Khintchine dichotomy for self-similar measures

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L 2 left-parenthesis double-struck upper R right-parenthesis slash upper S upper L 2 left-parenthesis double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL_{2}(\mathbb {R})/SL_{2}(\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a result of independent interest.