Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions.

By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points.

Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence.

We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes.

Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that in general settings, no alternative weighting function can yield a faster uniform rate.

In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.