Wirsching的正前驱密度程序：猜想1和3的证明
Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3
AI总结：
该研究证明了Wirsching正前驱密度程序中关于3n+1映射的猜想1和3，给出了相关函数的闭式表达式，严格限定了振荡范围，同时指出猜想2仍未解决。
AI中文摘要：
Wirsching（2003）将3n+1映射的一致正前驱密度归约为五个条件构成的链，这些条件被组织成三个猜想，我们证明了其中两个。猜想1涉及他的路径计数生成元：在将其映射到Elka函数的卷积中，划分权重呈次指数增长，而二项式比率呈几何衰减，因此宽度为O(√ℓ)的窗口占据主导地位，且其半径符合假设。猜想3涉及不变密度φ在0附近的渐近行为，φ是Berg和Krüppel提出的φ₀的以3为底的类似物。φ的精确对数拉普拉斯变换可分解为光滑部分、对数3周期修正项H以及双指数小余项。Berg和Krüppel于1998年将该修正项表示为无穷乘积，但其分析未确定该修正项是否为常数。我们将H表示为系数为Γ和ζ的闭式的傅里叶级数；ζ的经典无零点定理表明它不是常数，且我们严格限定了其振荡范围。Wirsching的比较类固定了H的一个相位，在此类中猜想3成立，极限为e^{H(0)}，经证明其值位于(0.53412203666478, 0.53412203666479)区间内。超出该类后，相位扫过整个周期，因此无限制渐近关系φ(t)~κφ₀(t)不成立。对于每个窗口半径，条件(⋆4)成立，其中μ=1/3：Wirsching的论证所需条件弱于猜想3本身，且同一鞍点链可直接解决该问题。结合猜想1，该链归约为单一条件(⋆3)。猜想2是通往该条件的途径，目前仍未解决。
英文摘要：
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 $φ$, 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 $φ(t)\simκφ_0(t)$ fails. Condition $(\star4)$ 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 $(\star3)$. Conjecture 2 is his route to it and remains open.
