Khintchine dichotomy for self-similar measures
Abstract
We extend Khintchine’s theorem to all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on S L 2 ( R ) / S L 2 ( Z ) SL_{2}(\mathbb {R})/SL_{2}(\mathbb {Z}) , a result of independent interest.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Stochastic processes and statistical mechanics
Random Matrices and Applications
Publication details
- Journal / source
- Journal of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 39
- Issue
- 3
- Pages
- 587–623
- Publication date
- 12 January 2026
- Language
- EN