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arxiv: 2605.08624 · v2 · pith:M3NN6BNFnew · submitted 2026-05-09 · 🧮 math.PR · math.NT

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

Dawid Tar{\l}owski This is my paper

Pith reviewed 2026-05-22 10:00 UTC · model grok-4.3

classification 🧮 math.PR math.NT
keywords sum-of-digits functionasymptotic densityCusick conjecturemartingaleplanar binary treesstopped random walkprobability measures on integersnonautonomous dynamics
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The pith

Reindexing odd integers creates martingale dynamics on binary trees that frame the Cusick conjecture as a special case of asymmetric tree evolution.

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

The paper studies probability measures that record how often the binary digit sum increases by a fixed amount d when a number is increased by t. Reindexing the odd integers according to a partial order turns the family of measures into nonautonomous dynamics on pairs of measures on the integers. These dynamics correspond to the growth of planar binary trees together with stopping times for a random walk that begins at zero. The stopped walk's marginal distributions are exactly the measures under study, and an associated martingale supplies explicit descriptions of their support, symmetries, variance, and limiting behavior. The median-preserving property of the martingale shows that the well-known Cusick conjecture follows from a broader statement about asymmetric evolution of the trees, a statement that receives numerical support.

Core claim

By reindexing the odd integers via a suitable partial order, the family of measures μ_t induces nonautonomous dynamics on pairs of probability measures on Z. These dynamics admit a natural interpretation in terms of the evolution of planar binary trees and the corresponding stopping times of a random walk starting at zero. The associated martingale gives a transparent structural description of the measures, including their support, symmetries, variance, and asymptotic behaviour. The median preserving property of this martingale shows 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.

What carries the argument

Reindexing of the odd integers by a partial order that produces nonautonomous dynamics on pairs of probability measures, interpreted as the evolution of planar binary trees together with stopping times for a stopped random walk.

If this is right

  • The measures admit explicit descriptions of support, symmetries, variance and asymptotic behaviour through the martingale.
  • The convolution relation μ_t = μ_1 * P_t holds where P_t arises from the stopped walk.
  • The process preserves the median of the measures.
  • The Cusick conjecture follows once the general asymmetric tree-evolution claim is established.

Where Pith is reading between the lines

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

  • The tree-growth viewpoint may supply an inductive proof of the generalized claim by tracking bias at each level of the trees.
  • Analogous partial-order reindexings could be applied to digit-sum problems in other bases or to related additive functions.
  • If the numerical evidence for asymmetric evolution holds in general, it would imply strict inequality for the mass on positive integers without needing separate case analysis for each t.

Load-bearing premise

The reindexing of the odd integers via a suitable partial order produces nonautonomous dynamics on pairs of probability measures that admit a natural interpretation in terms of evolution of planar binary trees and corresponding stopping times.

What would settle it

A numerical computation of the measure mass on positive integers for successively larger t that falls to 1/2 or below for some t, or an explicit counter-example configuration of binary trees whose asymmetric evolution fails to preserve the required bias.

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

2 major / 1 minor

Summary. The paper develops a martingale framework for the family of measures μ_t on ℤ, where μ_t(d) is the asymptotic density of n ∈ ℕ with s(n+t) − s(n) = d and s is the binary digit sum. Reindexing the odd integers under a partial order yields nonautonomous dynamics on pairs of measures, interpreted as the evolution of planar binary trees with associated stopping times. The μ_t appear as marginals of a stopped random walk starting at zero; the family P_t is obtained via the convolution μ_t = μ_1 ∗ P_t. The associated martingale furnishes structural information on support, symmetries, variance and asymptotics. The Cusick conjecture (μ_t(ℕ) > 1/2) is presented as a special case of a general claim on asymmetric tree evolution, the latter being supported by numerical checks at the end of the paper.

