Year: 2025
Type: preprint
Abstract: We extend the classical theorems of Khintchine and Schmidt in metric Diophantine approximation to the context of self-similar measures on $\mathbb{R}^d$. For this, we establish effective equidistribution of associated random walks on $\text{SL}_{d+1}(\mathbb{R})/\text{SL}_{d+1}(\mathbb{Z})$. This generalizes our previous work which requires $d=1$ and restricts Schmidt-type counting estimates to approximation functions which decay fast enough. Novel techniques include a bootstrap scheme for the associated random walks despite algebraic obstructions, and a refined treatment of Dani's correspondence. Along the way, we also establish non-concentration properties of self-similar measures near algebraic subvarieties of $\mathbb{R}^d$.
Source: arXiv (Cornell University)
Authors: Bénard, Timothée, He, Weikun, Zhang, Han
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416868332
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