Khintchine dichotomy and Schmidt estimates for self-similar measures on $\mathbb{R}^d$

TL;DR

This paper generalizes classical Khintchine and Schmidt theorems to self-similar measures in higher dimensions, using effective equidistribution and new techniques to handle algebraic obstructions and measure concentration.

Contribution

It extends metric Diophantine approximation results to self-similar measures on $\

Findings

Effective equidistribution of random walks on $ ext{SL}_{d+1}(R)/ ext{SL}_{d+1}(Z)$

Generalization of Khintchine and Schmidt theorems to higher dimensions

Non-concentration properties of self-similar measures near algebraic varieties

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 $\mathbb{R}^d$.