Quantitative time-averaged spherical means along equidistributed spirals: Diophantine rates and limits of uniformity

Abstract

We establish quantitative equidistribution estimates for spherical observables sampled along Kronecker flows. For the monotone measure-preserving inverse-CDF parametrization $\Phi_d:\mathbb{T}^d\to\mathbb{S}^d$ and a Diophantine frequency of exponent $\tau$, every $\varphi\in C^s(\mathbb{S}^d)$, $0 0$, the rate $O(T^{-s/[d(\tau+1)]+\varepsilon})$. Its power exponent is larger exactly when $\tau(d-1-s)<ds$; 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 $s/(\tau+d)$. 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 $\|\varphi\|_{C^1}\le1$ for which the error is not $O(R(T))$.