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 $\mu$ is a self-conformal measure in $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $\alpha>0$ such that for all $\beta>\alpha$ we have $$\mu\left(\left{\mathbf{x}\in \mathbb{R}^{d}:\max_{1\leq i\leq d}|x_{i}-p_i/q|\leq \frac{1}{q^{\frac{d+1}{d}}(\log q)^{\beta}}\textrm{ for i.m. }(p_1,\ldots,p_d,q)\in \mathbb{Z}^{d}\times \mathbb{N}\right}\right)=0.$$ 2. Using a result of Kleinbock and Weiss, we show that if $\mu$ is an affinely irreducible self-similar measure, then $\mu$ almost every $\mathbf{x}$ is not a singular vector.