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Zero-full law for well approximable sets in missing digit sets

Bing Li, Ruofan Li, Yufeng Wu

TL;DR

This work analyzes Diophantine approximation on missing-digit fractal sets $C(b,D)$ by rationals with denominators $t^n$, proving a zero-full law for the $f$-Hausdorff measure of the approximation set $W_t(\psi)\cap C(b,D)$ when $b$ and $t$ are multiplicatively dependent, and correcting a prior claim. It introduces a detailed endpoint-based intersection analysis, leverages the mass transference principle, and derives sharp dimension results: $\dim_{\rm H}(W_t(\psi)\cap C(b,D))=\gamma/\lambda_{\psi}$ in the dependent case, with partial bounds and conjectures for the independent case (same prime divisors and different prime divisors). The paper also establishes large intersection properties for particular $\psi$ and outlines conjectures guiding the independent-prime-divisor scenario. Overall, it advances the understanding of fractal Diophantine approximation on Cantor-type sets and provides a framework for further conjectures in lacunary approximation problems.

Abstract

Let $b \geq 3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose base $b$ expansion only consists of digits in a set $D \subseteq \{0,...,b-1\}$. We study how close can numbers in $C(b,D)$ be approximated by rational numbers with denominators being powers of some integer $t$ and obtain a zero-full law for its Hausdorff measure in several circumstances. When $b$ and $t$ are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) and generalize their theorem. When $b$ and $t$ are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.

Zero-full law for well approximable sets in missing digit sets

TL;DR

This work analyzes Diophantine approximation on missing-digit fractal sets by rationals with denominators , proving a zero-full law for the -Hausdorff measure of the approximation set when and are multiplicatively dependent, and correcting a prior claim. It introduces a detailed endpoint-based intersection analysis, leverages the mass transference principle, and derives sharp dimension results: in the dependent case, with partial bounds and conjectures for the independent case (same prime divisors and different prime divisors). The paper also establishes large intersection properties for particular and outlines conjectures guiding the independent-prime-divisor scenario. Overall, it advances the understanding of fractal Diophantine approximation on Cantor-type sets and provides a framework for further conjectures in lacunary approximation problems.

Abstract

Let be an integer and be the set of real numbers in whose base expansion only consists of digits in a set . We study how close can numbers in be approximated by rational numbers with denominators being powers of some integer and obtain a zero-full law for its Hausdorff measure in several circumstances. When and are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) and generalize their theorem. When and are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.
Paper Structure (12 sections, 19 theorems, 122 equations)

This paper contains 12 sections, 19 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hausdorff measure and dimension
  4. Mass transference principle
  5. Measure theoretic lemmas
  6. Intersection of balls and the missing digit set
  7. A special case
  8. Proof of \ref{['thm:main']}
  9. Further discussion
  10. Hausdorff dimension
  11. Multiplicatively independent case
  12. Large intersection property

Key Result

Theorem 1.1

Let $f$ be a dimension function such that $f(r) r^{-\log2 / \log 3}$ is monotonic and $\psi : \mathbb{N} \to (0,\infty)$ be a function. Then

Theorems & Definitions (32)

  • Theorem 1.1: LSV07
  • Claim 1.2: LSV07
  • Theorem 1.3
  • Conjecture 1.4: Velani's conjecture; see ACY20
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: Mass transference principle, BV06
  • Lemma 2.2: LSV07
  • Lemma 2.3: Sprindžuk79; see also LSV07
  • Example 3.1
  • ...and 22 more