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On the dimension of $αβ$-sets

Michael Hochman

Abstract

We show that the Feng-Xiong lower bound of $1/2$ for the box dimension of $αβ$-sets is tight. We also study how much of an $αβ$-orbit ``carries the dimension'': deleting an arbitararily small positive density set of times can cause the box dimension to drop to zero, but the Assouad dimension cannot drop below $1/4$.

On the dimension of $αβ$-sets

Abstract

We show that the Feng-Xiong lower bound of for the box dimension of -sets is tight. We also study how much of an -orbit ``carries the dimension'': deleting an arbitararily small positive density set of times can cause the box dimension to drop to zero, but the Assouad dimension cannot drop below .

Paper Structure

This paper contains 24 sections, 13 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Background and results
  3. Intersections of self-similar sets
  4. Organization
  5. Notation
  6. An $\alpha\beta$-set of box dimension $1/2$
  7. Sketch of the construction
  8. Katznelson's construction
  9. Analysis of the construction
  10. Growth condition
  11. Word lengths
  12. Letter frequencies and $\alpha\beta$-orbits
  13. Evolution of $\alpha_{n},\beta_{n}$
  14. Evolution of the sets $E_{n}$
  15. Choosing parameters and estimating the dimension
  16. ...and 9 more sections

Key Result

Theorem 1.1

There exist $\alpha,\beta\in\mathbb{R}$ with $1,\alpha,\beta$ rationally independen, and an $\alpha\beta$-set $E$, such that $\overline{\dim}_{B}E=1/2$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 5 more