Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3
Abstract
Wirsching (2003) reduces uniform positive predecessor density for the 3n+1 map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width O(√ℓ) dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density φ, a base-3 analogue of the Fabius density, against an explicit φ0 due to Berg and Krüppel. The exact log-Laplace transform of φ splits into a smooth part, a log 3-periodic correction H, and a doubly exponentially small remainder. Berg and Krüppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give H as a Fourier series with coefficients in closed form in Γ and ζ; a classical zero-free theorem for ζ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of H, and there Conjecture 3 holds with limit eH(0), certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic φ(t)∼κφ0(t) fails. Condition (⋆4) follows, at every window radius, with μ=1/3: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition (⋆3). Conjecture 2 is his route to it and remains open.
Related
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- Distributionally Faithful Imputation via Positive Semi-Definite Kernel Density Estimation
- Blow-ups of order types of positive density
BibTeX
@misc{tavares2026wirschings,
title = {{Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3}},
author = {Renato Augusto Tavares},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27617},
eprint = {2608.27617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27617},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3
AU - Renato Augusto Tavares
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27617
UR - https://arxiv.org/abs/2608.27617
AB - Wirsching (2003) reduces uniform positive predecessor density for the 3n+1 map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width O(√ℓ) dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density φ, a base-3 analogue of the Fabius density, against an explicit φ_0 due to Berg and Krüppel. The exact log-Laplace transform of φ splits into a smooth part, a log 3-periodic correction H, and a doubly exponentially small remainder. Berg and Krüppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give H as a Fourier series with coefficients in closed form in Γ and ζ; a classical zero-free theorem for ζ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of H, and there Conjecture 3 holds with limit e^H(0), certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic φ(t)∼κφ_0(t) fails. Condition (⋆4) follows, at every window radius, with μ=1/3: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition (⋆3). Conjecture 2 is his route to it and remains open.
ER -