Disintegration results for fractal measures and applications to Diophantine approximation

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.

💡 Research Summary

The paper establishes a novel disintegration framework for two important classes of fractal measures: self‑conformal measures on the real line and affinely irreducible self‑similar measures in higher dimensions. Traditionally, detailed geometric control of such measures required the underlying iterated function system (IFS) to satisfy the strong separation condition; when the IFS overlaps, many standard tools break down. The authors overcome this limitation by constructing a probability space ((\Omega,\mathcal A,P)) together with a family of measures ({\mu_\omega}{\omega\in\Omega}) such that the original stationary measure (\mu) can be written as the average (\mu=\int\Omega \mu_\omega,dP(\omega)).

Each component measure (\mu_\omega) enjoys three key properties:

1. Doubling – there exists a uniform constant (C_1) with (\mu_\omega(B(x,2r))\le C_1\mu_\omega(B(x,r))) for all (x) in the support and all radii (r>0). This guarantees a locally uniform distribution of mass.

2. Thin‑neighbourhood estimate – for any point (x) in the support, any affine subspace (W\subset\mathbb R^d), and any (\varepsilon>0), one has (\mu_\omega(W(\varepsilon r)\cap B(x,r))\le C_2\varepsilon^\alpha\mu_\omega(B(x,r))). Here (W(\varepsilon r)) denotes the (\varepsilon r)-neighbourhood of (W). This condition quantifies how little mass can concentrate near lower‑dimensional sets.

3. Dynamical self‑similarity – the measures satisfy a recursive relation analogous to the stationary equation (\mu=\sum p_a\varphi_a\mu), but now indexed by the random sequence (\omega). Explicitly, (\mu_\omega=\sum_{a\in A_{i_1}} q_{i_1}(a),\varphi_a\mu_{\sigma(\omega)}), where (\sigma) is the left shift on (\Omega).

These three properties are proved in Theorems 1.1 (self‑conformal case) and 1.2 (affinely irreducible self‑similar case). The construction is based on partitioning the alphabet of the IFS into non‑empty subsets (A_1,\dots,A_k), defining new probability vectors (q_i) on each subset, and then using the product space (\Omega=I^{\mathbb N}) to encode a random choice of sub‑alphabet at each iteration. The resulting random IFS (\Sigma_\omega) and measure (\mu_\omega) inherit the geometric regularity of a strongly separated system, even though the original IFS may have substantial overlaps.

Armed with this disintegration, the authors turn to two Diophantine applications.

1. Log‑weighted Diophantine approximation.
Pollington and Velani proved that a measure satisfying a doubling condition and a thin‑neighbourhood estimate gives zero measure to the set
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