-1)^{\text{something}} \times (k + 1)$
It's just the number $k+1$ with a plus or minus sign attached! This solves a puzzle that mathematicians Allouche and Shallit had been wondering about for a while.

The "Weighted" Twist

The authors didn't stop at the standard version. They introduced "weights," which are like giving different values to different positions in the binary code. Imagine if the first "1" you see is worth 1 point, the second is worth $t$ points, the third is worth $t^2$ points, and so on.

When they did this, the single numbers turned into polynomials (equations with variables like $t$ ). They found that for certain special values of $t$ —specifically when $t$ is related to "roots of unity" (numbers that circle back to 1 when multiplied by themselves enough times)—these polynomials vanish (become zero) for huge stretches of grid sizes.

Think of it like a piano. If you press certain keys (specific grid sizes), the note is silent. The authors proved that for these special $t$ values, there are long, structured intervals of silence. However, they also showed that the "non-silent" notes are rare and scattered, like finding a few specific islands in a vast, empty ocean.

What They Did and Didn't Find

It's important to know what this paper says is not the answer, and what it leaves open.

No Single Magic Formula for Everything: They did not find one simple formula that works for every possible grid size. The pattern is complex and depends on the binary structure of the number.
No Randomness: They explicitly show that the zeros aren't random. They happen in very specific, structured blocks.
Open Questions on "Multiple Roots": They did not prove that there are no other special numbers where the polynomial has a "double root" (a double zero). Instead, they explicitly list this as an open question (Question 6.2). They ask if such roots exist other than the ones they already know ( $t=0$ and $t=2\zeta$ ), but they haven't ruled them out yet.

How Sure Are They?

Proven Facts: The recursive recipe, the specific closed-form formulas for the special sequences (like the $k+1$ result), and the existence of long intervals where the determinants vanish are all mathematically proven. They are solid facts.
Simulations and Guesses: The authors ran computer simulations to check patterns for larger numbers. Based on these simulations, they suggest (but haven't proven yet) that the set of non-zero indices is even sparser than they calculated. They also conjecture (strongly guess) that the set of indices where the determinant is zero follows a very specific pattern for all cases, but they admit the "hard part" of proving the non-zero cases is still missing.

The "Period-Doubling" Connection

The paper also connects to a famous sequence called the "period-doubling sequence" (a pattern that keeps flipping and doubling). They found that their new, general method actually explains an old result about this sequence, but with a twist: they found a small sign error in the original paper's formula and fixed it. It's like finding a typo in an old map and realizing the treasure is actually in the spot you thought was empty.

The Big Picture

In short, this paper takes a messy, complicated mathematical object (Hankel determinants of digit sums) and shows that underneath the chaos, there is a rigid, beautiful structure. They found a way to predict the value for specific, infinite families of cases and proved that for certain special settings, the values disappear in long, predictable waves.

They leave us with a few open questions, like whether there's a universal limit to how big the roots of these polynomials can get (simulations suggest a limit around 2.618), and whether these patterns hold true for other number bases (like base 3 or base 10). But for now, they've handed us a powerful new tool to decode the secrets of binary sums.

Technical Summary: Hankel Determinants of Weighted Binary Sums of Digits

Problem Statement
The paper addresses the problem of evaluating Hankel determinants associated with digital sequences, specifically the binary sum-of-digits function. This problem was explicitly posed by Allouche and Shallit (Problem 4 in their monograph), asking for a simple closed form for the determinants $H(n) = \det[s(i+j)]_{0 \le i,j < n}$ , where $s(n)$ is the ordinary binary sum of digits. While previous research has successfully analyzed these determinants modulo integers (e.g., proving non-vanishing modulo 2 for the Thue-Morse sequence), obtaining exact closed-form expressions for arbitrary $n$ has remained rare. The authors note that restricting the scope to the ordinary sum of digits is often too limiting for recursive matrix reduction due to perturbations in digit contributions; thus, they generalize the problem to weighted binary sums of digits.