Significance. If the martingale construction is rigorous and the numerical evidence for asymmetric evolution is representative, the work supplies a novel probabilistic and combinatorial lens on sum-of-digits densities, linking them to stopped random walks and binary-tree dynamics without introducing free parameters. This perspective could open new routes toward the Cusick conjecture and related problems in analytic number theory.

major comments (2)
  1. [Final section] Final section (discussion of median-preserving property and Cusick conjecture): the reduction of the Cusick conjecture to the general claim on asymmetric evolution of the binary trees is supported only numerically. No analytical argument is supplied showing that the tree asymmetry and the stopping times force μ_t(ℕ) > 1/2 for every t; the numerical checks therefore remain the sole evidence for the broader statement on which the conjecture is said to depend.
  2. [Construction of nonautonomous dynamics] Section introducing the reindexing and nonautonomous dynamics: the claim that the partial order on odd integers produces dynamics whose marginals recover the original densities μ_t requires an explicit verification that the stopping-time construction does not alter the asymptotic densities; the current outline leaves open whether the convolution step with P_t preserves the defining property of μ_t without additional bias.
minor comments (1)
  1. [Abstract and introduction] The abstract states that the random walk 'starts from zero' and works with P_t via convolution, but the precise relation between the original μ_t and the convolved family P_t is not restated in the introduction; a short clarifying sentence would help readers track the notation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive evaluation of its significance. We address each major comment below.

read point-by-point responses

  1. Referee: [Final section] Final section (discussion of median-preserving property and Cusick conjecture): the reduction of the Cusick conjecture to the general claim on asymmetric evolution of the binary trees is supported only numerically. No analytical argument is supplied showing that the tree asymmetry and the stopping times force μ_t(ℕ) > 1/2 for every t; the numerical checks therefore remain the sole evidence for the broader statement on which the conjecture is said to depend.

    Authors: We acknowledge that the reduction of the Cusick conjecture to the general claim on asymmetric evolution of the binary trees is supported solely by numerical evidence in the manuscript, with no complete analytical argument provided to show that tree asymmetry and stopping times force μ_t(ℕ) > 1/2 for every t. The paper's main contribution is the martingale framework and the reduction itself; the numerical checks at the end illustrate the plausibility of the broader claim. An analytical proof of the general statement appears to lie beyond the scope of the current work and would require further combinatorial analysis of the tree dynamics. revision: no

  2. Referee: [Construction of nonautonomous dynamics] Section introducing the reindexing and nonautonomous dynamics: the claim that the partial order on odd integers produces dynamics whose marginals recover the original densities μ_t requires an explicit verification that the stopping-time construction does not alter the asymptotic densities; the current outline leaves open whether the convolution step with P_t preserves the defining property of μ_t without additional bias.

    Authors: We agree that an explicit verification is needed. In the revised manuscript we will add a dedicated subsection verifying that the partial order on odd integers and the associated stopping-time construction preserve the asymptotic densities that define the measures μ_t. We will also show explicitly that the convolution μ_t = μ_1 ∗ P_t recovers the original marginals without introducing bias, by computing the relevant marginal distributions of the stopped random walk directly from the tree evolution. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines μ_t via asymptotic densities of sum-of-digits differences, reindexes odd integers to obtain nonautonomous dynamics on pairs of measures, and interprets the construction as evolution of planar binary trees with associated stopping times for a random walk starting at zero. It then sets μ_t = μ_1 ∗ P_t and derives support, symmetries, variance and median-preserving properties directly from the martingale structure using standard external probability theory. The claim that Cusick's conjecture is a special case of asymmetric tree evolution is stated explicitly and supported only by numerical checks at the end; this does not reduce any equation or central result to a definitional tautology, fitted parameter, or self-citation chain. The framework remains self-contained against external martingale benchmarks with no load-bearing step that collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on the existence of the asymptotic densities as probability measures and on the modeling assumptions that allow the random-walk and tree interpretations.

axioms (2)
  • domain assumption The asymptotic densities μ_t(d) exist and form probability measures on Z for each t
    Stated as well known at the beginning of the abstract.
  • ad hoc to paper The random walk starts from zero, yielding the family P_t via convolution
    Explicitly assumed to work with the convolution representation μ_t = μ_1 * P_t.
invented entities (1)
  • Stopped random walk on planar binary trees with associated stopping times no independent evidence
    purpose: To represent the measures μ_t as marginal distributions and to enable the martingale description
    Introduced as the central modeling device in the abstract.

pith-pipeline@v0.9.0 · 5839 in / 1413 out tokens · 40130 ms · 2026-05-22T10:00:22.114648+00:00 · methodology

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Works this paper leans on

17 extracted references · 17 canonical work pages

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