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

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 ${\mathbb{R}}^d$.

arXiv comment65 pages. The irreducibility assumption in Theorem 1.1 has been simplified. A new Section 3 details moment estimates for $\sigma$ and the $\lambda$-random walk. Explanations of certain proofs, in particular in Section 7, have been improved