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Cone upgrade for effective equidistribution and weighted Khintchine--Schmidt theorem on fractals
Abstract
We prove a fractal analogue of the weighted theorems of Khintchine and Schmidt for products of self-similar measures on the real line whose defining iterated function systems share a common contraction ratio. The proof translates the counting problem into homogeneous dynamics and builds on recent work of Bénard, He and Zhang, who treated the unweighted case. Our main new ingredient is a \emph{cone upgrade}: an effective double equidistribution theorem for translates of product self-similar measures, uniform over diagonal elements whose weights range over a fixed cone in the positive Weyl chamber, deduced from effective equidistribution for a single weight. This yields a partial effective form of a theorem of Khalil, Luethi and Weiss.
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BibTeX
@misc{aggarwal2026cone,
title = {{Cone upgrade for effective equidistribution and weighted Khintchine--Schmidt theorem on fractals}},
author = {Gaurav Aggarwal and Alexander Gorodnik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04953},
eprint = {2609.04953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04953},
}
RIS
TY - UNPB
TI - Cone upgrade for effective equidistribution and weighted Khintchine--Schmidt theorem on fractals
AU - Gaurav Aggarwal
AU - Alexander Gorodnik
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04953
UR - https://arxiv.org/abs/2609.04953
AB - We prove a fractal analogue of the weighted theorems of Khintchine and Schmidt for products of self-similar measures on the real line whose defining iterated function systems share a common contraction ratio. The proof translates the counting problem into homogeneous dynamics and builds on recent work of Bénard, He and Zhang, who treated the unweighted case. Our main new ingredient is a \emph{cone upgrade}: an effective double equidistribution theorem for translates of product self-similar measures, uniform over diagonal elements whose weights range over a fixed cone in the positive Weyl chamber, deduced from effective equidistribution for a single weight. This yields a partial effective form of a theorem of Khalil, Luethi and Weiss.
ER -