Methodology
The authors introduce a sequence of complex weights $w = (w_j)_{j \ge 0}$ and define the weighted sum-of-digits function $s_w(n)$ . The associated Hankel determinants are denoted $H_w(n)$ . The core methodology relies on exploiting the self-similarity of the sum-of-digits function through block matrix decompositions and Kronecker products.

Recursive Reduction: The paper establishes a general recursive framework (Theorem 4.2) that relates $H_w(n)$ and an auxiliary determinant $\tilde{G}_w(n)$ (associated with the first difference of $s_w$ ) to determinants of smaller size. This recursion depends on the binary expansion of $n$ .
Auxiliary Determinants: A significant portion of the technical work involves analyzing $G_w(n)$ , the Hankel determinant of the first difference sequence $d_w(n) = s_w(n+1) - s_w(n)$ . The authors derive a closed product formula for $G_w(n)$ (Theorem 3.4) based on the run-length structure of the binary expansion of $n$ .
Polynomial Specialization: The authors specialize the weights to the form $w_j = t^j$ , transforming the determinants into polynomials $H(n, t)$ . This allows for the study of roots, degrees, and leading coefficients.
Kernel Construction: To investigate vanishing conditions for specific values of $t$ (specifically $t = 2\zeta$ where $\zeta$ is a root of unity), the authors construct explicit non-zero vectors in the kernel of the Hankel matrices, utilizing tensor products and properties of the matrix $F(n, t)$ .

Key Contributions and Results

General Recursion: The central result (Theorem 4.2) provides a set of recursive formulas allowing the effective computation of $H_w(n)$ for all $n$ . This offers a partial solution to the Allouche-Shallit problem in a generalized setting.
Closed Forms for Specific Indices: For the ordinary binary sum of digits ( $w_j=1$ $w_{j} = 1$ ), the authors derive explicit closed forms for several infinite families of indices:
- For $n_k = \lceil 2^{k+2}/3 \rceil$ , the determinant is given by the remarkably simple formula:
  $H(n_k) = (-1)^{\frac{(k+2)(k+3)}{2}} (k+1)$
- Formulas are also provided for indices of the form $2^m$ , $2^m \pm 1$ , and $2^m + \epsilon$ (Proposition 4.5).
Polynomial Structure: For the polynomial specialization $w_j = t^j$ , the authors determine the degree and leading coefficient of $H(n, t)$ . They prove that $H(n, t)$ factors as $\sigma(n) t^{\beta(n)} (2-t)^{n-2} \hat{H}(n, t)$ , where $\hat{H}(n, t)$ is a monic polynomial with positive integer coefficients (Theorem 4.10).
Vanishing and Non-Vanishing:
- The paper investigates the set of indices $n$ for which $H(n, 2\zeta) = 0$ for roots of unity $\zeta$ . It is shown that for any root of unity $\zeta$ of order $d \ge 2$ , the determinants vanish on long intervals centered at multiples of large powers of 2 (Theorem 5.4).
- The set of non-vanishing indices is proven to be sparse (density 0) but infinite (Theorem 5.5).
- An infinite family of indices distinct from $n_k$ is identified where $H(n, 2\zeta) \neq 0$ (Theorem 5.10).
Period-Doubling Sequence: By choosing weights as Jacobsthal numbers, the authors recover and correct a result by Fokkink, Kraaikamp, and Shallit regarding the Hankel determinants of the period-doubling sequence (Proposition 3.9).
Sign Behavior: The sequence of signs of $H(n, 1)$ is shown to be non-periodic, and its generating function is proven to be transcendental over $\mathbb{Q}(x)$ (Theorem 4.13).

Significance
The paper claims to provide a "partial solution" to the problem posed by Allouche and Shallit by extending the scope to weighted sums, which is necessary to derive effective recursions. The primary significance lies in the derivation of exact closed-form evaluations for specific infinite families of indices, a rarity in the field where results are typically limited to modular arithmetic. Furthermore, the work generalizes previous findings on the period-doubling sequence and offers a detailed structural analysis of the vanishing behavior of these determinants when viewed as polynomials. The authors note that while they do not provide a single closed form for arbitrary $n$ , the recursive structure allows for effective computation and reveals deep connections between the 2-adic structure of indices and the algebraic properties of the determinants.