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 $O(\sqrt{\ell})$ dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density $\varphi$, a base-3 analogue of the Fabius density, against an explicit $\varphi_0$ due to Berg and Kr\"uppel. The exact log-Laplace transform of $\varphi$ splits into a smooth part, a log 3-periodic correction $H$, and a doubly exponentially small remainder. Berg and Kr\"uppel 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 $\Gamma$ and $\zeta$; a classical zero-free theorem for $\zeta$ 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 $\varphi(t)\sim\kappa\varphi_0(t)$ fails. Condition $(\star4)$ follows, at every window radius, with $\mu=1/3$: 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 $(\star3)$. Conjecture 2 is his route to it and remains open.