Disintegration results for fractal measures and applications to Diophantine approximation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911502315814912 |
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| author | Baker, Simon |
| author_facet | Baker, Simon |
| contents | 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 $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $α>0$ such that for all $β>α$ we have $$μ\left(\left\{\mathbf{x}\in \mathbb{R}^{d}:\max_{1\leq i\leq d}|x_{i}-p_i/q|\leq \frac{1}{q^{\frac{d+1}{d}}(\log q)^β}\textrm{ for i.m. }(p_1,\ldots,p_d,q)\in \mathbb{Z}^{d}\times \mathbb{N}\right\}\right)=0.$$
2. Using a result of Kleinbock and Weiss, we show that if $μ$ is an affinely irreducible self-similar measure, then $μ$ almost every $\mathbf{x}$ is not a singular vector. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09599 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Disintegration results for fractal measures and applications to Diophantine approximation Baker, Simon Dynamical Systems Number Theory 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 $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $α>0$ such that for all $β>α$ we have $$μ\left(\left\{\mathbf{x}\in \mathbb{R}^{d}:\max_{1\leq i\leq d}|x_{i}-p_i/q|\leq \frac{1}{q^{\frac{d+1}{d}}(\log q)^β}\textrm{ for i.m. }(p_1,\ldots,p_d,q)\in \mathbb{Z}^{d}\times \mathbb{N}\right\}\right)=0.$$ 2. Using a result of Kleinbock and Weiss, we show that if $μ$ is an affinely irreducible self-similar measure, then $μ$ almost every $\mathbf{x}$ is not a singular vector. |
| title | Disintegration results for fractal measures and applications to Diophantine approximation |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2501.09599 |