Disintegration results for fractal measures and applications to Diophantine approximation

TL;DR

This paper establishes disintegration results for certain fractal measures and applies them to prove new Diophantine approximation properties, showing that typical points in these measures are not well-approximable or singular.

Contribution

It introduces disintegration results for self-conformal and affinely irreducible self-similar measures and applies these to Diophantine approximation, extending previous results.

Findings

Self-conformal measures have disintegrations resembling measures satisfying strong separation.

Almost all points in these measures are not singular vectors.

Certain Diophantine inequalities hold for measures under specified conditions.

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.