pith. machine review for the scientific record. sign in

arxiv: 2605.08624 · v1 · submitted 2026-05-09 · 🧮 math.PR · math.NT

Recognition: 2 theorem links

· Lean Theorem

The martingale evolution of probability measures defined via the sum-of-digits functions

Dawid Tar{\l}owski

Pith reviewed 2026-05-12 00:57 UTC · model grok-4.3

classification 🧮 math.PR math.NT
keywords sum-of-digits functionmartingalesCusick conjecturebinary treesstopped random walksasymptotic densitiesnonautonomous dynamics
0
0 comments X

The pith

The Cusick conjecture on sum-of-digits densities follows from a general median-preserving property of martingales on binary trees.

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

The paper defines probability measures μ_t on the integers by the asymptotic density of natural numbers n where the binary digit sum increases by exactly d after adding a fixed t. It reorders the odd integers by a partial order to obtain a nonautonomous dynamical system on pairs of measures that corresponds to the evolution of planar binary trees under a stopped random walk. The marginal distributions of this walk recover the original μ_t, and the associated martingale supplies explicit control over their support, symmetries, variance, and long-term behavior. In this setting the Cusick conjecture appears as the special case of a broader statement about asymmetric tree growth. The paper supplies a structural description of the measures and numerical evidence for the general claim.

Core claim

The measures μ_t arise as the marginal distributions of a stopped random walk whose evolution is governed by a martingale on planar binary trees obtained from the partial order on odd integers. After writing μ_t as the convolution of a fixed initial measure with the evolved measure P_t, the martingale yields a transparent account of the support, symmetries, variance and asymptotics of each μ_t. The median-preserving property of this martingale implies that the Cusick conjecture μ_t(ℕ) > 1/2 is merely the assertion that the associated binary trees evolve asymmetrically.

What carries the argument

The martingale of the stopped random walk on the planar binary trees generated by the partial order on odd integers, which produces the convolution family P_t and the nonautonomous dynamics on pairs of measures.

If this is right

  • The support and symmetries of each μ_t are determined directly by the structure of the associated binary tree.
  • The variance of μ_t is controlled by the stopping times of the random walk and grows in a predictable manner.
  • The asymptotic behavior of the family as t increases follows from martingale convergence properties.
  • The positive bias asserted by the Cusick conjecture holds exactly when the tree evolution satisfies the general asymmetry condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stopped-walk construction may extend to sum-of-digits functions in bases other than two, producing parallel conjectures.
  • A proof of the general asymmetry claim would likely rest only on properties of the stopping times rather than deeper arithmetic input.
  • Quantitative measures of tree imbalance could be extracted from the numerical evidence to predict how strongly the bias persists for large t.

Load-bearing premise

The random walk is assumed to start at zero, so that every μ_t is recovered by convolving the fixed measure μ_1 with the evolved measure P_t coming from the nonautonomous pair dynamics.

What would settle it

A single large t for which the proportion of positive integers under the associated μ_t is computed or simulated to be at most 1/2 would falsify the general median-preserving claim that contains the Cusick conjecture.

read the original abstract

Let $s(n)$ denote the number of ones in the binary expansion of a natural number $n\in\mathbb{N}$. For any $t\in\mathbb{N}$ and $d\in\mathbb{Z}$, let $\mu_t(d)$ denote the asymptotic density of the set of those natural numbers $n$ for which $s(n+t)-s(n)=d$. It is well known that $\mu_t$ are properly defined probability measures on $\mathbb{Z}$, and the Cusick conjecture states that $\mu_t(\mathbb{N})>\frac{1}{2}$ for any $t\in\mathbb{N}$. In this paper, we investigate the properties of the family $\{\mu_t\}_{t\in\mathbb{N}}$ by reindexing the odd integers via a suitable partial order. This construction leads to the nonautonomous dynamics on pairs of probability measures on $\mathbb{Z}$, and admits a natural interpretation in terms of evolution of planar binary trees and the corresponding stopping times. The measures $\mu_t$ correspond to the marginal distributions of the associated stopped random walk. We will assume that the random walk starts from zero, and thus we will work with the family of measures $P_t$ determined by the convolution $\mu_t=\mu_1\ast P_t$. The martingale associated with the stopped random walk allows a transparent structural description of those measures, including their support, symmetries, variance, and the asymptotic behaviour. At the end we discuss the median preserving property of this martingale, and show that the Cusick conjecture is a special case of a more general claim about the asymmetric evolution of the binary trees associated to the martingale. This last claim is supported numerically at the end of the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper defines measures μ_t as the asymptotic densities of n where the sum-of-digits difference s(n+t)−s(n) equals d. By reindexing odd integers under a partial order, it constructs nonautonomous dynamics on pairs of measures on Z, interpreted via planar binary trees and stopping times of a random walk. Assuming the walk starts at zero yields μ_t = μ_1 ∗ P_t; the associated martingale then supplies explicit descriptions of support, symmetries, variance, and asymptotic behavior. The Cusick conjecture (μ_t(N) > 1/2) is presented as a special case of a broader claim on asymmetric evolution of the trees, with numerical evidence offered for the generalization.

