AI 中文总结
该研究针对Kronecker流生成的等分布螺旋，证明了球面可观测量时间平均的多项式收敛速率，处理了参数化退化问题，还得到收缩螺旋下的显式速率及径向剖面二分性，并指出无统一速率适用于所有有理独立频率。
AI 中文摘要
我们研究由Kronecker流生成的等分布螺旋上采样的球面可观测量的定量时间平均。对于丢番图频率向量与球面上的Hölder可观测量，我们证明了其向球面均值的窗口一致多项式收敛速率。主要难点在于球面的保测参数化在环面上不连续且存在极点退化，我们通过局部化和de la Vallée Poussin逼近方案处理该问题。随后，我们将该估计应用于收敛到中心的收缩螺旋，在温和的径向正则性条件下得到球面均值的显式速率。我们还处理了可积角数据与中心处的齐次奇点，证明了幂律与指数收缩径向剖面之间的二分性。最后，我们表明即使对于球面上光滑均值为零的数据，也无法对所有有理独立频率保持一致速率。
英文摘要
We establish quantitative equidistribution estimates for spherical observables sampled along Kronecker flows. For the monotone measure-preserving inverse-CDF parametrization $Φ_d:\mathbb{T}^d\to\mathbb{S}^d$ and a Diophantine frequency of exponent $τ$, every $φ\in C^s(\mathbb{S}^d)$, $0<s\le1$, has time averages converging to the spherical mean at rate $O(T^{-\vartheta})$, where $\vartheta=\frac{s}{(1+s)(τ+d)}$, uniformly in the initial phase and averaging window. Since $Φ_d$ is discontinuous across a polar cut, the proof localizes the pullback away from the polar degeneracies and combines de la Vallée Poussin approximation, Fourier decay, and the Diophantine lower bound. The same exponent holds for a continuous measure-preserving variant. For comparison, mollification followed by Koksma--Hlawka gives, for every $\varepsilon>0$, the rate $O(T^{-s/[d(τ+1)]+\varepsilon})$. Its power exponent is larger exactly when $τ(d-1-s)<ds$; at equality the nominal exponents coincide, while the localization estimate has no $T^\varepsilon$ loss. On data vanishing near the polar degeneracies, the cutoff-free argument gives exponent $s/(τ+d)$. We apply the estimates to shrinking spiral-type trajectories, integrable angular data, and homogeneous singularities, including a contrast between power-law and exponential radial contraction. Finally, on $\mathbb{S}^2$, for every prescribed $R(T)\to0$ we construct a rationally independent frequency and a smooth mean-zero observable with $\|φ\|_{C^1}\le1$ for which the error is not $O(R(T))$.
Comments46 pages. Revised and expanded version; title and abstract updated; added comparison with mollified Koksma--Hlawka rates and clarified the limits of uniformity
