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Jarník-type theorem for self-similar sets

Yubin He, Lingmin Liao

TL;DR

The paper develops a Jarník-type theory for the intersection of a compact fractal $K$ carrying a $δ$-Ahlfors regular measure with inhomogeneous well-approximable sets $W_d(\psi,\boldsymbol{\theta})$, under a local counting property. It derives a sharp upper bound and a nontrivial lower bound for the Hausdorff dimension of $K\cap W_d(\psi,\boldsymbol{\theta})$, with refined lower bounds in special regimes and a measure-theoretic version in one dimension. The results yield full dimension for homogeneous VWA inside strongly irreducible self-similar sets and resolve inhomogeneous VWA questions for sufficiently thick missing digits sets, extending prior work of Chen and Yu. The methodology combines large intersection theory, mass distribution, effective equidistribution in $\mathrm{SL}_{d+1}$, and careful local counting arguments to connect fractal geometry with Diophantine approximation on fractals.

Abstract

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $δ$-Ahlfors regular measure $μ$. For any $τ>1/d$ and any ``inhomogeneous'' vector $\boldsymbolθ\in\mathbb R^d$, let $W_d(ψ_τ,\boldsymbolθ)$ denote the set of $(ψ_τ,\boldsymbolθ)$-well approximable numbers, where $ψ_τ(q)=q^{-τ}$. Assuming a local estimate for the $μ$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf p+\boldsymbolθ)/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(ψ_τ,\boldsymbolθ)$, together with some nontrivial lower bounds when $τ$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $θ=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(ψ_τ,0)$ has full $(δ+\frac{2}{1+τ}-1)$-Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in $\mathbb R^d$, extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within sufficiently thick missing digits sets in $\mathbb R$, affirmatively answering a question posed by Yu [arXiv:2101.05910]. Our applications build on the seminal works of Yu [arXiv:2101.05910] and Bénard, He and Zhang [arXiv:2508.09076].

Jarník-type theorem for self-similar sets

TL;DR

The paper develops a Jarník-type theory for the intersection of a compact fractal carrying a -Ahlfors regular measure with inhomogeneous well-approximable sets , under a local counting property. It derives a sharp upper bound and a nontrivial lower bound for the Hausdorff dimension of , with refined lower bounds in special regimes and a measure-theoretic version in one dimension. The results yield full dimension for homogeneous VWA inside strongly irreducible self-similar sets and resolve inhomogeneous VWA questions for sufficiently thick missing digits sets, extending prior work of Chen and Yu. The methodology combines large intersection theory, mass distribution, effective equidistribution in , and careful local counting arguments to connect fractal geometry with Diophantine approximation on fractals.

Abstract

Let be a compact subset equipped with a -Ahlfors regular measure . For any and any ``inhomogeneous'' vector , let denote the set of -well approximable numbers, where . Assuming a local estimate for the -measure of the intersections of with the neighborhoods of ``rational'' vectors , we establish a sharp upper bound for the Hausdorff dimension of , together with some nontrivial lower bounds when is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case (, ) for a class of sufficiently thick self-similar sets , and moreover has full -Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in , extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within sufficiently thick missing digits sets in , affirmatively answering a question posed by Yu [arXiv:2101.05910]. Our applications build on the seminal works of Yu [arXiv:2101.05910] and Bénard, He and Zhang [arXiv:2508.09076].
Paper Structure (10 sections, 16 theorems, 183 equations)

This paper contains 10 sections, 16 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Hausdorff measure and content
  3. Proof of Theorem \ref{['t:main']}: upper bound of $\mathop{\mathrm{\dim_H}}\nolimits(K\cap W_d(\psi,\theta))$
  4. Proof of Theorem \ref{['t:main']}: lower bound of $\mathop{\mathrm{\dim_H}}\nolimits(K\cap W_d(\psi,\theta))$
  5. Estimate of the $\nu$-measure of balls for general cases
  6. Estimate of the $\nu$-measure of balls for $\delta \ge d-1$, $\bf\theta= 0$, and $\tau < \frac{\delta + 2 - d}{2d - \delta - 1}$
  7. Proof of Corollary \ref{['c:corollary']}
  8. Some Examples
  9. Irreducible self-similar sets and proof of Theorem \ref{['t:example1']}
  10. Missing digits sets and proof of Theorem \ref{['t:example2']}

Key Result

Theorem 1.2

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular probability measure $\mu$. Suppose that $\mu$ satisfies the $(\alpha,\beta,\boldsymbol{\theta})$-local counting property for some $1/d<\alpha<\frac{1+\delta-\beta\delta}{d-\delta}$, $0<\beta<1$ and ${\bm\theta}\in and Moreover, if $\delta>d-1$, ${\bm\theta}=\mathbf{0}$ and $\psi(q)\ge cq^{-\tau}$ with $1/d<\tau

Theorems & Definitions (38)

  • Definition 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Remark 3
  • Remark 4
  • Corollary 1.3
  • Theorem 1.4
  • Remark 5
  • Remark 6
  • ...and 28 more