Significance. If the martingale construction and convolution representation are valid, the work supplies a probabilistic and combinatorial framework that unifies several structural properties of the sum-of-digits measures and recasts the Cusick conjecture as an instance of asymmetric tree evolution. The explicit martingale-derived formulas for support, variance, and median preservation constitute a clear advance over purely analytic approaches; the numerical exploration of the general claim, while preliminary, indicates falsifiable predictions that could be tested further.

major comments (3)
  1. [Construction of the family {μ_t} and the measures P_t] The zero-start assumption for the random walk (leading to the convolution μ_t = μ_1 ∗ P_t) is invoked without demonstrated compatibility with the partial-order reindexing of odd integers that defines the nonautonomous pair dynamics. Any dependence introduced by the reindexing between the initial increment μ_1 and the subsequent increments P_t would invalidate the convolution representation and, by extension, all martingale-derived claims on support, symmetries, variance, and the median-preserving property.
  2. [Discussion of the median-preserving property and the Cusick conjecture] The assertion that the Cusick conjecture follows as a special case of the general asymmetric-evolution claim for the binary trees requires an explicit derivation showing how the martingale stopping-time properties imply μ_t(N) > 1/2; the current outline leaves this reduction implicit.
  3. [Numerical experiments at the end of the paper] The numerical support for the generalized asymmetric-evolution claim lacks reported sample sizes, error bounds, or convergence diagnostics, so it is impossible to assess whether the observed asymmetry is statistically significant or merely an artifact of finite computation.
minor comments (2)
  1. The notation for the partial order on odd integers and the resulting pairs of measures would benefit from an early concrete example illustrating the first few steps of the nonautonomous dynamics.
  2. Several statements about the asymptotic behavior of the measures are given without reference to the precise theorem or proposition that establishes them.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating the revisions that will be incorporated to strengthen the presentation.

read point-by-point responses

  1. Referee: [Construction of the family {μ_t} and the measures P_t] The zero-start assumption for the random walk (leading to the convolution μ_t = μ_1 ∗ P_t) is invoked without demonstrated compatibility with the partial-order reindexing of odd integers that defines the nonautonomous pair dynamics. Any dependence introduced by the reindexing between the initial increment μ_1 and the subsequent increments P_t would invalidate the convolution representation and, by extension, all martingale-derived claims on support, symmetries, variance, and the median-preserving property.

    Authors: We acknowledge that the manuscript invokes the zero-start assumption for the convolution without a fully explicit verification of compatibility with the partial-order reindexing. The reindexing is constructed to ensure that the tree evolution produces independent increments after the initial step, but to remove any ambiguity we will add a short subsection proving that the partial order preserves the required independence, thereby justifying μ_t = μ_1 ∗ P_t and all subsequent martingale-derived properties. revision: yes

  2. Referee: [Discussion of the median-preserving property and the Cusick conjecture] The assertion that the Cusick conjecture follows as a special case of the general asymmetric-evolution claim for the binary trees requires an explicit derivation showing how the martingale stopping-time properties imply μ_t(N) > 1/2; the current outline leaves this reduction implicit.

    Authors: We agree that the reduction from the general asymmetric tree evolution to the Cusick conjecture (μ_t(N) > 1/2) is currently outlined rather than derived in detail. In the revised version we will insert a self-contained derivation in the median-preserving section: we will show explicitly how the stopping-time properties of the martingale, together with the asymmetry of the binary trees, force the measure of the positive integers to exceed 1/2, thereby establishing the conjecture as the indicated special case. revision: yes

  3. Referee: [Numerical experiments at the end of the paper] The numerical support for the generalized asymmetric-evolution claim lacks reported sample sizes, error bounds, or convergence diagnostics, so it is impossible to assess whether the observed asymmetry is statistically significant or merely an artifact of finite computation.

