Dyadic Approximation in the Middle-Third Cantor Set

TL;DR

This paper investigates how numbers in the middle-third Cantor set can be approximated using dyadic rationals, revealing distinct behaviors compared to triadic approximation and linking to dynamical systems phenomena.

Contribution

It provides a new metric theory of dyadic approximation in the Cantor set, highlighting differences from previous triadic approximation studies and connecting to Furstenberg's dynamical systems conjecture.

Findings

Dyadic approximation behavior differs significantly from triadic approximation.

The results relate to Furstenberg's times 2 and 3 phenomenon.

The study extends understanding of number approximation in fractal sets.

Abstract

In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley, Salp, and Velani (2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured.