We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class and every $γ>0$, uniform convergence at scale $γ$, agnostic learnability at scale $γ/2$, and finiteness of the fat-shattering dimension at every scale $γ'>γ$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS 1998), which incur such a loss. The key technical ingredient is a direct bound on empirical $\ell_\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $γ$: an $O(\log^2 n)$ bound holds already at scale $γ/2$, while an $O(\log n)$ bound holds at scale $2γ$. We further show that the $O(\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM 1997) and Rudelson and Vershynin (Ann. of Math. 2006). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML 2026). We also highlight several open questions on quantitative sample complexity and evaluability.

翻译：我们研究实值函数类在最优尺度上展现一致收敛性与可学习性的问题。主要结果建立了PAC学习基本定理的尺度敏感性推广：对于每个有界实值函数类及任意$\gamma>0$，在尺度$\gamma$上的一致收敛性、在尺度$\gamma/2$上的不可知可学习性，以及在每个尺度$\gamma'>\gamma$上胖粉碎维数的有限性三者等价。这解决了Anthony与Bartlett（剑桥大学出版社，1999）关于控制可学习性精确尺度的疑问，否证了该书归因于Phil Long的猜想（即乘法2倍差距不可避免），并改进了Bartlett与Long（JCSS，1998）中存在的此差距的上界。关键技术成分是经验$\ell_\infty$覆盖数的直接界，避免了通过包装数的标准迂回。作为推论，我们获得了以胖粉碎尺度$\gamma$表示的渐近度量熵界：在尺度$\gamma/2$上即达$O(\log^2 n)$界，而在尺度$2\gamma$上可达$O(\log n)$界。进一步证明$O(\log^2 n)$界有时是紧的。这些结果解决了Alon等人（JACM，1997）以及Rudelson与Vershynin（数学年刊，2006）的开放问题。作为应用，我们建立了有界积分概率度量的尖锐二分法：每个此类IPM要么可估计，要么无法在任何乘法因子$c<3$内被弱评估，而3-弱评估性始终成立，解决了Aiyer等人（ICML，2026）的开放问题。我们还强调了关于定量样本复杂度与可评估性的若干开放问题。