The Doeblin-Lenstra conjecture: effective results and central limit theorems

Abstract

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 B\'enard, He and Zhang. As a consequence, we also obtain central limit theorems for these Diophantine statistics in both classical and fractal settings.