A note on dyadic approximation in Cantor's set

TL;DR

This paper investigates how well points in the middle-third Cantor set can be approximated by dyadic rationals, showing that beyond a certain approximation rate, almost no points are well approximable with respect to the natural measure.

Contribution

It refines previous results by establishing thresholds for dyadic approximation in the Cantor set, highlighting the measure-zero nature of well-approximable points beyond certain rates.

Findings

Almost no points are dyadically $ au$-well approximable for large $ au$

Refines previous approximation thresholds in the Cantor set

Uses measure-theoretic methods for approximation analysis

Abstract

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, $K$, for approximation functions of the form $\psi_{\tau}(n) = n^{-\tau}$ ($\tau \ge 0$). In particular, we show that for values of $\tau$ beyond a certain threshold we have that almost no point in $K$ is dyadically $\psi_{\tau}$-well approximable with respect to the natural probability measure on $K$. This refines a previous result in this direction obtained by the first, third, and fourth named authors (arXiv, 2020).