Disintegration results for fractal measures and applications to Diophantine approximation

2025/01/16 by Baker, Simon
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.09599

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 μ is a self-conformal measure in ℝ or an affinely irreducible self-similar measure, then there exists α>0 such that for all β>α we have μ(\x∈ ℝ^d:max_{1≤ i≤ d}|x_i-p_i/q|≤ \frac1q^(d+1)/(d)(log q)^β\textrm for i.m. (p₁,…,p_d,q)∈ ℤ^d× ℕ\)=0. 2. Using a result of Kleinbock and Weiss, we show that if μ is an affinely irreducible self-similar measure, then μ almost every x is not a singular vector.

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