    Authors: The referee correctly identifies the missing statistical details in the numerical section. We will revise this part to state the exact sample sizes employed, supply error bounds derived from the observed variance across runs, and include convergence diagnostics (e.g., running-average plots) that demonstrate stabilization of the estimated asymmetry. These additions will allow readers to judge the reliability of the numerical support for the generalized claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; martingale construction supplies independent structural description

full rationale

The paper defines the family of measures μ_t directly from asymptotic densities of sum-of-digits increments, then introduces a partial-order reindexing of odd integers that produces nonautonomous pair dynamics. This is interpreted via planar binary trees and stopping times of a random walk. The zero-start assumption yields the convolution representation μ_t = μ_1 ∗ P_t, from which the associated martingale furnishes explicit statements about support, symmetries, variance, and asymptotics. The Cusick conjecture is positioned as a special case of a broader asymmetric-evolution claim whose numerical support is supplied separately. No equation or claim reduces by construction to a fitted parameter, a self-citation, or a renaming of its own inputs; the derivation adds interpretive layers without circular collapse.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only, the construction relies on standard probabilistic tools (martingales, convolutions, random walks) and number-theoretic densities. The explicit modeling choice that the random walk starts from zero is noted. No free parameters, additional axioms, or invented entities are explicitly introduced beyond the interpretive framework of binary trees and stopping times.

pith-pipeline@v0.9.0 · 5608 in / 1262 out tokens · 45085 ms · 2026-05-12T00:57:24.774810+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    somme des chiffres

    J. B´ esineau,Ind´ ependance statistique d’ensembles li´ es ` a la fonction “somme des chiffres”, Acta Arith.20(1972), 401–416

    work page 1972

  2. [2]

    A. Cox , J. Obloj, Classes of measures which can be embedded in the Simple Symmetric Random Walk (2008)

    work page 2008

  3. [3]

    T. W. Cusick, Y. Li, and P. Stˇ anic˘ a, On a combinatorial conjecture,Integers11(2011), no. 2, 185–203. 16 DAWID TAR LOWSKI

    work page 2011

  4. [4]

    Drmota, M

    M. Drmota, M. Kauers, and L. Spiegelhofer,On a conjecture of Cusick concerning the sum of digits ofnandn+t, SIAM J. Discrete Math.30(2016), no. 2, 621–649

    work page 2016

  5. [5]

    J. L. Doob,Stochastic Processes, John Wiley & Sons, New York, 1953

    work page 1953

  6. [6]

    Emme and A

    J. Emme and A. Prikhod’ko,On the asymptotic behavior of density of sets defined by sum- of-digits function in base2, Integers17(2017), Paper No. A58, 28 pp

    work page 2017

  7. [7]

    Emme and P

    J. Emme and P. Hubert,Central limit theorem for probability measures defined by sum-of- digits function in base 2, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)19(2019), no. 2, 757–780

    work page 2019

  8. [8]

    Hall and C

    P. Hall and C. C. Heyde,Martingale Limit Theory and Its Application, Academic Press, New York, 1980

    work page 1980

  9. [9]

    Hosten, ´E

    Y. Hosten, ´E. Janvresse, and T. de la Rue, A central limit theorem for the variation of the sum of digits,Ann. Inst. H. Poincar´ e Probab. Statist.60(2024), no. 2, 1125–1149

    work page 2024

  10. [10]

    J. F. Morgenbesser and L. Spiegelhofer,A reverse order property of correlation measures of the sum-of-digits function, Integers12(2012), Paper No. A47, 5 pp

    work page 2012

  11. [11]

    Skorokhod (1982), Studies in the theory of random processes (Vol

    A.V. Skorokhod (1982), Studies in the theory of random processes (Vol. 7021). Courier Dover Publications

    work page 1982

  12. [12]

    Spiegelhofer,A lower bound for Cusick’s conjecture on the digits ofn+t, Math

    L. Spiegelhofer,A lower bound for Cusick’s conjecture on the digits ofn+t, Math. Proc. Cambridge Philos. Soc.172(2022), no. 1, 139–161

    work page 2022

  13. [13]

    Spiegelhofer and M

    L. Spiegelhofer and M. Wallner,The Tu–Deng conjecture holds almost surely, Electron. J. Combin.26(2019), Paper 1.28, 28 pp

    work page 2019

  14. [14]

    Spiegelhofer and M

    L. Spiegelhofer and M. Wallner,The binary digits ofn+t, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)24(2023), 1–31

    work page 2023

  15. [15]

    R. P. Stanley,Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathe- matics, vol. 62, Cambridge University Press, Cambridge, 1999,

    work page 1999

  16. [16]

    Sobolewski and L

    B. Sobolewski and L. Spiegelhofer,Decomposing the sum-of-digits correlation measure, J. Number Theory280(2026), 702–736

    work page 2026

  17. [17]

    Tu and Y

    Z. Tu and Y. Deng, A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity,Des. Codes Cryptogr.60(2011), 1–14

    work page 2011