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Badly approximable points on non-linear carpets

Roope Anttila, Jonathan M. Fraser, Henna Koivusalo

Abstract

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urbański from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest.

Badly approximable points on non-linear carpets

Abstract

The badly approximable points in are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urbański from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest.
Paper Structure (12 sections, 17 theorems, 131 equations, 1 figure)

This paper contains 12 sections, 17 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-linear carpets
  3. Main results
  4. Symbolic spaces
  5. Symbolic Barański carpets
  6. Symbolic self-conformal sets
  7. Symbolic non-linear carpets
  8. Proofs of the main results
  9. Hausdorff dimension of non-linear carpets
  10. Modified lower dimension of non-linear carpets
  11. Badly approximable numbers on non-linear carpets
  12. Parabolic Cantor sets

Key Result

Proposition 1.1

Suppose $X \subseteq \mathbb{R}^d$ is closed and hyperplane diffuse in the sense that there exists $\beta>0$ such that for all $R \in (0,1), x \in X$ and affine hyperplanes $V \subseteq \mathbb{R}^d$, where $V_\varepsilon$ denotes the open $\varepsilon$-neighbourhood of $V$. Then

Figures (1)

  • Figure 1: Two non-linear carpets which fit in our framework

Theorems & Definitions (28)

  • Proposition 1.1
  • Remark 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • ...and 18 more