Disintegration results for fractal measures and applications to Diophantine approximation

Abstract

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. As an application of our results, we prove the following Diophantine statements: 1. Using a result of Pollington and Velani, we show that if $\hat{I}¼$ is a self-conformal measure in $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $\hat{I}\pm >0$ such that for all ${\hat{I}}^2>\hat{I}\pm$ we have $$\hat{I}¼\left(\left\{\mathbf{x}\in {\mathbb{R}}^d:\underset{1\le i\le d}{\max }|x_i-p_i/q|\le \frac{1}{q^{\frac{d+1}{d}}(\log q{)}^2}\text{ for i.m. }(p_1,\dots ,p_d,q)\in {\mathbb{Z}}^d\times \mathbb{N}\right\}\right)=0.$$ 2. Using a result of Kleinbock and Weiss, we show that if $\hat{I}¼$ is an affinely irreducible self-similar measure, then $\hat{I}¼$ almost every $\mathbf{x}$ is not a singular vector.