Doeblin–Lenstra 猜想：有效结果与中心极限定理
The Doeblin-Lenstra conjecture: effective results and central limit theorems
AI总结：
本文通过幺模格对角流的有效遍历结果，证明经典、高维及分形测度下 Doeblin–Lenstra 定律的定量收敛，并推出相关 Diophantine 统计量的中心极限定理。
摘要：
我们在 Doeblin–Lenstra 定律中建立了有效收敛速率，该定律描述了由典型实数的连分数渐近分数所产生的逼近系数的极限分布。更一般地，我们证明了高维最佳逼近情形下 Doeblin–Lenstra 定律的定量版本，同时也针对从实直线上的分形测度（包括三分 Cantor 测度）中抽样的点证明了相应版本。我们的方法将问题归结为证明幺模格空间上对角流的 Birkhoff 平均值的有效收敛。关键步骤是表明，尽管所关注的可观测量具有不连续性，但它平均而言满足有效遍历定理所需的正则性条件。在分形情形下，我们建立了自相似测度在对角流下的有效多重等分布性质，推广了 Bénard、He 和 Zhang 最近关于单一等分布的工作。因此，我们还在经典情形和分形情形中获得了这些 Diophantine 统计量的中心极限定理。
英文摘要：
We establish effective convergence rates in the Doeblin-Lenstra law, describing the limiting distribution of approximation coefficients arising from continued fraction convergents of a typical real number. More generally, we prove quantitative versions of the Doeblin-Lenstra law for best approximates in higher dimensions, as well as for points sampled from fractal measures on the real line, including the middle-third Cantor measure. Our method reduces the problem to proving effective convergence of Birkhoff averages for diagonal flows on spaces of unimodular lattices. A key step is to show that, despite the discontinuity of the observable of interest, it satisfies the regularity conditions on average required for effective ergodic theorems. For the fractal setting, we establish effective multi-equidistribution properties of self-similar measures under diagonal flow, extending recent work on single equidistribution by Bénard, He and Zhang. As a consequence, we also obtain central limit theorems for these Diophantine statistics in both classical and fractal settings.
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